Undergraduate Lecture Notes in Physics

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Neil Ashby

University of Colorado, Boulder, CO, USA

William Brantley

Department of Physics, Furman University, Greenville, SC, USA

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Physics Program, Bard College, Annandale-on-Hudson, NY, USA

Michael Fowler

Department of Physics, University of Virginia, Charlottesville, VA, USA

Morten Hjorth-Jensen

Department of Physics, University of Oslo, Oslo, Norway

Michael Inglis

Department of Physical Sciences, SUNY Suffolk County Community College, Selden, NY, USA

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Rajendra K. Bera

The Amazing World of Quantum Computing

Rajendra K. Bera

Acadinnet Education Services India, Bangalore, Karnataka, India

ISSN 2192-4791e-ISSN 2192-4805

Undergraduate Lecture Notes in Physics

ISBN 978-981-15-2470-7e-ISBN 978-981-15-2471-4

https://doi.org/10.1007/978-981-15-2471-4

© Springer Nature Singapore Pte Ltd. 2020

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Preface

Our World Consists of Both Real and Imagined Things

This book has a limited purpose—to introduce to inquisitive students the amazing world of quantum computing. About the relationship between physics and mathematics, Richard Feynman said:

Mathematics is a language plus reasoning; it is like a language plus logic. Mathematics is a tool for reasoning. … [I]t is impossible to explain honestly the beauties of the laws of nature in a way that people can feel, without their having some deep understanding of mathematics. ¹

This is certainly true for the modern physicist, but it was not true for Michael Faraday (1791–1867), who in his days, virtually ignorant of mathematics, discovered important pieces of the laws governing electricity and magnetism, ² to which James Clerk Maxwell, later in 1865, gave an elegant mathematical formulation. ³ Indeed, it was so elegant that Maxwell unified the electric force and the magnetic force into what we call the electromagnetic force. As McMullin says, “It was left to Michael Faraday to propose the ‘physical existence’ of lines of force and to James Clerk Maxwell to add as criterion the presence of energy as the ontological basis for a full-blown ‘field theory’ of electromagnetic phenomena.” ⁴ Maxwell’s mathematical formulation inspired Albert Einstein to have a fresh look at the nature of space time and thus was born the theory of relativity ⁵ that turned our intuitive understanding of space and time upside down.

Of Faraday, Maxwell said,

When I had translated what I considered to be Faraday’s ideas into a mathematical form, I found that in general the results of the two methods coincided, so that the same phenomena were accounted for, and the same laws of action deduced by both methods, but that Faraday’s methods resembled those in which we begin with the whole and arrive at the parts by analysis, while the ordinary mathematical methods were founded on the principle of beginning with the parts and building up the whole by synthesis. ⁶

To see the elegant beauty of quantum mechanics and its application to computation, it is not so much the underlying mathematics but the unusual concepts (begin with the whole) that one needs to get used to. These concepts are precisely understood in elegant mathematical structures. Of course, to compute, one must finally resort to mathematics and the churning of numbers. In this, we have generally tried to stay close to the formal statements and notations used by Nielsen and Chuang [9] in their excellent bookQuantum Computation and Quantum Information to enable readers to easily transition to that book.

If this book encourages the reader to seriously pursue a career in science, technology, engineering, and mathematics (STEM), I would consider my time well spent in writing it.

Acknowledgements

I started preparing notes for writing this book as I was learning the subject about eighteen years ago. It was just as well because the book tries to anticipate the places where someone new to the subject may have difficulty; it has affected what I chose to present and omit in the book for the first-time reader getting introduced to quantum computing.

I am, of course, immensely grateful to the many people—students, friends, and colleagues—who have attended my lectures on quantum computing. Their compliments eventually persuaded me to try my hand in writing this book. I thank them all for their courtesy, comments, and friendship. While this may sound completely strange, I learnt the subject entirely on my own, without mentors, without attending lectures, and without any formal training in quantum mechanics. This should be interesting to students who want to learn the subject and are in awe of quantum mechanics. My learning through self-study was possible because of my pleasant experience as a solo researcher in aerospace engineering for more than three decades. The mental discipline acquired over those years in dealing with mathematics was immensely useful in my attempt to try and understand a subject quite alien to me when I first approached it. For this reason, I do not have a list of people from the profession whom I could thank. But I do thank Prof. Richard Feynman , whom I could not meet in my only visit to Caltech in February 1980 and never knew him personally, but whose three volumes—The Feynman Lectures on Physics—were not just inspiring but which finally opened my eyes to the awe-inspiring beauty of physics, and Prof. Sir James Lighthill, whom I got to know personally, through exchange of letters and a brief meeting, because of a rather unusual event in my research career as an aeronautical engineer, and whose encouragement in my research activities came at a time when I was getting pretty disillusioned about continuing as a researcher in aerospace engineering in India. When I finally took a career decision to quit aerospace in August 1995, I joined IBM in India (it was then Tata Information Systems Ltd. with IBM shareholding; it later became IBM) and a few years later found a manager, Dr. Uday Shukla, who went beyond my expectations and his call of duty to give me the freedom I needed to delve into quantum computing. None of the three I have mentioned are with us anymore. To them I owe much and offer my homage.

During the writing of this book, I owe a debt of gratitude to three people—Vikram Menon and Ayan Chatterjee who carefully read the manuscript and made many useful suggestions that might help the first-time reader and to Sunish Raj who always smilingly ensured that I had the infrastructure a researcher depends on to be productive. Finally, I express my heartfelt thanks to the anonymous reviewer of the manuscript whose suggestions have led to a more cohesive book.

The sole responsibility for any errors and omissions that may have crept in, within the scope and ambition of this book, lie with me.

My Expectations from the Reader

Do not take the lecture too seriously . . . just relax and enjoy it. I am going to tell you what nature behaves like. If you will simply admit that maybe she does behave like this, you will find her a delightful, entrancing thing. Do not keep saying to yourself, if you can possibly avoid it, “But how can it be like that ?” because you will get . . . into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that. ⁷

—Richard Feynman, speaking about quantum theory.

I expect readers, not trained in quantum mechanics, to be confused, especially if they have heard of such weird objects as Schrödinger’s cat being simultaneously dead and alive, the phenomenon of teleportation, a photon concurrently travelling along two paths, and so on. I hope such readers will heed Feynman’s famous advice quoted above. They may take comfort from the fact that even established researchers in quantum mechanics have great difficulty developing an intuition about the subject. Indeed, Einstein once remarked,Quantum Mechanics: Real Black Magic Calculus . And Niels Bohr said, “If quantum mechanics hasn’t profoundly shocked you, you haven’t understood it yet”. ⁸

It helps if one views quantum mechanics as a mathematical framework for constructing physical theories and then concentrates on grasping the rules of the game. It pays to make a conscious effort to block out the real world (as we understand it on the basis of our day-to-day experiences) and explore the subject as if we were trying to understand an alien world. Patience and diligence will be handsomely rewarded for, in the end, what will be revealed to you is the finest set of scientific conjectures created by the human mind to date.

A good understanding of linear algebra and a moderate understanding of Fourier analysis is definitely expected. You must know some mathematics to start learning quantum mechanics and to compute! And finally, you must develop the discipline to use the postulates of quantum mechanics with unquestioning obedience and definitely not apply common sense in the process. Just stick to the postulates and the underlying mathematics that support them with unswerving loyalty.

The book is meant to be foundational reading for a senior-level and graduate-level course in quantum computing. I hope the reader, after reading this book, will be able to move on to more advanced texts in quantum computing with greater ease and confidence. The galaxy of outstanding researchers the subject has attracted, both from industry and academia, is amazing. You will see some of the finest scientific minds at work, and consequently you will find deep questions being asked and explored. It is therefore necessary that students of the subject continuously keep track of current research literature and use the book as a guide to get a foothold on the subject. Periodically, excellent reviews on quantum computing appear in the literature. Students are well advised to read them carefully.

I bring the view of an outsider to the subject and owe no allegiance to any particular interpretation of quantum mechanics. I, however, marvel at the remarkable achievements of quantum mechanics and the enormous potential of quantum computing.

References

[1]

A. Einstein, On a heuristic view concerning the production and transformation of light. Annalen der Physik17 , 132–148 (1905a). (English translation from German). https://​einsteinpapers.​press.​princeton.​edu/​vol2-trans/​101

[2]

A. Einstein, Ist die Trägheit eines Körpers von seinem Energiegehalt abhängig? Annalen der Physik18 , 639–641 (1905b), https://​doi.​org/​10.​1002/​andp.​19053231314 . English translation: Does the inertia of a body depend upon Its energy-content? at http://​www.​fourmilab.​ch/​etexts/​einstein/​E_​mc2/​www/​ . Accessed 09 Jan 2020

[3]

A. Einstein,Relativity: The Special and General Theory , Digital Reprint, Elegant Books, First published in 1920. English translation by Robert W. Lawson, https://​www.​ibiblio.​org/​ebooks/​Einstein/​Einstein_​Relativity.​pdf . Accessed 09 Jan 2020

[4]

R. Feynman,The Character of Physical Law , Modern Library Edition, 1994, Originally published by BBC in 1965, (and in paperback by MIT Press, 1967)

[5]

I.H. Hutchinson,The Genius and Faith of Faraday and Maxwell (The New Atlantis, Number 41, Winter, 2014), pp. 81–99, http://​www.​thenewatlantis.​com/​docLib/​20140702_​TNA41Hutchinson.​pdf . Accessed 09 Jan 2020

[6]

J.C. Maxwell,A Treatise on Electricity and Magnetism , vol. I, 3rd edn. (Clarendon Press, 1891), http://​www.​aproged.​pt/​biblioteca/​MaxwellII.​pdf . Republished by Dover in 1954. Accessed 09 Jan 2020

[7]

E. McMullin, The origins of the field concept in physics, Phys. Perspect.4 (1), 13–39 (2002). http://​doi.​org/​10.​1007/​s00016-002-8357-5

[8]

J.C. Maxwell, A dynamical theory of the electromagnetic field. Philos. Trans. R. Soc. Lond.155 , 459–512 (1865)

[9]

M.A. Nielsen, I.L. Chuang,Quantum Computation and Quantum Information (Cambridge University Press, 2000). [Errata at http://​www.​squint.​org/​qci/​ ]

Rajendra K. Bera

Bengaluru (formerly Bangalore), India

Contents

1 Quantum Cryptography and Quantum Teleportation 1

1.​1 Introduction 1

1.​2 Hello to Some Weirdness in Quantum Mechanics 2

1.​3 Time for Some Mathematics 4

1.​3.​1 Quantum Operators that Act on a Qubit 5

1.​3.​2 A Quantum Operator that Acts on a Qubit Pair 7

1.​4 Encryption and Key Distribution 8

1.​5 Teleportation 11

1.​6 Concluding Remarks 14

References 14

2 Distinguishing Features and Axioms of Quantum Mechanics 17

2.​1 Introduction 17

2.​2 Two-Layer Description of the World 18

2.​2.​1 The Observer in Physics 19

2.​2.​2 Complementarity (Wave-Particle Duality) 19

2.​2.​3 Causality and Determinism 22

2.​3 Superposition, Measurement, and Entanglement 23

2.​4 Classical Mechanics Powers Our Intuition 25

2.​5 The Birth of Modern Quantum Mechanics 26

2.​5.​1 Serendipity at Work 28

2.​6 Cautionary Note on Notations in Quantum Mechanics 30

2.​7 Postulates of Quantum Mechanics Formally Stated 30

2.​7.​1 A Quantum System’s State Space Is a Hilbert Space 31

2.​7.​2 A Quantum System Evolves via Unitary Transformations 31

2.​7.​3 A Quantum System Collapses When Measured 32

2.​7.​4 Hilbert Space Grows Rapidly with the Size of a Quantum System 33

2.​7.​5 Born’s Probabilistic Interpretation 35

2.​7.​6 Heisenberg’s Uncertainty Principle 36

2.​8 Observables and Operators 37

2.​8.​1 Observables in Quantum Mechanics Are Operators 38

2.​8.​2 The Need for Observable-Operators 39

2.​8.​3 Remarks on Vector Spaces 40

2.​9 Weirdness of Quantum Mechanics (In Summary) 41

2.​10 Interpretations of Quantum Mechanics 43

2.​10.​1 Copenhagen Interpretation 44

2.​10.​2 Everett’s Many-World Interpretation 44

2.​10.​3 Bohm’s Interpretation 45

2.​11 From Galileo–Newton to Schrödinger–Born 46

2.​12 Concluding Remarks 47

References 48

3 Mathematical Elements Needed to Compute 53

3.​1 Introduction 53

3.​1.​1 Propositional Calculus (Propositional Logic) 54

3.​1.​2 First-Order Predicate Calculus (First Order Logic) 55

3.​2 Elements of Linear Algebra 56

3.​2.​1 Various Representations of a State Vector 56

3.​2.​2 Bases and Linear Independence 58

3.​3 Linear Operators and Matrices 61

3.​3.​1 Inner Product 62

3.​3.​2 Outer Product 63

3.​3.​3 Tensor Product 64

3.​4 Eigenvalue, Eigenvector, Spectral Decomposition, Trace 67

3.​4.​1 Eigenvalues and Eigenvectors 67

3.​4.​2 Diagonal Representation of an Operator or Orthonormal Decomposition 68

3.​4.​3 Normal Operators and Spectral Decomposition 68

3.​4.​4 Unitary Operators 69

3.​4.​5 Positive Operator 70

3.​4.​6 Trace of a Matrix 70

3.​4.​7 Commutator and Anti-Commutator 71

3.​4.​8 Polar and Singular Value Decompositions 71

3.​4.​9 Completeness Relation 72

3.​5 Cauchy–Schwarz Inequality 72

3.​6 Pauli Matrices 73

3.​7 Concluding Remarks 74

References 75

4 Some Mathematical Consequences of the Postulates 77

4.​1 Introduction 77

4.​2 No-Cloning Theorem 78

4.​2.​1 Consequences of the No-Cloning Theorem 80

4.​3 No-Deleting Theorem 80

4.​4 No-Hiding Theorem 81

4.​5 EPR Paradox and Bell Inequalities 82

4.​5.​1 An Analogy for Factorizable States 83

4.​5.​2 Einstein, Podolsky, Rosen Pose a Paradox 83

4.​5.​3 What Does Hidden Variable Theory Mean?​ 85

4.​5.​4 Bell Inequality 86

4.​5.​5 An Intriguing Question 88

4.​5.​6 Returning to the Bell Inequality 89

4.​5.​7 Would Newton Have Approved of Entanglement?​ 90

4.​6 Superposition and Indeterminacy 90

4.​7 Mathematical Consequences 91

4.​8 Concluding Remarks 94

References 95

5 Waves and Fourier Analyses 99

5.​1 Introduction 99

5.​2 Waves 99

5.​2.​1 The Wave Equation 103

5.​2.​2 Travelling Waves 103

5.​2.​3 Standing or Stationary Waves 103

5.​2.​4 Wave Packets 104

5.​2.​5 Probability Waves 105

5.​3 Fourier Analysis 106

5.​4 Wave Packets in Some Detail 107

5.​4.​1 Group and Phase Velocities 108

5.​5 Concluding Remarks 109

References 109

6 Getting a Hang of Measurement 111

6.​1 Introduction 111

6.​2 Measurement of Quantum Systems 112

6.​2.​1 Cascaded Measurements Are Single Measurements 114

6.​2.​2 Projective Measurements; Observable-Operators 115

6.​2.​3 Distinguishing Quantum States 118

6.​2.​4 When Measurement Basis States Differ from Computational Basis States 118

6.​2.​5 Positive Operator-Valued Measure (POVM) Measurements 119

6.​2.​6 The Effect of Phase on Measurement 120

6.​2.​7 Can Every Observable Be Measured?​ 121

6.​2.​8 Measurement with Photons and Electrons 121

6.​2.​9 Whither Causality?​ 122

6.​3 Heisenberg’s Uncertainty Principle (Revisited) 123

6.​4 Concluding Remarks 126

References 127

7 Quantum Gates 129

7.​1 Introduction 129

7.​2 Operators (A Summary) 131

7.​3 The Qubit 132

7.​3.​1 Global Phase Factor 134

7.​3.​2 Relative Phase Factor 134

7.​3.​3 Unitary Operators 134

7.​3.​4 Hermitian Operators 136

7.​4 Important Qubit Gates 136

7.​4.​1 Pauli Gates and Other 1-Qubit Gates 136

7.​4.​2 2-Qubit Controlled-not Gate 138

7.​4.​3 Creating Entangled Bell States 140

7.​4.​4 Bit Copying—An Application of the Controlled-not Gate 141

7.​4.​5 3-Qubit Toffoli Gate 141

7.​4.​6 3-Bit Fredkin Gate 143

7.4.7 Controlled-U Gate 144

7.​5 Universal Set of Gates 145

7.​5.​1 Universal Set of Classical Gates 145

7.​5.​2 Universal Set of Quantum Gates 146

7.​6 Some Basic Quantum Operations 148

7.​6.​1 Random Number Generation 148

7.6.2n- Qubit Hadamard Gate 148

7.​6.​3 A 3-Qubit Gate for AND and NOT Operations 149

7.​7 Taking Stock of Gates 149

7.​8 Concluding Remarks 152

References 153

8 Unusual Solutions of Usual Problems 155

8.​1 Introduction 155

8.​1.​1 Mach–Zehnder Interferometer 156

8.​2 Some Simple Quantum Algorithms 158

8.2.1 Computingx ∧y 158

8.2.2 Computingx +y 159

8.​2.​3 Swapping States 159

8.​2.​4 The Deutsch Algorithm 160

8.​2.​5 The Deutsch–Jozsa Algorithm 162

8.2.6 Computingf (x ) in Parallel 163

8.​2.​7 Hardy’s Reprieve 164

8.​2.​8 The Elitzur–Vaidman Bomb Problem 166

8.​2.​9 Securing Banknotes 168

8.​3 Concluding Remarks 169

References 169

9 Fundamental Limits to Computing 171

9.​1 Introduction 171

9.​2 Hilbert’s Second Problem 172

9.​2.​1 Recursive Set 174

9.​3 Hilbert’s Tenth Problem 175

9.4 Turing and theEntscheidungsproblem 177

9.​4.​1 Turing’s Halting Problem 179

9.​4.​2 The Church–Turing Thesis 183

9.​4.​3 Deutsch on the Church–Turing Thesis 184

9.​4.​4 Can Quantum Computers Prove Theorems?​ 185

9.​5 Thermodynamic Considerations 185

9.​5.​1 The One-Molecule Gas 187

9.​5.​2 Knowledge and Entropy 188

9.​5.​3 Information Is Physical 188

9.​5.​4 Toffoli Gate 190

9.​5.​5 Bennett’s Solution for Junk Bits 191

9.​5.​6 Reversible Classical Computation Set the Stage for Quantum Computing 192

9.​5.​7 Maxwell’s Demon 192

9.​6 Computational Complexity 195

9.​6.​1 Classification of Complexity 199

9.​6.​2 NP-Complete Problems Stand or Fall Together 203

9.​7 Concluding Remarks 203

References 204

10 The Crown Jewels of Quantum Algorithms 207

10.​1 Introduction 207

10.​2 General Remarks on Quantum Algorithms 208

10.​3 Modulo Arithmetic 209

10.​3.​1 Some Important Properties of Congruence 209

10.​3.​2 Congruence Classes 210

10.​3.​3 Modulo 2 Arithmetic 211

10.​4 Bits and Qubits 211

10.​4.​1 Bitwise Operators 212

10.​4.​2 String Manipulation Leads to Algorithms 212

10.​5 UTM, DTM, PTM, and QTM 214

10.​5.​1 Are Quantum Computers More Powerful?​ 215

10.​6 The Quantum Fourier Transform 216

10.​6.​1 Background 216

10.​6.​2 Quantum Fourier Transform 217

10.​7 Computing the Period of a Sequence 221

10.​8 Shor’s Factoring Algorithm 224

10.​8.​1 Shor’s Algorithm Implemented 226

10.​8.​2 Computational Complexity of Shor’s Algorithm 226

10.​9 Phase Estimation Problem 227

10.​10 Grover’s Search Algorithm 229

10.​10.​1 Grover’s Algorithm Verified 233

10.​10.​2 Computational Complexity of Grover’s Algorithm 234

10.​10.​3 Remarks on Grover’s Algorithm 234

10.​11 Dense Coding and Teleportation 234

10.​11.​1 Dense Coding 235

10.​11.​2 Teleportation 236

10.​12 Concluding Remarks 237

References 238

11 Quantum Error Corrections 241

11.​1 Introduction 241

11.​2 Protecting the Computational Hilbert Space 242

11.​2.​1 Dissipation 243

11.​2.​2 Decoherence 244

11.​2.​3 Algorithmic Error Correction Is Possible 246

11.​3 Calderbank–Shor–Steane Error Correction 246

11.​3.​1 Encoding-Decoding 246

11.​3.​2 Steps of Error Correction 247

11.​4 Decoherence-Free Subspace 250

11.​5 Concluding Remarks 251

References 251

12 Time-Multiplexed Interpretation of Measurement 253

12.​1 Introduction 253

12.​2 A Conjectured Sub-planck Mechanism 255

12.​3 Application of the Basic Model 258

12.​3.​1 Measurement of a Two-Particle Entangled System 258

12.​3.​2 Quantum Adder 259

12.​4 Teleporting a Qubit of an Unknown State 259

12.​5 Concluding Remarks 262

References 262

Index 263

About the Author

Rajendra Bera, Ph.D.,

is Chief Mentor at Acadinnet Education Services, Bengaluru, India since 2010. He received his B.Tech., M.Tech., and Ph.D. degrees in Aeronautical Engineering from the Indian Institute of Technology Kanpur, India. From 1979 to 1980, he was Visiting Assistant Professor of Aerospace, Mechanical, and Nuclear Engineering at the University of Oklahoma, USA, and in 1988 Visiting Faculty of Aerospace Engineering at the Indian Institute of Technology Kanpur, India, where he taught fighter aircraft design. From 2006 to 2011, he was Honorary Professor at the International Institute of Information Technology, Bengaluru (formerly Bangalore), India, where he taught quantum computing and intellectual property rights. From 2013 to 2014, he was Visiting Professor at the Department of Aerospace Engineering, Jain University, Bengaluru (formerly, Bangalore), India, where he taught fighter aircraft design and intellectual property rights. During his student days, he was an active amateur pilot.

From 1971 to 1995, Dr. Bera served at the National Aerospace Laboratories, Bangalore, where he worked in aerodynamics, flight dynamics, theory of elasticity, neural networks, science and technology policy, and technology transfer to industry. From 1995 to 2005, he worked at IBM Software Labs, Bangalore, where he developed an R&D group focusing on new technologies and mentored young researchers and inventors. He is the sole inventor on 28 US patents, all assigned to IBM. His patenting areas include compiler optimization, resource allocation, pattern recognition, and static analysis of computer codes. A former member of the New York Academy of Sciences, Dr. Bera is a fellow of the Institution of Engineers (India) and is listed in several editions of Marquis Who’s Who. The sole author of more than 40 research publications in prominent journals, his current research interests include pattern recognition in molecular biology, quantum computing, intellectual property rights, and nonlinear dynamical systems.

Footnotes

1

Feynman [4].

2

Williams (n.d.). Williams, L. P., Michael Faraday, Encyclopaedia Britannica Online, http://​www.​britannica.​com/​biography/​Michael-Faraday (Accessed 31 March 2016).See also : Hutchinson [5] .

3

Maxwell [8].

4

McMullin [7].

5

Einstein [1]. This paper developed an account of space and time that toppled Newton’s system, and mathematically showed that rapidly moving rods shrink and clocks slow and the speed of light is an impassable barrier. The unexpectedly non-intuitive differences between processes at high speeds and those at lower speeds were fully captured by Maxwell’s equations of electrodynamics. Einstein [2].See also : Einstein [3].

6

Maxwell [6].

7

Feynman, R., Probability and Uncertainty—the Quantum Mechanical view of Nature, “Messenger Lectures” at Cornell, November 1964. Available in Feynman, R.,The Character of Physical Law , The Modern Library, New York, 1994, p. 123.

8

As it appears in Juliana K. Vizzotto, Thorsten Altenkirch, and Amr Sabry, Yale CS Colloquium, 20 January 2005, http://​www.​cs.​indiana.​edu/​~sabry/​papers/​quantum-effects-yale.​pdf .

© Springer Nature Singapore Pte Ltd. 2020

R. K. BeraThe Amazing World of Quantum ComputingUndergraduate Lecture Notes in Physicshttps://doi.org/10.1007/978-981-15-2471-4_1

1. Quantum Cryptography and Quantum Teleportation

Rajendra K. Bera1  

(1)

Acadinnet Education Services India, Bangalore, Karnataka, India

Rajendra K. Bera

Email: rajendrabera@yahoo.com

Abstract

This chapter is meant to be an appetizer and lightly relies on the reader’s intuition to understand the mathematical steps involved. The chapter directly introduces two quantum algorithms: (1) How to encrypt messages (cryptography), which if snooped upon during transmission to a recipient, will be detected; and (2) how to teleport the state of a quantum object. Along the way just enough intuitively understandable but weird and exclusive aspects of quantum mechanics as compared to classical mechanics are introduced.

1.1 Introduction

To understand quantum mechanics and quantum computing, you will need to have some working knowledge of complex numbers, linear algebra, familiarity with complex matrices, and matrix operations. It would be a good idea for you to refresh your ability to calculate eigenvalues and eigenvectors of a matrix after you have read this chapter and are still interested in reading the remaining chapters of this book.

In this chapter, we will directly introduce you to two quantum algorithms: (1) How to encrypt messages (cryptography ), which if snooped upon during transmission to a recipient, will be detected (in no case will the snooper be able to read the message correctly); and (2) how to teleport the state of a quantum object. Along the way, you will be introduced to some intuitively understandable aspects of quantum mechanics, just enough for you to understand the two algorithms. Interestingly, classical physics has no solution for either problem. So, quantum mechanics does allow you to live in a magical world. To practice this magic, you will need to get used to some weird ways in which Nature works exclusively at the quantum level, a level most often seen at the atomic and sub-atomic levels. Before proceeding further, let me introduce you to some words of wisdom from the great physicist Richard Feynman . Prefacing a lecture on quantum mechanics, he said,

Do not take the lecture too seriously… just relax and enjoy it. I am going to tell you what nature behaves like. If you will simply admit that maybe she does behave like this, you will find her a delightful, entrancing thing. Do not keep saying to yourself, if you can possibly avoid it, ‘But how can it be like that?’ because you will get… into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that.¹

His words put me at ease and encouraged me to learn quantum mechanics and quantum computing. The fact is, human intuition, not intellect, is ill-equipped to deal with the mysteries of the quantum world; it is best understood in the abstract language of mathematics. Why? Feynman answers:

Mathematics is a language plus reasoning; it is like a language plus logic. Mathematics is a tool for reasoning.²

In quantum mechanics, if your mathematics is right, you can ignore common sense and intuition. If in doubt, check your mathematics. This checking is not an intelligent activity, because, in principle, it can be mechanized. This we know from Alan Turing and his amazing paper³ in 1936 in which he showed that a machine, the Universal Turing Machine (UTM), can mimic an unintelligent person, who tirelessly and with absolute concentration performs calculations as instructed, in a step-by-step manner, i.e., according to a given algorithm. This Turing Person has at its disposal unlimited time, paper, pencil, and energy. Note that the UTM does only those tasks that a human might do in executing an algorithm without the help of insight. There is a strong belief, so far unrefuted, among computer scientists that what is human-computable is machine-computable. This is the famous Church–Turing Thesis : “The class of functions computable by a Turing machine corresponds exactly to the class of functions which we would naturally regard as being computable by an algorithm.” It is also a statement about the limitations of the human mind.

1.2 Hello to Some Weirdness in Quantum Mechanics

We now come to a few weird things in quantum mechanics. Quantum physicists use the terms superposition and measurement in very specific ways. You are perhaps already aware that in physics there are a set of key words, e.g., force, work, temperature, entropy, enthalpy, etc., which all qualified physicists understand in the same way, and these words and their various relationships are expressible by precise mathematical formulas. Hence, if in doubt, check the math before jumping to the conclusion that you have made a Nobel Prize-winning discovery.

In quantum mechanics, the term superposition is used to state that matter or energy at the quantum level can be in two different states at the same time, e.g., an electron can be in spin-up and spin-down states at the same time or a photon can be vertically polarized and horizontally polarized at the same time. When you measure the state of a quantum entity (say, the electron or the photon) in a superposed state, you will see only one or the other of the two states it is concurrently in. For example, you will see the electron in the spin-up state or the spin-down state, but you will not be able to predict beforehand what the measurement will be. You will never see some combination of spin-up and spin-down states. Likewise, for the photon. You will never be able to measure a photon in some combined state of vertical and horizontal polarization and you will not be able to predict the measurement outcome beforehand with certainty. That is because, and as far as we can tell, Nature decides the measurement outcome at the last moment by tossing a biased coin (i.e., probabilistically). The biasness of the coin has a precise mathematical relationship with how the quantum states are superposed in the quantum entity being measured. All the probabilities we talk about in quantum mechanics has to do with measurement and not with the way an undisturbed quantum system evolves. It is possible to take a quantum entity, e.g., an electron or a photon and put it into a state of superposition of our choice. But if we are given a quantum entity of unknown state, we can never determine its state by any means or make an exact replica of it.⁴ All this is very strange, but is it?

See Fig. 1.1. On the left is shown an electron in its two possible spin-up and spin-down states and in its state of superposition. It is difficult to get a feel for quantum superposition and measurement because of its emotionless abstraction. The picture on the right is very different. If you look at it several times, you (the measurement apparatus) will randomly see (measure) either a pretty girl or an old wrinkled woman (the collapsed state of the picture). It is the same picture! Your brain will ensure that at no time will you see both the pretty girl and the old woman or some weird combination of the two. This picture, not in detail, provides a glimpse of quantum superposition and measurement through the interplay of eye and mind.

Fig. 1.1

(Left) Lieven Vandersypen, Dot-to-Dot Design. IEEE Spectrum, September 2007. pp. 42–47. https://​sites.​cs.​ucsb.​edu/​~cappello/​IEEE/​Spec_​20070901_​Sep_​2007.​pdf; (Right) German postcard from 1888. Wikimedia Commons. https://​commons.​wikimedia.​org/​wiki/​File:​German_​postcard_​from_​1888.​png. Popularly titled as My wife and my mother-in-law. (In public domain). If you look at the eyes in the figure, you will see the mother-in-law; if you look at the bonnet you will see the wife.

There is a third term in quantum mechanics that refers to an enigmatic quantum phenomenon called entanglement. This is an intriguing state of being in which two quantum entities are so deeply correlated that they behave as one composite entity, no matter how far apart they are in space. Indeed, distance has no meaning for entangled entities. If the state of one is changed, the state of the other is instantly adjusted to be consistent with quantum mechanical laws. If a measurement is made on one, the other automatically and instantaneously collapses to a predefined state. Einstein derisively called such action at a distance “spooky.” He wrote to Max Born in March 1947, “I cannot seriously believe in [quantum theory] because it cannot be reconciled with the idea that physics should represent a reality in time and space, free from spooky actions at a distance.”⁵ After Einstein’s death, it was proven that he was wrong.⁶ Entanglement is real, and it is a joint characteristic of two or more quantum entities when entangled. Just as a quantum entity can be put in a desired state of superposition, so can two quantum entities be put in a state of entanglement, which means they acquire a group property. In quantum computing, entanglement plays an unusually big role.

1.3 Time for Some Mathematics

To proceed further, we need some mathematics. The important thing about mathematics is that it is a language that allows you to communicate with precision and compactness in a systematic manner. The most important characteristic of modern mathematics is that it is presented as an axiomatic study, whereas the sciences are axiomatic only to the extent they utilize mathematics. The key to axiomatic reasoning is the idea that the truth of a statement must be shown to follow logically from the truth of other statements, which have already been shown to be true by this method. Deductive reasoning dominates mathematics. The biggest dilemma in creating an axiomatic system is how do we get started? We crank up the system by accepting one or more statements to be true without demanding a proof of their truthfulness, i.e., we rely upon our intuition (the Achilles heel of mathematics). These statements are called postulates or axioms. We try to keep such statements as few as possible and do our best to see that the statements are extremely unlikely to lead to contradictions. Briefly, by a deductive system we mean:

We have selected one or more concepts that we feel are very primitive and that we agree to accept them without definition. These are the undefined concepts of the system (e.g., point and straight line in Euclidean geometry).

We have selected some statements concerning the undefined concepts that we feel express very primitive truths about the undefined concepts and that we are going to accept without proof. These are the axioms of the system.

Using undefined concepts and axioms we can begin the process of defining new concepts in terms of the undefined concepts. We call these defined concepts.

And establish the truth of new statements about these concepts based on the axioms. We call the new true statements theorems.

By an axiomatic (also called formal) system, we mean a system comprising a set of symbols; a grammar for combining the symbols into statements; a set of axioms, or statements that are accepted without proof; and rules of inference for deriving new statements (theorems). A proof is a listing of the sequence of inferences that derive a theorem. It is vital that a proof be formally (i.e., mechanically) verifiable. Thus, there is a correspondence between an axiomatic system and a computational system whereby a proof is essentially a string (usually a binary string) processing computation. This is shown in Table 1.1.

Table 1.1

Correspondence between axiomatic and computational systems

Axiomatic system

Computational system

Axioms

Program input or initial state

Rules of inference

Program interpreter

Theorem(s)

Program output

Derivation

Computation

Source Lewis [19]

A proof in axiomatic mathematics is an impeccable argument that uses only the methods of pure logical reasoning. The reasoning is such that it enables one to infer the validity of a given mathematical assertion from the pre-established validity of other mathematical assertions or the axioms. Once a mathematical assertion has been established by this procedure, it is called a theorem. Axiomatic mathematics is about axioms, theorems, and proofs.

To implement an axiomatic system, we set up a typographical system so that statements appear as mere strings of symbols according to some typographical rules. Through an appropriately chosen translation rules, symbol strings can be rewritten as binary strings and manipulated by digital computers (essentially, Universal Turing Machines).

1.3.1 Quantum Operators that Act on a Qubit

In the world of quantum mechanics, physical systems are described by an abstract mathematical object called the state vector (or the wave function) $$ \left| \psi \right\rangle $$. (Note the unusual notation $$ \left| \psi \right\rangle $$ first introduced by Paul Dirac. The symbols: $$ \left\langle \cdot \right| $$ is called bra and represents a column vector; $$ \left| \cdot \right\rangle $$ is called ket and represents a row vector; the dot inside the symbols is a placeholder for labels. The nomenclature is more fully described in Chap. 3, Tables.) People are still trying to understand the exact status of $$ \left| \psi \right\rangle $$ in quantum theory. However, such ignorance has not prevented physicists from moving forward. Further, for the purposes of this chapter, it is enough to know two (of four⁷) postulates or axioms of quantum mechanics . First is the natural evolution of a quantum system, i.e., of the state function $$ \left| \psi \right\rangle $$. It evolves in a deterministic manner according to the linear Schrödinger equation . Second is the measurement of the system by a process called wave packet reduction (or more dramatically as wave function collapse). The problem with measurement is that nobody yet knows how to precisely define the act of measurement. Whenever it happens, the quantum system collapses according to a probabilistic postulate. This makes measurements of quantum systems a source of major conceptual difficulties, but not enough to impede quantum mechanics from making breathtaking advances.

The unusual difference between quantum mechanics and classical mechanics is that in the former we have two different postulates for the evolution of the same mathematical object, and in the latter we have only one. So, in quantum mechanics we sometimes have difficulty in knowing which postulate to apply. This leads us to the problem of decoherence—the problem of the instability of coherence—since we are not always quite sure what constitutes a measurement, it is possible that during its evolution (say while doing computations) a quantum system may get decohered and create errors in the computations. The reason why quantum computers still have a long way to go in terms of robustness is that superposition and entanglement are extremely fragile states. Any interaction with the environment (i.e., anything external to the quantum system being studied) and the quantum system may decohere. Preventing decoherence from taking hold before a calculation is completed remains the biggest challenge.

The quantum mechanical counterpart of the classical binary bit, which at any time is in state 0 or 1, is the qubit (short for quantum bit, so named by Schumacher in [26]).⁸ A qubit can be in a superposition of 0 and 1, i.e., it can concurrently be 0 and 1 (like “my wife and my mother-in-law in Fig. 1.1). The state 0 of a qubit is represented by $$ \left| 0\right\rangle $$ and state 1 by $$ \left| 1\right\rangle $$, called its eigenstates, and the general superposed state of a qubit is represented by the unit vector $$ \left| \psi \right\rangle $$  = a$$ \left| 0\right\rangle $$ + b$$ \left| 1\right\rangle $$, where a and b are complex numbers constrained by the relation |a|² + |b|² = 1. If such a superposition is measured with respect to the basis {$$ \left| 0\right\rangle $$, $$ \left| 1\right\rangle $$}, the probability that $$ \left| \psi \right\rangle $$ will collapse to $$ \left| 0\right\rangle $$ is |a|² and the probability that it will collapse to $$ \left| 1\right\rangle $$ is |b|².

The state of a qubit (the simplest quantum entity we can think of) given by $$ \left| \psi \right\rangle $$ = a$$ \left| 0\right\rangle $$ + b$$ \left| 1\right\rangle $$ can be altered using one or more unitary operators . These operators have some simple and easy-to-remember properties. First, by definition, a unitary operator U has an inverse, which means that you can undo an operation, and that inverse is the same as its conjugate transpose U†, hence U†U = I = UU†, where I is the identity operator, i.e., it leaves things it operates on unchanged, much like multiplying a number by 1. You really need to know only 4 unitary operators . These are:

1-qubit unitary operators

Operating on $$ \left|\psi \right\rangle $$ = a$$ \left| 0\right\rangle $$ + b$$ \left| 1\right\rangle $$

Identity

I:

$$ \left| 0\right\rangle $$ → $$ \left| 0\right\rangle $$

$$ \left| 1\right\rangle $$ → $$ \left| 1\right\rangle $$

I$$ \left| \psi\right\rangle $$ = a$$ \left| 0\right\rangle $$ + b$$ \left| 1\right\rangle $$

Negation

X:

$$ \left| 0\right\rangle $$ → $$ \left| 1\right\rangle $$

$$ \left| 1\right\rangle $$ → $$ \left| 0\right\rangle $$

X$$ \left| \psi\right\rangle $$ = a$$ \left| 1\right\rangle $$ + b$$ \left| 0\right\rangle $$

ZX

Y:

$$ \left| 0\right\rangle $$ → −$$ \left| 1\right\rangle $$

$$ \left| 1\right\rangle $$ → $$ \left| 0\right\rangle $$

Y$$ \left| \psi \right\rangle $$ = −a$$ \left| 1\right\rangle $$ + b$$ \left| 1\right\rangle $$

Phase shift

Z:

$$ \left| 0\right\rangle $$ → $$ \left| 0\right\rangle $$

$$ \left| 1\right\rangle $$ → −$$ \left| 1\right\rangle $$

Z$$ \left| \psi \right\rangle $$ = a$$ \left| 0\right\rangle $$ − b$$ \left| 1\right\rangle $$

The rightmost column shows how each operator affects $$ \left| \psi \right\rangle $$. Any other unitary operator M required to operate on a qubit can be created by a linear combination of these four operators as

where α, β, γ, and δ are complex constants of one’s choice. Second, U is diagonalizable, and its eigenvectors are orthogonal. U only rotates the vector $$ \left| \psi \right\rangle $$ it operates on; it does not change the length of the vector, which remains 1, i.e., in $$ \left| \psi \right\rangle $$ = a$$ \left| 0\right\rangle $$ + b$$ \left| 1\right\rangle $$, a and b may change but only by maintaining |a|² + |b|² = 1.

There is one frequently used gate, called the Hadamard gate, H = (X + Z)/√2 that operates on a qubit to accomplish the following:

Note that the length of H$$ \left| \psi \right\rangle $$ remains unity since |a + b|²/2 + |a − b|²/2 = 1.

1.3.2 A Quantum Operator that Acts on a Qubit Pair

The general state of a 2-qubit system is given by the linear combination

Here, the notation $$ \left| xy \right\rangle $$ depicts the state where the first qubit’s state is $$ \left| x \right\rangle $$ and of the second is $$ \left| y \right\rangle $$. The 2-qubit system has only four eigenstates: $$ \left| 00 \right\rangle $$, $$ \left| 01 \right\rangle $$, $$ \left| 10 \right\rangle $$, and $$ \left| 11 \right\rangle $$. Further, |a|² + |b|² + |c|² + |d|² = 1 to ensure that $$ \left| \psi \right\rangle $$ remains a unit vector. There is a 2-qubit unitary operator , Cnot, called the controlled-not , that acts on $$ \left| \psi \right\rangle $$ such that

The Cnot operator flips the second (target) qubit if the first (control) qubit is $$ \left| 1 \right\rangle $$ and does nothing if the control qubit is $$ \left| 0 \right\rangle $$ This operation also entangles the two qubits (more on this in Chap. 7). Note that the length of Cnot$$ \left| \psi \right\rangle $$ remains unity, i.e., |a|² + |b|² + |c|² + |d|² = 1.

Important remark: It can be shown that by stringing together 1-qubit operations and the 2-qubit controlled-not operation, it is possible to build a quantum computer capable of doing anything a classical computer can do.⁹ We also note that there are only two ways to manipulate a quantum system: (1) make a measurement which would irreversibly and probabilistically collapse the system into one of its eigenstate or (2) use unitary operators to deterministically evolve the system.

1.4 Encryption and Key Distribution

For our limited purpose of cryptography , we need the 1-qubit unitary Hadamard operator, H, that operates on one qubit at a time and to remember that measurement is a non-unitary operation that collapses the quantum system being measured in a probabilistic and irreversible way. Further, the inability to copy an unknown quantum state is a key difference between ordinary and quantum information. This fact has made quantum information theory very attractive to cryptographers.

The exchange of secret messages in a completely secure manner using non-quantum mechanical means requires a perfect cypher. Such a cypher known as the Vernam cypher¹⁰ or one-time pad was invented in 1917 by Gilbert S. Vernam. Unfortunately, it requires a key equal in size to the plaintext message. Shannon’s information theory¹¹ shows that we cannot do better. For keys to be shorter, the cyphertext must compromise and contain some information about the plaintext message. Thus, for perfect security we have the problem of distributing the key itself, which must be done over a secure channel such as by a trusted courier. In many situations, such as banking transactions where the volume of information is very large, this is impractical. Therefore, alternative, but less secure methods such as the RSA public key cryptosystem¹² are often used. Exchanging keys securely is therefore a truly crucial step in cryptography . It is the secure exchange of keys that we discuss here.

In 1984, Charles H. Bennett and Gilles Brassard described the first completely secure quantum key distribution algorithm, now known as the BB84 protocol ,¹³ in which quantum states are used to establish a random secret key for cryptography . Thus, they were able to circumvent the restriction of Shannon’s theory and permit keys shorter than the message. The BB84 protocol exploits two unique quantum mechanical aspects—the ability to generate perfectly random numbers (using the Hadamard operator) and the fact that, in general, any observation (measurement) disturbs (collapses) the quantum system being observed. Thus, if there is an eavesdropper attempting to intercept a message being transmitted, his or her presence will be felt as a disturbance in the communication channel. BB84 does not use quantum entanglement. Since the protocol makes it possible to generate keys which are perfectly random, it is impossible for anyone who does not know it to decode any message, even if it is sent publicly.

BB84 protocol

Suppose the proverbial quantum denizens Alice and Bob want to communicate privately, and Eve is interested in eavesdropping. The available means of communication are an ordinary bidirectional open channel (e.g., a telephone) and a unidirectional quantum channel. Both channels can be eavesdropped by Eve. The quantum channel allows Alice to send individual particles (say, photons) to Bob who can measure their quantum state. Eve can attempt to measure the state of these photons and can resend them to Bob. To establish the key, Alice begins by sending Bob a sequence of encoded photons. To encode the photons, she randomly uses one of the following two bases:

where the arrows indicate the polarized state of the photon used to encode the binary digits 0 (up arrow or left inclined arrow) and 1 (horizontal arrow or right inclined arrow). (Physicists do have a sense of humor in picking symbols for quantum states!) Bob measures the state of the photons he gets by randomly picking either basis. After the photons have been transmitted and measured, Alice and Bob communicate over the open channel, the basis they used for coding and decoding of each photon. (This amounts to sending a string of symbols, which are meaningless without the results themselves.) On average, 50% of the time their bases will match. Alice and Bob use those photons as the key for which their bases agree and discard the other photons. So far there is no quantum advantage.

Can Eve steal the key? Suppose Eve measures the state of the photons sent by Alice and resends new photons with the measured state to Bob. However, Eve will get her measurement basis wrong, on average, 50% of the time, since Eve does not know the basis sequence used by Alice. Thus, when Bob measures a resent photon with the correct basis (Alice’s basis), there will be a 25% probability that he will measure the wrong value. This is because Eve, by measuring the photons en route, would have collapsed them to her measured value. Thus, Eve is bound to introduce a high rate of error that Alice and Bob can detect by communicating a sufficient number of parity bits of their keys over the open channel. So, not only will Eve’s version of the key will be, on average, 25% incorrect, but that someone is eavesdropping will be apparent to Alice and Bob. If eavesdropping is detected, Alice and Bob simply discard the key and send a new one. Only when both are certain that their key was not compromised do they use it for encryption. Their encrypted messages can now be sent over bidirectional open channels. Quantum cryptography’s great advantage is that it solves the key distribution problem by taking advantage of the fact that measurement of a quantum system, no matter how delicately made, causes a collapse of the system’s wave function in an unpredictable manner. There was hardly any serious mathematics involved here! Indeed, one small step in mathematics was one giant leap in cryptography !

On April 24, 2014, Nature reported, “This week, China will start installing the world’s longest quantum-communications network, which includes a 2,000-km link between Beijing and Shanghai. And a study jointly announced this week by the companies Toshiba, BT and ADVA, with the UK National Physical Laboratory in Teddington, reports ‘encouraging’ results from a network field trial, suggesting that quantum communications could be feasible on existing fibre-optic infrastructure.”¹⁴ On September 29, 2018, the Chinese satellite Micius successfully beamed down a small data packet of encryption keys encoded in photons to a ground station in Xinglong, a couple of hours’ drive to the northeast of Beijing. Within an hour Micius, as it passed over Austria, delivered another such data packet to a station near the city of Graz. “The video encryption was conventional, not quantum, but because the quantum keys were required to decrypt it, its security was guaranteed. This made it the world’s very first quantum-encrypted intercontinental video link.”¹⁵ China’s ambition is to become a global leader in secure quantum communication by 2030.

1.5 Teleportation

Teleportation is the ability to transmit the quantum state of an entity, say, a particle, using classical bits and to re-construct the exact quantum state at the receiver. In 1993, Charles H. Bennett led a group which showed how a particle of unknown quantum state can be teleported.¹⁶

For our limited purpose of teleportation , we need the Hadamard operator that operates on one qubit at a time, and the Cnot operator that acts on two qubits at a time and in the process entangles them. Note also that it is impossible to make a duplicate of a quantum entity without knowing its complete state, but we can prepare a quantum entity in as many copies as we like in a state of our choice. Teleportation allows the transfer of an unknown state of a first quantum entity to a second quantum entity but only by changing the state of the first entity. Instinctively, one perhaps realizes that teleportation may be realized by manipulating a pair of entangled particles; if we could impose a specific quantum state on one member of an entangled pair of particles, then we would be instantly imposing a predetermined quantum state on the other member of the entangled pair.

Briefly, this is how it works. Initially, two entangled photons (p2 and p3) propagate toward two remote regions of space. Photon p2 reaches Alice, while photon p3 reaches Bob. A third photon p1 in state $$ \left| \phi \right\rangle $$ is then provided to Alice. Alice now possesses p1 and p2. The goal is to put Bob’s photon p3 into state $$ \left| \phi \right\rangle $$ without transporting any photon between Bob and Alice. (Recall p2 and p3 are entangled; hence, any change in the state of one will bring about an instantaneous change in the other.)

Obviously, Alice cannot perform any measurement on photon p1 currently in state $$ \left| \phi \right\rangle $$ because it would destroy the state of the photon. So, she entangles the two photons p1 and p2 in her possession (this, of course, entangles all three photons) and then performs a “combined measurement” on them. (Rest assured this can be done.) She, then, communicates the result of her measurement to Bob (using classical means—telephone, email, etc.).

Bob now applies to his photon a unitary operation depending on the classical information he has received from Alice. This operation puts his photon in exactly the state $$ \left| \phi \right\rangle $$, the initial state of p1 and thus realizes teleportation . Since all the photons are entangled, the two photons in Alice’s possession are no longer in their original states, and hence there is no duplicate of $$ \left| \phi \right\rangle $$ in existence! Note that the whole operation is mixed because it involves a combination of transmission of quantum information (through the entangled state) and classical information (phone call from Alice to Bob), which cannot travel faster than the speed of light. So, teleportation cannot be done faster than the speed of light.

Let us now see the mathematics behind it. Alice has qubit p1 of unknown state $$ \left| \phi \right\rangle $$ = a$$ \left| 0\right\rangle $$ + b$$ \left| 1\right\rangle $$. She wishes to send the state of this qubit to Bob through classical channels. Alice and Bob, respectively, possess qubit p2 and p3 which are entangled in the state $$ \left| \psi_{0} \right\rangle $$ = (1/√2) ($$ \left| 00 \right\rangle $$+ $$ \left| 11 \right\rangle $$) (it has the special name Bell state after John Bell; see Chap. 4, Sect. 4.​5). The first qubit here is p2 and the second p3. At any time, the state of the 3-qubit system is a linear combination of some 3-qubit eigenstates, each of form $$ \left| xyz \right\rangle $$ where x, y, z, respectively, belong to qubit p1, p2, and p3. The initial state, χ₀ of our 3-qubit system is

of which Alice controls the first two qubits and Bob controls the third qubit . Alice now applies Cnot to the first two qubits (p1 and p2) in her possession using p1 as the control. This puts the 3-qubit system in the state (note the changes in the second qubit ):

She now applies the Hadamard operator on the first qubit . This puts the 3-qubit system in the state

Alice then makes a combined measurement of the first two qubits to get one of the 2-qubit eigenstates $$ \left| 00 \right\rangle $$, $$ \left| 01 \right\rangle $$, $$ \left| 10 \right\rangle $$, or $$ \left| 11 \right\rangle $$ with equal probability. Depending on the result of the measurement, the quantum state of Bob’s entangled qubit is projected to

Bob must now wait for Alice to send her measurement, which she does by encoding it in two classical bits using a classical communication system which cannot communicate faster than light.

Note that Alice’s measurement has irretrievably altered the state of p1 from its original state $$ \left| \phi \right\rangle $$, which she is trying to send to Bob. When Bob receives Alice’s 2 bits, he knows how the state of his half of the entangled pair compares to the original state of Alice’s qubit (see Table below).

Result sent

Bob’s qubit

Decoder

Output

00

a$$ \left| 0 \right\rangle $$ + b$$ \left| 1 \right\rangle $$

I

a$$ \left| 0 \right\rangle $$ + b$$ \left| 1 \right\rangle $$

01

a$$ \left| 1 \right\rangle $$ + b$$ \left| 0 \right\rangle $$

X

a$$ \left| 0 \right\rangle $$ + b$$ \left| 1 \right\rangle $$

10

a$$ \left| 0 \right\rangle $$ − b$$ \left| 1 \right\rangle $$

Z

a$$ \left| 0 \right\rangle $$ + b$$ \left| 1 \right\rangle $$

11

a$$ \left| 1 \right\rangle $$ − b$$ \left| 0 \right\rangle $$

Y

a$$ \left| 0 \right\rangle $$ + b$$ \left| 1 \right\rangle $$

Operator

Operation

Identity

I

$$ \left| 0 \right\rangle $$ → $$ \left| 0 \right\rangle $$

$$ \left| 1 \right\rangle $$ → $$ \left| 1 \right\rangle $$

Negation

X

$$ \left| 0 \right\rangle $$ → $$ \left| 1 \right\rangle $$

$$ \left| 1 \right\rangle $$ → $$ \left| 0 \right\rangle $$

ZX

Y

$$ \left| 0 \right\rangle $$ → −$$ \left| 1 \right\rangle $$

$$ \left| 1 \right\rangle $$ → $$ \left| 0 \right\rangle $$

Phase shift

Z

$$ \left| 0 \right\rangle $$ → $$ \left| 0 \right\rangle $$

$$ \left| 1 \right\rangle $$ → −$$ \left| 1 \right\rangle $$

Bob can now reconstruct the original state $$ \left| \phi \right\rangle $$ of the unknown qubit p1 by applying the appropriate decoder to p3 in his possession as shown in the table’s decoder column.

The amazing thing about quantum teleportation is that it permits the transfer of quantum information into inaccessible space and into a quantum memory without revealing or destroying the stored quantum information.

The following sample of technological advances related to teleportation and cryptography indicate the tremendous role they will play in the future of secure communications. Teleportation of a laser beam consisting of millions of photons was achieved in 1998. In June 2002, an Australian team reported a more robust method of teleporting a laser beam. Teleportation of trapped ions (calcium and beryllium) was achieved in June 2004 by two groups and reported in Nature.¹⁷ Teleportation of single molecules may take some time. In 2006, a team at the Niels Bohr Institute, Copenhagen, teleported information stored in a laser beam into a cloud of atoms. Thus, for the first time teleportation between light and matter was achieved. One is the carrier of information and the other is the storage medium.¹⁸ In 2014, physicists demonstrated a device that can teleport quantum information to a solid-state quantum memory over telecom fiber, a crucial capability required of any future quantum Internet.¹⁹ On June 28, 2019, researchers from Yokohama National University in Japan reported teleporting quantum information securely inside a diamond.²⁰ Using quantum teleportation , they were able to transfer the state of a photon polarization into a carbon spin in diamond. The breakthrough could help us better share and store sensitive information.

China launched the world’s first quantum satellite Micius (mentioned in Sect. 1.4, is named after the ancient Chinese scientist and philosopher Micius) on August 15, 2016.²¹ At the heart of the satellite is a crystal that produces pairs of entangled photons, whose properties remain entwined; however far apart they are separated. The satellite can fire the partners in these pairs to ground stations in Beijing and Vienna, and use them to generate a secret key. In June 2017, Chinese researchers announced that they had beamed photons between the satellite Micius and the two distant ground stations, and were successful in maintaining the entangled quantum state at a record-breaking distance of more than 1,200 km.²² Quantum theory predicts that entanglement can persist at any distance.

1.6 Concluding Remarks

This chapter was meant to be an appetizer and lightly relied on your intuition to understand the mathematical steps involved. If it has made you curious, do move on to the next chapter where the real stuff begins. But before doing so, do have a quick look at a text book that describes linear algebra, complex numbers, complex matrices and their operations, and the significance of eigenvalues and eigenvectors²³ of a matrix. Mathematics and computation are about abstract symbol manipulation that unintelligent computers can be programmed to do. So, turn yourself into a robot, learn these manipulations, then switch on your intelligent self, and go to Chap. 2. Imagine if you can teleport with so little mathematics, what more you might achieve if you really knew mathematics.

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C.H. Bennett, G. Brassard, Quantum Cryptography: Public Key Distribution and Coin Tossing, in Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, IEEE, New York (The conference was held at Bangalore, India, Dec 1984), pp. 175–179

8.

C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70, 1895–1899 (1993)

9.

P.E. Black, D. Richard Kuhn, C.J. Williams, in Quantum Computing and Communications, Advances in Computers, vol. 56 (Academic Press, 2002)

10.

G. Brassard, in Brief History of Quantum Cryptography: A Personal Perspective (2006). arXiv:​quant-ph/​0604072v1 11 Apr 2006. https://​arxiv.​org/​pdf/​quant-ph/​0604072.​pdf

11.

G. Brassard, C. Crépeau, in Cryptology Column—25 Years of Quantum Cryptography (ACM SIGACT News · Sept 1996), pp. 1–12. https://​www.​researchgate.​net/​publication/​220556114_​25_​years_​of_​quantum_​cryptography

12.

F. Bussieres et al., in Quantum Teleportation from A Telecom-Wavelength Photon To A Solid-State Quantum Memory (2014). arxiv.​org/​abs/​1401.​6958, 27 Jan 2014. http://​arxiv.​org/​abs/​1401.​6958

13.

D. Castelvecchi, China’s quantum satellite clears major hurdle on way to ultrasecure communications. Nature, 15 June 2017. https://​www.​nature.​com/​news/​china-s-quantum-satellite-clears-major-hurdle-on-way-to-ultrasecure-communications-1.​22142

14.

A. Einstein, B. Podolsky, N. Rosen, Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 41, pp. 777–780 (1935). http://​www.​drchinese.​com/​David/​EPR.​pdf. Known as the EPR paper, it claimed that QM was an incomplete theory. After Einstein died in 1955, John Bell and others would prove him wrong

15.

R. Feynman, in The Character of Physical Law, Modern Library Edition, 1994 (Originally published by BBC in 1965, and in paperback by MIT Press, 1967, 1965)

16.

E. Gibney, Chinese satellite is one giant step for the quantum internet. Nature 27 July 2016 (updated 16 Aug 2016). http://​www.​nature.​com/​news/​chinese-satellite-is-one-giant-step-for-the-quantum-internet-1.​20329

17.

M. Giles, The man turning china into a quantum superpower. MIT Technol. Rev. 19 Dec 2018 (1965). https://​www.​technologyreview​.​com/​s/​612596/​the-man-turning-china-into-a-quantum-superpower/​

18.

O. Goldreich, Foundations of Cryptography, vol. 2 (Cambridge University Press, 2004). http://​www.​wisdom.​weizmann.​ac.​il/​~oded/​foc-book.​html

19.

J.P. Lewis (2001) Large limits to software estimation. ACM Softw. Eng. Notes 26(4), 54–59

20.

N.D. Mermin (1985) Is the moon there when nobody looks? Real. Quant. Theory, Phys. Today 38–47. http://​maltoni.​web.​cern.​ch/​maltoni/​PHY1222/​mermin_​moon.​pdf

21.

S. Muralidharan et al., in Efficient Long Distance Quantum Communication (2015). arXiv:​1509.​08435v1 [quant-ph], 28 Sept 2015, http://​arxiv.​org/​pdf/​1509.​08435v1.​pdf

22.

E.S. Polzik et al., Quantum teleportation between light and matter. Nature 443, 557–560 (2006)

23.

J. Qiu, Quantum communications leap out of the lab. Nature 508, 441–442 (2014). http://​www.​nature.​com/​polopoly_​fs/​1.​15093!/​menu/​main/​topColumns/​topLeftColumn/​pdf/​508441a.​pdf

24.

M. Riebe et al., Deterministic quantum teleportation with atoms. Nature 429, 734–737 (2004)

25.

R.L. Rivest, A. Shamir, L.M. Adleman, A method of obtaining digital signatures and public-key cryptosystems. Comm. ACM 21(2), 120–126 (1978)

26.

B. Schumacher, Quantum coding. Phys. Rev. A 5(4), 2738–2747 (1995)

27.

C.E. Shannon, Communication theory of secrecy systems. Bell Syst. Tech. J. 28(4), 656–715 (Oct 1949) (The material in this paper appeared originally in a confidential report “A Mathematical Theory of Cryptography” dated Sept. 1, 1945, which has now been declassified.). http://​pages.​cs.​wisc.​edu/​~rist/​642-spring-2014/​shannon-secrecy.​pdf

28.

P. W. Shor, J. Preskill, Simple proof of security of the BB84 quantum key distribution protocol. Phys. Rev. Lett. 85, 441–444 (2000). arXiv quant-ph/0003004, 2000. http://​arxiv.​org/​PS_​cache/​quant-ph/​pdf/​0003/​0003004.​pdf

29.

G. Strang (2014) in Differential Equations and Linear Algebra (Wellesley-Cambridge Press, MA, 2014)

30.

K. Tsurumoto, R. Kuroiwa, H. Kano, Y, Sekiguchi, H. Kosaka, Quantum teleportation-based state transfer of photon polarization into a carbon spin in diamond. Commun. Phys. 2(74) (2019). https://​www.​nature.​com/​articles/​s42005-019-0158-0

31.

A. Turing (1936) On computable numbers, with an application to the Entscheidungs problem, in Proceedings of the London Mathematical Society, Series 2, vol. 42, pp. 230–265; Errata (1937), vol. 43, pp. 544–546. http://​www.​abelard.​org/​turpap2/​tp2-ie.​asp

32.

S. Wiesner, Conjugate coding. Sigact News 15(1), 78–88 (1983). (Original manuscript written circa 1969)

33.

W.K. Wootters, W.H. Zurek, A single quantum cannot be cloned. Nature, 299, 802–803 (1982). http://​puhep1.​princeton.​edu/​~mcdonald/​examples/​QM/​woottersnature29​980282.​pdf

Footnotes

1

Feynman [15], Chap. 6.

2

Feynman [15].

3

Turing [31].

4

Wootters and Zurek [33].

5

As quoted in: Mermin [20]. “Einstein maintained that quantum metaphysics entails spooky actions at a distance; experiments have now shown that what bothered Einstein is not a debatable point but the observed behaviour of the real world.”

6

See the following three papers to develop a perspective on how physicists came to understand entanglement: Einstein et al. [14], Bell [6], and Aspect et al. [3].

7

All four postulates are formally described in Chap. 2, Sect. 2.​7.

8

Schumacher [26].

9

See Barenco et al. [4]. See also Chap. 7 of this book.

10

Developed by Gilbert S. Vernam of AT&T. This is the only known totally secure cypher. Vernam was granted a patent protecting the cypher: Secret Signaling System. US Patent No. 1,310, 719, patented July 22, 1919.

11

Shannon [27]. See also: Goldreich [18]; Black et al. [9], pp. 189–244, Sect. 5.

12

Rivest et al. [25]. See also: Allenby and Redfern [2]. Rivest, Shamir, and Adleman received the Turing award for 2002 for their contributions to public key cryptography . http://​www.​acm.​org/​announcements/​turing_​2002.​html.

13

Bennett and Brassard [7]. The first quantum cryptography ideas were proposed by Stephen Wiesner in the late 1960s, but unfortunately were not accepted for publication at the time! It was eventually published in 1983, Wiesner [32]. Bennett and Brassard built upon Wiesner’s work. A simple proof of the security of the BB84 protocol was provided by Shor and Preskill [28]. See also: Brassard and Crépeau [11] and Brassard [10].

14

Qiu [23]. See also: Muralidharan [21].

15

Giles [17].

16

Bennett et al. [8].

17

Riebe et al. [24] and Barrett et al. [5].

18

Polzik et al. [22].

19

Bussieres et al. [12].

20

Tsurumoto et al. [30].

21

See, e.g., Alba [1] and Gibney [16].

22

Castelvecchi [13].

23

For an easy to understand reference for eigenvalues and eigenvectors, see Chap. 6, http://​math.​mit.​edu/​~gs/​linearalgebra/​ila0601.​pdf in Strang [29].

© Springer Nature Singapore Pte Ltd. 2020

R. K. BeraThe Amazing World of Quantum ComputingUndergraduate Lecture Notes in Physicshttps://doi.org/10.1007/978-981-15-2471-4_2

2. Distinguishing Features and Axioms of Quantum Mechanics

Rajendra K. Bera1  

(1)

Acadinnet Education Services India, Bangalore, Karnataka, India

Rajendra K. Bera

Email: rajendrabera@yahoo.com

Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language comprehensible to everyone.

—Albert Einstein (Einstein and Infeld [28], p. 17.)

Abstract

This chapter describes certain fundamental differences between classical and quantum mechanics, their different postulates, the role of the observer, what is meant by local and non-local interactions, causality and determinism , and the role of force, energy, and momentum. A short introduction to the purely quantum mechanical aspect of superposition, measurement, and entanglement is provided to mentally prepare the reader for the chapters ahead.

2.1 Introduction

Quantum mechanics is a remarkable creation of the human mind. Quantum means a specific amount of something, and mechanics means the study of motion. In quantum mechanics—the study of motion of quantities—the core idea is that Nature operates on bits and pieces (i.e., quanta) of matter or energy, or their classically measurable physical properties in a mathematically definable way. The language of quantum mechanics is mathematics.

Since its creation in the early 1920s, quantum mechanics has come to represent our deepest understanding of how Nature behaves. It has been verified and applied with enormous success in many areas—the activities inside the Sun, the atom, and the nuclear fusion of stars; superconductors; the structure of DNA (deoxyribonucleic acid); the behavior of elementary particles of Nature—just to name a few. It describes the interactions of electrons, photons, neutrons, etc. at atomic and subatomic scales with accuracies better than any other physical theory in the history of science. Our aim in this book is to discuss the application of quantum mechanics to computing. Even though the modern or new quantum mechanics (as axiomatically described in Sect. 2.7) is now the standard theory for dealing with atomic and subatomic level phenomena, it took computer scientists another half-century to even wonder whether quantum effects might be harnessed for computation. The answer was far from obvious till David Deutsch provided it in 1985.¹

Quantum computing is about computing with quantum systems (computers) using the rules of quantum mechanics rather than the rules of classical mechanics.² This is a crucial difference since the postulates of quantum mechanics are intrinsically different from the postulates of classical mechanics.³ Conventional computers (essentially universal Turing machines), based on classical physics, are in one particular state at any given time. Because quantum computers depend upon such mysterious quantum phenomena as superposition and entanglement of states, each computer can perform exponentially increasing computations in parallel with increasing number of computing qubits (the quantum counterpart of classical bits), a feat impossible on any conventional computer.

Before proceeding further, note that in physics, certain words, such as force, momentum, energy, mass, action, work, etc., have precise meanings, although they may not have such precise meanings in everyday usage of the English language. In fact, some of these words are understood differently even between classical and quantum physics. These differences are crucial and carry deep significance.

2.2 Two-Layer Description of the World

Since the appearance of Maxwell’s equations of electromagnetism⁴ in 1865, theoretical physicists have become accustomed to dealing with abstract fields and consequently with the idea of describing the world in two connected layers.⁵ The first layer consists of fields satisfying simple linear equations. The objects of this layer are abstract and hence not directly accessible to our senses. The second layer deals with things that can be measured directly, such as mechanical stresses, energy, forces, position, and velocity. The quantities in the second layer are quadratic or bilinear combinations of quantities in the first layer. For example, to calculate energies or stresses, you take the square of the electric field strength or multiply one component of the field by another.

This two-layer mathematical description of the laws of Nature adds both mystery (abstraction always makes things look mysterious) and simplicity (linear equations are simpler to deal with than nonlinear equations). The two-layer description was the basic reason why Maxwell’s theory seemed mysterious and difficult when first enunciated. In quantum mechanics, the first layer describes the abstract wave function; the relationship of the second layer to the first is rather bizarre as we shall soon see and involves bilinear combination of the wave function in the first layer. While in most abstract classical models of the world, there is a direct correspondence between elements of the abstraction and the real world, such is not the case in the quantum world. This is because the effect of the observer (or equivalently a measuring apparatus) when probing one or the other world is fundamentally different.

2.2.1 The Observer in Physics

To observe is to make a measurement and quantify it in some units of measurement. Measurement requires an apparatus and/or a human observer to interact with the phenomenon being measured. In Newtonian mechanics, it is implicitly assumed that this interaction, in principle, can be made as non-interfering as we please. This notion underwent a drastic change in Einstein’s theory of relativity, and a dramatic change in quantum mechanics. It turns out that observation in physics is context-observer dependent.

In quantum mechanics, the evolution of a quantum system and the results of measurements on that system are governed by two different postulates, and we are not always sure which postulate to apply when; that is, we do not know the criteria by which Nature decides when a quantum system is being observed. Measurement is a special process governed by a separate law of Nature and a statistical one at that! This is elaborated in Sects. 2.7.3, 2.7.5, and 2.7.6.

2.2.2 Complementarity (Wave-Particle Duality)

In classical physics, an object can either have the properties of a particle or of a wave, but not both. Thus, there are separate equations for wave motion and particle motion. Intriguingly, in the quantum world an object has both properties and is governed by a single equation famously known as the Schrödinger equation . It appears as a fundamental law of Nature and hence serves as an axiom. In fact, Bohr’s complementarity principle (or wave-particle duality; 1928) states that a quantum object can behave either as a particle or as a wave, but never as both at the same time.⁶ A quarter-century later, John Wheeler remarked:

Bohr’s principle of complementarity is the most revolutionary scientific concept of this century and the heart of his fifty-year search for the full significance of the quantum idea.⁷

Jan Faye too makes an observant comment:

In general, Bohr considered the demands of complementarity in quantum mechanics to be logically on a par with the requirements of relativity in the theory of relativity. He believed that both theories were a result of novel aspects of the observation problem, namely the fact that observation in physics is context-dependent. This again is due to the existence of a maximum velocity of propagation of all actions in the domain of relativity and a minimum of any action in the domain of quantum mechanics. And it is because of these universal limits that it is impossible in the theory of relativity to make an unambiguous separation between time and space without reference to the observer (the context) and impossible in quantum mechanics to make a sharp distinction between the behavior of the object and its interaction with the means of observation.⁸

In nineteenth-century classical physics, it was well settled that light had wave properties (Young 1801)⁹ and not of matter, and electrons had particle properties and not of waves (Thomson [65]).¹⁰ In 1900, Max Plank, reluctantly and without conviction, had proposed that if one assumed heat radiation is emitted and absorbed in distinct units, or quanta, then he could provide an excellent curve-fit to the experimentally observed blackbody spectrum.¹¹ He had used the quanta only as an implausible mathematical artifice and nothing more. He was quite unhappy about the assumption he had to make. At the time he did not suspect that his quanta and the constant named after him that was critical to the curve-fit would lay the foundation for quantum mechanics as we know it today.¹²

In 1905, Albert Einstein boldly assumed that Planck’s quantization of energy was true and hence treated light as comprising streaming particles and successfully explained the photoelectric effect.¹³ This implied that light may have both wave and particle properties and that wave-particle duality may be necessary to understand light. In 1924, taking a cue from Einstein, Louis de Broglie boldly suggested his matter-wave theory that not only waves are particles, but particles are waves too.¹⁴ Starting with the known formulas E = hν (Planck’s formula ), and $$ E = mc^{2} $$ (Einstein’s formula ), where E is the energy of the particle under consideration, m its mass, ν a frequency associated with it, h the Planck’s constant, and c the speed of light, he calculated that matter has wavelength λ = h/p (de Broglie’ formula), where p is the momentum of the matter. λ is now called the de Broglie wavelength. The derivation of the de Broglie formula is disarmingly simple. Starting with $$ E = mc^{2} $$, we note that for a photon, mc is the photon’s momentum p, therefore $$ E = \left( {mc} \right)c = pc = p(\lambda \nu ) $$, where c, the speed of light, is simply wavelength λ times frequency ν for waves. On the other hand, Planck had concluded, while explaining the black-body radiation phenomenon, that $$ E = h\nu $$. Hence, $$ E = h\nu = mc^{2} = p(\lambda \nu ) $$ leads to h/p = λ or p = h/λ for photons.

Thus, de Broglie boldly connected the momentum of a particle to its wavelength. It was not clear then or now, if the electron is a wave, what is waving? Not surprisingly, these ideas astounded and confounded his thesis examiners. Einstein was consulted by Paul Langevin, one of the examiners. Einstein wrote to Langevin, “Louis de Broglie’s work has greatly impressed me. He has lifted a corner of the great veil. In my work I obtain results which seem to confirm his. If you see him, please tell him how much esteem and sympathy I have for him.”¹⁵ De Broglie’s thesis was accepted for the Ph.D. In 1927, Davisson and Germer¹⁶ experimentally showed the diffraction of electrons by a crystal of nickel and so confirmed the existence of de Broglie’s matter waves. The same year G. P. Thomson (J. J. Thomson’s son) too carried out experiments on the behavior of electrons going through very thin films of metals, which, once again, showed that electrons behave as waves despite being particles.¹⁷ Nobel Prizes followed.¹⁸

Of course, no one knows what these waves actually are! But de Broglie did have a mental model of the waves. What he did was to assign a frequency, not directly to any internal periodic behavior, if any, of the particle but to a wave which accompanied the particle through space and time, in a manner that it was always in phase with whatever were the “internal” processes going on inside the particle. He called these waves “pilot” waves, which guide the particle in its motion. These pilot waves had two velocities associated with them. One was the phase velocity—the speed at which a wave crest moves—and the second, a group velocity—the speed of the reinforcement regions (wave packets ) formed when many waves are superimposed. He identified the group velocity as the velocity of the corresponding particle and showed that the reinforcement region has all the mechanical properties, e.g., energy and momentum, normally associated with a particle (see also Chap. 5, Sect. 5.​4.​1). Nevertheless, de Broglie did connect two amazing phenomena in the physics of subatomic particles—the quantum nature of energy and wave-particle duality .

Which behavior (wave or particle) one observes depends on the choice of measurement apparatus. This is true whether applied to electrons or photons or any quantum object. Under complementarity , it is completely possible, when experimenting with a given quantum system, to interpret the results of one set of experiments based on wave properties and another set based only on particle properties. The two sets of experiments are not contradictory. Since the experiments were conducted under different experimental conditions, their results cannot be combined in a single picture but must be regarded as complementary. An analogy with the Möbius strip appears helpful in seeing how wave-particle duality may be reconciled. The two properties analogously appear like the “two sides” of a Möbius strip. Locally, on a Möbius strip, a region appears to be a Euclidean surface with two sides, but when one steps back to perceive the whole, its non-Euclidean nature becomes obvious (the Möbius strip has only one surface and one edge and hence it is not orientable). To perceive wave-particle duality , one needs to be able to see a bigger picture.

Wave-particle duality is not the only complementarity in quantum mechanics. It shows up in terms of dynamical variables (e.g., position and momentum) in the form of Heisenberg’s uncertainty principle (see Sect. 2.7.6), in terms of continuity and discontinuity (such as the discontinuous transition of electrons between discrete energy levels in an atom while its corresponding wave function moves continuously from the region of space between the initial orbit (energy level) and the final orbit (energy level). The essence of complementarity is that, while the two elements of a complementary pair stand in apparent opposition to each other, both are needed for a complete description of quantum processes, never mind that the complete precision of definition of either is incompatible with that of the other (see also Sect. 2.9).

2.2.3 Causality and Determinism

To humans, causality and determinism are important in understanding the Universe, our place in it, and the limits of our ability to control our fate. Our primitive notion of causality comes from the mechanical concept of force and work. A precise definition of the causal aspects of matter in motion requires a specification of the energies and momenta of all the relevant constituent parts of a system.

A complete behavioral description of a system requires two distinct but related elements. First is the description of what happens, i.e., the space-time order of events, and second is the description of why it happens, i.e., the causal description of the relationship between events. In classical physics, the causal factors are the forces acting on each particle in the system to cause changes in velocity.¹⁹ These forces may be internal (as between parts of the same system) or external. In any case, Newtonian mechanics implies that if the forces are specified for all time, and if the initial positions and velocities are known of all particles in the universe, then we can calculate their behavior at any other time, in the past or the future. This famous observation of Pierre Simon de Laplace (1749–1827) is known as the principle of determinism.²⁰ Quantum theory emphatically tells us that the very premise of the Laplacian principle of determinism that position and velocity of each particle can be precisely known is itself false. Under Heisenberg’s uncertainty principle (see Sect. 2.7.6), if the position of a (quantum) particle is determined precisely, then we can have no idea of its velocity, and vice versa! This severely limits the applicability of classical physics in the quantum world.

In short, quantum mechanics operates under a very different set of postulates or laws or axioms from that of classical mechanics. Indeed, the core concepts of quantum mechanics involve the assumptions of incomplete continuity, incomplete determinism, indivisible unit of energy, the transfer of a quantum from one system to another as an indivisible process, and the indivisible unity of the entire universe.²¹ Bohr’s principle of complementarity (see Sect. 2.2.2) declares that the properties and motions of matter are expressed in terms of opposing but complementary pairs of potentialities, either of which can be realized in a more definite form in an appropriate experiment or observation, but only at the expense of a corresponding loss of information regarding the other. In quantum mechanics, even commonly used scientific words such as wave, particle, energy, momentum, measurement, etc. as they are generally understood in classical mechanics cannot be applied without reservation and some modification. The reader is advised to be always conscious of this fact when dealing with quantum mechanics.

In classical mechanics, energy and momentum may be viewed as a convenient way of thinking because these quantities are conserved. In quantum mechanics, energy and momentum are expressed in an entirely different way, although both quantities do remain conserved. One may assume that quantities that remain conserved in both would play an important role in eventually illuminating how transitions between the two take place. One such attempt was recently made by Peter Renkel.²²

2.3 Superposition, Measurement, and Entanglement

We briefly encountered the conceptual essence of superposition in Chap. 1—Fig. 1.​1 where one could see either a pretty girl or an old women (the two interpreted eigenstates of the picture) at a given instant but not in some bizarre combination of the two. Quite independent of what one sees, from multiple observations we deduce that the picture contains two concurrent eigenstates or independent interpretations at any time for the same arrangement of pixels in the picture. Further, any editing of the picture will concurrently affect both interpretations. Clearly, not every picture we create or see has such superposition built into it. Further, observing the picture does not change the picture, e.g., if you saw a pretty girl, the old woman does not permanently vanish leaving only the pretty girl to be observed. The picture does not lose information when it is observed.

Unlike such rare pictures which allow multiple interpretations when observed, in the quantum world, superposition of certain attributes coexisting in multiple states is the rule, not the exception. For example, an electron can be simultaneously in two states—spin-up and spin-down. But it cannot be in a superposed state of being positively charged and negatively charged, or existing and not existing. Further, when a quantum system is in a state of superposition and observed, we see only one of its eigenstates randomly chosen by Nature, and the observation process rather abruptly freezes the system into the observed state and erases all other eigenstates present till then in the superposition. That is, till the system is manipulated into a different state in some way after the first observation, all subsequent observations by anyone will reveal only the frozen eigenstate of the system. Observation of a quantum system generally implies loss of information, i.e., the system undergoes decoherence. In a limited sense, Fig. 1.​1 provides an intuitive feel for what we mean by superposition of quantum eigenstates and collapse of the quantum system when observed. A quantum system, when observed or measured, reveals only one of its many observable eigenstates and no more. And the state one observes is chosen randomly by Nature according to a probabilistic rule first enunciated in 1926 by Max Born.²³ It is measurement alone which introduces probability in quantum mechanics.

Entanglement, another baffling aspect of quantum mechanics, is a strange state of being (a form of quantum superposition) where two quantum entities, when paired, function as a unit. Entanglement of two photons can occur when, e.g., a single photon is turned into two complementary photons by an optical beam splitter (such as a β-BaB₂O₄ (BBO) crystal, which absorbs the incoming photon and creates two new photons, each of lesser energy than the incoming photon). If the state of one is changed, the state of the other changes instantly as dictated by quantum mechanical rules, no matter how far apart in the Universe they are at the time (i.e., distance is irrelevant). If one is measured, both collapse instantly in a mathematically defined way.²⁴ Einstein had famously derided this non-local phenomenon as “spooky action at a distance,” (see Sect. 1.​2). Posthumously, he was proven wrong! The phenomenon has been well confirmed in laboratory experiments.²⁵ They confirm that when two or more particles are entangled, a measurement on any one particle or a combined measurement on a subgroup of particles will cause a “collapse” to occur instantly on the remaining particles no matter where they are in the Universe. A group of entangled particles thus have a distributed existence yet function as an integrated unit. Indeed, entanglement is a uniquely quantum mechanical resource which plays a key role in the design of efficient quantum algorithms. In fact, entanglement is a fundamental resource of Nature on par with information, energy, and entropy. So far, there is no complete theory of entanglement; it is perhaps a fundamental characteristic of elementary particles that cannot be further analyzed. Entanglement does not exist in classical physics.

Superposition and entanglement distinguish quantum logic from classical logic in certain ways, e.g., the distributive law of propositional logic fails.²⁶ Quantum computers, instead of acting on bits, operate on qubits . While qubits have some similarity with bits (e.g., qubits can be in one of two alternate states, say, 0 and 1, like a classical bit), they can also exist in a fuzzy superposition of 0 and 1. Qubits can be entangled with other qubits too, but classical bits cannot. Superposition and entanglement enrich quantum logic (and quantum physics in general). While a classical computer will sequentially explore potential solutions, say, to a mathematical optimization problem, a quantum computer can look at every potential solution simultaneously, i.e., in parallel.

2.4 Classical Mechanics Powers Our Intuition

Newton’s laws imply that all motion has a cause; that is, if a body moved one could always determine the cause of its motion. This is what we know as the “commonsensical” cause and effect (causality) doctrine. In quantum physics, the situation is vastly different because quantum states have a different meaning than that in classical physics, and the logic is non-Boolean. Knowing a quantum state means “knowing as much as can be known about how the system was prepared.”²⁷ At the moment, we do not know if anything more than this can be known. Much worse, “quantum evolution of states only allows us to compute the probabilities of the outcomes of later experiments.”²⁸ It is here that we find a fundamental difference between classical mechanics and quantum mechanics. In classical mechanics, there is no material difference between states and measurements, but in quantum mechanics the difference is profound and fundamental; a quantum measurement, no matter how delicately done, invariably changes the state of the system, if the possibility exists, via a “collapse” mechanism. If the state vector was measurable, then quantum mechanics would have been deterministic, but that is not the case. We can measure only observables (see Sect. 2.8), and here, even if we knew the state vector exactly, in general, we still would not know the result of any measurement till the measurement is made. However, between measurements, a quantum system evolves in a perfectly deterministic way according to the time-dependent Schrödinger equation (see Sects. 2.5.1 and 2.7.2). After a measurement is made, the system is left in an eigenstate of the observable-operator.

Quantum mechanics is non-intuitive not because of the mathematics involved (it is mechanizable) but in the way the mathematics must be interpreted. The study of the simplest classical system, the basic logical unit for computer science, the two-state bit, provides no difficulty to human intellect. The same basic logical unit in a quantum system, now called a qubit , poses enormous intellectual challenges to the same human intellect. Chapter 1 gave you a preview of this.

2.5 The Birth of Modern Quantum Mechanics

Heisenberg’s formulation of quantum theory was the first to appear in 1925.²⁹ It was called matrix mechanics. It is this version of quantum mechanics that is used in quantum computing. His other contribution to quantum mechanics is the iconic uncertainty principle (see Sect. 2.7.6), which holds that certain pairs of non-commuting variables, e.g., (position, momentum) and (energy, time), suffer from an unusual restriction—the more accurately one in a pair is measured, the less accurately the other can be measured. This effect is very visible at quantum mechanical levels, but not even measurable at classical mechanical level. For example, if an object with a mass of 1 g has its position measured to an accuracy of 1 μm, then the uncertainty in measuring its velocity is 10⁻²⁵ m/s. But if we localize an electron to within an atom of diameter 10⁻¹⁰ m, then the uncertainty in velocity would be 10⁶ m/s!³⁰ His views on science are as follows: (1) “[W]e have to remember that what we observe is not nature in itself but nature exposed to our method of questioning.”³¹ (2) “Natural science, does not simply describe and explain nature; it is part of the interplay between nature and ourselves.”³²

In 1926, Erwin Schrödinger provided the central equation in quantum mechanics that is known by his name.³³ Since this equation exhibits wave-like properties, it is also known as wave mechanics. In one-dimension, the equation describes the wave function’s evolution in time by:

Here, ψ is the iconic wave function of quantum mechanics. Even though this equation does not look like the familiar wave equation (e.g., that of describing a vibrating string), it does have solutions that represent waves propagating through space. For example, for a free particle for which V(x) = 0, and of momentum p and energy E,

is a solution. Of course, there are other situations where the solution is distinctly non-wavelike and the form, inter alia, depends on the nature of the potential V(x). The time-independent version is used, e.g., to calculate the allowed energies of a particle.³⁴

The wave equation has the advantage that the “laws of motion and the quantum conditions are deduced simultaneously from one simple Hamiltonian principle.”³⁵ While the Schrödinger equation serves as the counterpart of Newton’s dynamical equation (momentum conservation) in classical physics, where we can determine the position of a particle as a function of time, the former produces a wave function which tells us (after we square the wave function) how the probability of finding a particle in some region of space varies as a function of time. The probability does not come from the mathematics, it comes from Max Born³⁶ in the way he gives the wave function a meaning in physics.

No physical interpretation of $$ \psi $$, the wave function, is known. It contains all the information about the quantum system it represents; it is expressible as the sum of an infinite series of periodic functions, namely the Fourier series. Even without understanding what $$ \psi $$ is, it was found that the eigenvalues of the equation were equal to the energy of the quantum mechanical system, e.g., the energy levels of the hydrogen atom agreed with those provided by Rydberg. The energy levels turn out to be the eigenvalues of the quantum system. Thus, when we measure a quantum system, we measure (i.e., collapse the wave function to) one of its eigenvalue according to Born’s probabilistic rule.

Dirac’s seminal paper on quantum mechanics came out in 1925³⁷ soon after Heisenberg’s matrix formulation of quantum mechanics was brought to his notice the same year; he was a doctoral student at the time. His independent version comprised a non-commutative algebra for calculating atomic properties. He also invented the “bra-ket” notation, now the standard notation for describing quantum states. His 1930 book Principles of Quantum Mechanics is a classic in physics. In this book, he incorporates Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics into a single formalism. In 1926, he got his Ph.D. from Cambridge University; in 1932, he became the Lucasian Professor of mathematics at Cambridge.

The three pioneers were awarded the Nobel Prize in Physics: Werner Heisenberg in 1932, “for the creation of quantum mechanics, the application of which has, inter alia, led to the discovery of the allotropic forms of hydrogen”; Erwin Schrödinger and Paul Adrien Maurice Dirac in 1933, “for the discovery of new productive forms of atomic theory.”

In our new understanding of the Universe, the wave function carries all the information one can know of a particle, both its position, and its speed. If you know the wave function at one time, then its values at other times are determined by the Schrödinger equation . The evolution of the wave function is deterministic, but not in the way Laplace envisaged. As we have already noted, even though the wave function is deterministic, in quantum mechanics, particles cannot be ascribed well defined positions and speeds. What we know is the abstract wave function; its value at each point in space is a complex number. The magnitude of the wave function gives the probability that the particle will be found in that position. If the wave function is strongly peaked in a small region, it means that the uncertainty in the position of the particle is small but then the wave function will necessarily vary very rapidly near the peak, up on one side and down on the other. This means that the uncertainty in the speed will be very large. The converse will be true if the wave function is spread out in a wide region.³⁸ This is the substance of Heisenberg’s inviolable uncertainty principle.

Clearly, the relations between $$ \left| \psi \right\rangle $$ and physical properties are much less direct than in classical mechanics. This has led to several interpretations of quantum theory. Although the exact status of $$ \left| \psi \right\rangle $$ in the theory remains unclear, in mathematical terms, $$ \left| \psi \right\rangle $$ expresses the entire weighted sum, with complex number weighting factors $$ w_{i} $$ of all the possible alternative states $$ \left| {s_{i} } \right\rangle $$ that are open to the system, expressed as (you had a glimpse of it in Chap. 1, Sect. 1.​3)

It is required that the states themselves always be normalized, so that we can distinguish the coefficients $$ w_{i} $$ from the states. Further, $$ \left| {s_{i} } \right\rangle $$ and $$ w_{i} \left| {s_{i} } \right\rangle $$ both represent the same physical state. A quantum entity is completely described by its wave function. The physical properties of the entity can be extracted from the wave function using quantum operators.

2.5.1 Serendipity at Work

Schrödinger’s equation was conjured, not derived. Schrödinger pushed de Broglie’s idea further even though he too was not sure what it was that was waving!³⁹ He chose to represent the wave function by the Greek letter $$ \psi $$ and it became an iconic symbol of quantum mechanics. He was convinced his equation described real things not mathematical abstractions.⁴⁰ In quantum mechanics, operators serve as surrogates for certain classical operands; hence, commutability of operators plays such an important role in measurement. Here are a few important operators:

where $$ \hbar = h/2\pi $$ is called the reduced Planck constant. A quick look at how Schrödinger intuitively discovered his wave equation is instructive. In classical physics, at any time t, the total energy E of a particle is the sum of its kinetic energy and its potential energy. The kinetic energy per unit mass is given by $$ p^{2} /2m $$ and the potential energy per unit mass by V(r, t) as some function of position r and time t. The total energy E is therefore

When this is written in operator form, we get the Schrödinger equation

or

This was amazing intuition at work. The link between the Schrödinger equation and the classical wave equation is the de Broglie relation (see Sect. 2.2.2).

2.6 Cautionary Note on Notations in Quantum Mechanics

Quantum mechanics has its own unusual notations. The reader should spend some time getting familiar with them. This is absolutely crucial. The unusual Dirac notation ⁴¹ adopted by physicists may well be playing a role in terrifying students who are new to quantum mechanics. The subject is littered with the ubiquitous appearances of symbols like $$ \left\langle \psi \right| $$, $$ \left| \psi \right\rangle $$, $$ \left\langle {\phi |\psi } \right\rangle $$, $$ \left| \psi \right\rangle \left\langle \psi \right| $$, $$ \left| \phi \right\rangle \otimes \left| \psi \right\rangle $$, and so on. The basic problem is unfamiliarity with such symbols. Once you get used to them, life becomes easy and an admiration for Dirac develops. The use of Hilbert space in quantum mechanics creates additional aura around the subject. Since quantum computing deals only with finite systems, the Hilbert space is quite easy to construct (see Chap. 3). Instead of the Dirac notation , one may use an alternative matrix notation (see Chap. 3). Therefore, not surprisingly, quantum mechanics requires a thorough understanding of linear algebra, which is the study of vector spaces and of linear operations on those vector spaces. Researchers use both Dirac and matrix notations interchangeably, as convenience dictates. The mathematical tools needed to deal with quantum mechanics are provided in Chap. 3 for linear algebra and in Chap. 5 for Fourier analysis . In the meantime, the reader should pay careful attention to the meanings attached to certain terms, e.g., Hilbert space, physical system, state, wave function (eigenfunction, eigenstate), eigenvalue, physical quantities, and observables (something measurable by a classical measuring system).

2.7 Postulates of Quantum Mechanics Formally Stated

A postulate is a statement of a belief for which we do not have an explanation as to why it is true. We accept it in good faith. Before we state the postulates of quantum mechanics , a word of caution. The mathematics is not particularly difficult but mentally adjusting to the weirdness of quantum mechanics is. In the early days of scientific research, it was experience-based intuition that suggested reasonable explanations. With curiosity-driven success, the scope of scientific research progressively expanded as did the desire to seek consistent and ever more fundamental explanations. As the explanations increased in sophistication—they became laws of Nature. Their sophistication now defies intuition and common sense. We have travelled far from a disjointed set of observations to mathematically stated Newton’s laws of motion to Maxwell’s equations of electromagnetism to general theory of relativity to quantum mechanics. It now takes great imagination to comprehend the Universe at the quantum level. So, here are the four formal postulates of quantum mechanics as stated in Nielsen and Chuang.⁴²

2.7.1 A Quantum System’s State Space Is a Hilbert Space

Postulate 1: Associated with any isolated physical system is a complex vector space with inner product (that is, a Hilbert space in finite space) known as the state space of the system. The system is completely described by its state vector, which is a unit vector in the system’s state space.

Quantum mechanics, however, does not tell us what the state space of a given physical system is nor does it tell us how to find the state vector of a system. Figuring that out for a given system remains a difficult problem. Fortunately, in quantum computing, the state vector for the qubit is well known. Experiments show that there is no need for other descriptions than the Hilbert space since all interactions, such as momentum transfer, electric fields, and spin conservation, can be included within this framework.

2.7.2 A Quantum System Evolves via Unitary Transformations

Postulate 2: The evolution of a closed quantum mechanical system is described by a unitary transformation. That is, the state $$ \left| {\psi \left( {t_{1} } \right)} \right\rangle $$ of the system at time $$ t_{1} $$ is related to the state $$ \left| {\psi \left( {t_{2} } \right) } \right\rangle $$ of the system at time $$ t_{2} $$ by a unitary operator U which depends only on the times $$ t_{1} $$ and $$ t_{2} $$,

It turns out that just as quantum mechanics does not tell us the state space of a particular quantum system, it also does not tell us which unitary operators ⁴³ U to use in real-world quantum dynamics. The postulate merely assures us that the evolution of any closed quantum system may be described in such a way. A corollary is that closed quantum systems are reversible. Once again, for single qubits , it conveniently turns out that any unitary operator at all can be realized in realistic systems.

In passing we note that an alternative version of Postulate 2 is the Schrödinger equation , which describes the evolution of a closed quantum system in continuous time,

Here, h is the Planck’s constant, and $$ \widehat{H} $$ is a fixed Hermitian operator (and hence self-adjoint) known as the Hamiltonian operator of the closed system. In fact, $$ \widehat{H} $$ is the operator that represents the system’s total energy. This is a linear equation in which the real challenge is discovering the Hamiltonian needed to describe a given physical system. We will not use the Schrödinger equation . Instead, we will use the equivalent matrix formulation, $$ \left| {\psi \left( {t_{2} } \right)} \right\rangle = U\left| {\psi \left( {t_{1} } \right)} \right\rangle $$ due to Werner Heisenberg. It presents a discrete-time description of quantum dynamics using unitary operators . However, for the reader’s benefit we show that U and $$ \widehat{H} $$ are related.

The solution to the Schrödinger’s wave equation is:

From which we define

It can be shown that this operator is unitary, and that any unitary operator U can be realized in the form U = exp(iK) for some Hermitian operator K. Thus, there exists a one-to-one correspondence between the discrete-time description of quantum systems using unitary operators (matrix mechanics formulation of Heisenberg), and the continuous time description using Hamiltonians (wave mechanics formulation of Schrödinger).

2.7.3 A Quantum System Collapses When Measured

Postulate 3: Quantum measurements are described by a collection $$ \left\{ {M_{m} } \right\} $$ of measurement operators. These are operators, which act on the state space of the system being measured. The index m refers to the measurement outcomes that may occur in the experiment. If the state of the quantum system is $$ \left| \psi \right\rangle $$ just before the measurement, then the probability that result m occurs is given by

and the state of the system after the measurement is

where the measurement operators satisfy the completeness condition

This condition fulfills the essential requirement that the respective probabilities associated with each state of $$ \left| \psi \right\rangle $$ must sum to one:

Our objects of interest are the qubits , which will usually be measured in the computational basis $$ \left\{ {\left| 0 \right\rangle ,\left| 1 \right\rangle } \right\} $$ whose measurement operators are $$ M_{0} = \left| 0 \right\rangle \left\langle 0 \right| $$ and $$ M_{1} = \left| 1 \right\rangle \left\langle 1 \right|. $$ It is easily verified that they satisfy the completeness equation. Let $$ \left| \psi \right\rangle = a\left| 0 \right\rangle + b\left| 1 \right\rangle $$. Then the probability that the measurement will yield 0 is

and that it will yield 1 is

Notice that we can only determine the complex coefficients, such as a and b describing the wave function $$ \left| \psi \right\rangle $$, in distinction to the states, up to an arbitrary phase factor $$ e^{i\theta } $$ $$ \left( {{\text{i}} . {\text{e}} . ,\;\left\langle \psi \right|M\left| \psi \right\rangle = \left\langle \psi \right|e^{ - i\theta } Me^{i\theta } \left| \psi \right\rangle } \right) $$ only. The post-measurement state of the system, respectively, will be fixed at

Essentially, only certain sets of measurements can be made at any one time, and arbitrary quantum states cannot be measured with arbitrary accuracy. Measurement is a non-unitary and irreversible operation in general, unless the system is already in the observable eigenstate. No matter how delicately done, the very first measurement forever alters the state of the system.

2.7.4 Hilbert Space Grows Rapidly with the Size of a Quantum System

Postulate 4: The state space of a composite physical system is the tensor product of the state spaces of the component physical systems. Moreover, if we have systems numbered 1 through n, and system number i is prepared in the state $$ \left| {\psi_{i} } \right\rangle $$, then the joint state of the total system is

Why is the tensor product the mathematical structure used to describe the state space of a composite physical system? We do not know. Remember it is a postulate. No wonder quantum mechanics is so mysterious!

Comments on the postulates

The four postulates are independent of one another. The state vector $$ \left| \psi \right\rangle $$ is an abstract mathematical entity; its origin and any underlying sub-quantum structure it might have is unknown. The processes that comprise measurement too are unknown. In our experience, certain physical interactions are recognizably “measurements,” but there are unknown others which are not. Their incognito presence makes the building of robust quantum computers difficult because they can surreptitiously destroy the quantum superposition of the state vector $$ \left| \psi \right\rangle $$. There is deeply mysterious, unexplained physics.

Postulates 1 and 4 describe the mathematical state space in which quantum mechanical systems live and act. It asserts that a quantum system is completely described by its state vector, $$ \left| \psi \right\rangle $$, which is a unit vector in the system’s state space. Postulate 2 describes the evolution of a quantum system as described by its state vector $$ \left| \psi \right\rangle $$. Postulate 3 describes the post-measurement, collapsed state of $$ \left| \psi \right\rangle $$.

The evolution of $$ \left| \psi \right\rangle $$ under Postulate 2 is governed by the Schrödinger equation and it is completely deterministic, continuous, and smooth. Its value, even using proxies, cannot be determined by any classical measurement system. In fact, the heart of quantum theory is Postulate 2 once we know how to interpret $$ \left| \psi \right\rangle $$ according to Postulate 3. There are no probabilities involved in Postulate 2.

Postulate 3 asserts that any legitimate classical measurement on $$ \left| \psi \right\rangle $$, in general, will yield only partial, non-deterministic information about the system’s state immediately prior to the measurement. The very first measurement on a system will irreversibly “collapse” $$ \left| \psi \right\rangle $$ and do so in a probabilistic manner. Measurement overrides Postulate 2. It is not possible to know $$ \left| \psi \right\rangle $$ of an unknown quantum system completely by making multiple measurements on the same system or even on multiple copies of the system. Since all the probabilities associated with quantum mechanics are contained in Postulate 3, a knowledge of probability is essential to analyze measured data. Measurement is an irreversible process; once made, the previous history of $$ \left| \psi \right\rangle $$ is lost. Postulates 2 and 3 appear conflicting.

Why Nature conforms to the above four postulates at the quantum level, and how they may link to classical physics (Newton’s laws of motion, Maxwell’s equations of electromagnetism, and Einstein’s theory of general relativity) is a huge mystery. Nevertheless, a strong belief exists in the form of the “correspondence principle” (see Sect. 2.9) due to Niels Bohr that such a link may one day be found.

Finally, it is intriguing that the state vector $$ \left| \psi \right\rangle $$ is postulated (Postulate 1) to be a complex entity with real and imaginary parts. While complex functions do appear in classical physics, they are invariably interpreted either as (1) an indicator that the solution is unphysical, e.g., in the Lorentz transformation when the speed of a physical object is greater than the speed of light, or (2) as a shortcut to dealing with two independent and equally valid solutions of the equations, one real and one imaginary, as in the case of two dimensional inviscid flow governed by the Laplace equation where the stream function and the velocity potential appear jointly as a complex variable. In quantum mechanics, $$ \left| \psi \right\rangle $$ as a whole is of interest. Yet, since $$ \left| \psi \right\rangle $$ is not directly observable and is not a real physical entity, its complex character can be ignored, especially in view of Postulate 3, which in any case extracts a real value in a measurement. All measurable quantities depend on the absolute squares of the components of the state vector and such absolute squares are always real.

2.7.5 Born’s Probabilistic Interpretation

Max Born concluded that it was neither necessary nor feasible to visualize the waves because they are not real things; in fact, they are probability waves! He said:

… the whole course of events is determined by the laws of probability; to a state in space there corresponds a definite probability, which is given by the de Broglie wave associated with the state.⁴⁴

This probability is simply the square of the amplitude of the matter-wave associated with the state, $$ \left| {\left| \psi \right\rangle } \right|^{2} $$. To Born, $$ \left| \psi \right\rangle $$ could not be a real thing if it existed in more than three dimensions. He said:

We have two possibilities. Either we use waves in spaces of more than three dimensions … or we remain in three-dimensional space, but give up the simple picture of the wave amplitude as an ordinary physical magnitude, and replace it by a purely abstract mathematical concept … into which we cannot enter.⁴⁵ … Physics is in the nature of the case indeterminate, and therefore the affair of statistics.⁴⁶

Born’s bold interpretation of Schrödinger’s wave equation finally enabled quantum mechanics to predict probabilities in a predictable deterministic manner!

The appearance of “exact” causal laws in the macroscopic world can be explained by noting that when only the probability of each elementary quantum transfer is determined where many quanta are involved, the probability becomes almost a certainty. An analogy is helpful. Insurance companies operate successfully under uncertainty because they have been successful in being able to predict the mean lifetime of a person within a large group, even though an exact prediction of the lifetime of a single individual in the group is not possible. It is the mean lifetime that assumes importance. Similarly, mean values play an important role in quantum mechanical predictions. It is in the sense of the “mean” that Bohr’s correspondence principle has any meaning. At play is the law of large numbers, i.e., the average of the results from a large number of trials of the same experiment should be close to the expected value. The expected value of a random variable is the weighted average of all possible values that the random variable can take. The weights used are the probabilities in the case of a discrete random variable, or the probability density function in the case of a continuous random variable.

2.7.6 Heisenberg’s Uncertainty Principle

In 1927, Heisenberg showed that both the position and the velocity of a quantum object cannot be simultaneously measured exactly, even in theory.⁴⁷ Further, this restriction cannot be overcome by refining the measuring apparatus or techniques. The classical concept of exact position and exact velocity being measured simultaneously is barred in quantum mechanics. It is a limit imposed by Nature as we penetrate the subatomic world. Heisenberg originally stated his uncertainty principle, based on how waves behave in classical physics, as follows: If ∆q is the error in the measurement of any coordinate and ∆p is the error in its canonically conjugate momentum, then

He said, “The more precise the measurement of position, the more imprecise the measurement of momentum, and vice versa.”⁴⁸ We can never expect to measure both precisely. This means that a statement such as “The particle has position x and the particle has momentum p” is completely meaningless (not even wrong) in quantum mechanics but not in classical mechanics. The classical and quantum concepts of the state of a system are entirely different.

The modern and correct interpretation of Heisenberg’s uncertainty principle is in terms of the standard deviation and the commutator operator rather than the earlier misconceived notion that the uncertainty principle is about disturbances created by the measurement process. Mathematically, given a multitude of identical quantum systems each in state $$ \left| \psi \right\rangle $$, suppose we measure C on some of those systems, and D in others, where C and D are represented by two observables (represented by operators), then the standard deviation ∆(C) of the C results times the standard deviation ∆(D) of the D results will satisfy the inequality.⁴⁹

Here, [C, D] is the commutator operation. This is the content of Heisenberg’s Uncertainty Principle .⁵⁰ It imposes a fundamental limit on the sharpness of the dynamics (courtesy the Planck’s constant) by preventing us from even talking about initial conditions since we cannot know the exact position and velocity of a particle simultaneously. Hence, it prevents us from even talking about trajectories. Thus, indeterminism is built into the very structure of matter where certain pairs of variables (such as position and momentum) cannot even exist simultaneously with perfectly defined values. In the most extreme case, absolute precision of one variable in a pair would entail absolute imprecision regarding the other. This is in sharp contrast to classical physics where “if we know the present exactly, we can calculate the future.” The Uncertainty Principle says the opposite: It is not the conclusion that is wrong but the premise. One cannot calculate the precise future motion of a particle, but only a range of possibilities for its future motion (However, the probabilities of each motion, and the distribution of particles following these motions in the many-particle case, can be calculated exactly from Schrödinger’s wave equation .).

The Uncertainty Principle implies that no matter how cleverly you design a thought experiment to circumvent it, you will not be able to execute it. You will not be able to simultaneously measure non-commuting observables or simultaneous measurements on entangled particles! In the entangled case, you will not know exactly the position of two entangled particles at the same instant (if you cannot find them with precision, you cannot position your measuring apparatus) or know that you are measuring at the same instant.

2.8 Observables and Operators

Scientific theories are only concerned with observable (i.e., measurable) things. Examples are a particle’s coordinates, energy, momentum, or angular momentum; the electric field at a point in space; etc. In classical mechanics, an observable always has a value for any state of the system. We can observe (or make a measurement on) an object only by letting it interact with some external influence. Only carefully measured results have real significance in physics. Quantum mechanics places physical limits on measurements that cannot be exceeded by improved techniques or skills and there is a minimum unavoidable disturbance that will be imparted to the observed object.

Measurements return real numbers, so the focus is on real dynamical variables. A complex dynamical variable would require two measurements—one for the real part and the other for the imaginary part. This would not matter if measurements imparted only negligible disturbances to the measured system (e.g., as in classical two-dimensional, incompressible, inviscid fluid flow experiments, which can be theoretically modeled using complex variables).⁵¹ But in quantum mechanics, measurements, in general, interfere with one another. It is generally impermissible to assume that two measurements can be made exactly and simultaneously. Even if measurements are made in rapid succession, the first measurement will inevitably collapse the quantum state of the system. Hence, measurable dynamical variables must be real. In addition, there are other conditions attached to measurable quantum variables that are intimately related to eigenvalues and eigenvectors of measurement operators that represent the variable because they have a physical significance other than just being an orthonormal (orthogonal and normalized) basis for measurement. See Sect. 2.8.1.

2.8.1 Observables in Quantum Mechanics Are Operators

States in quantum mechanics are vectors in a vector space. Observables are described by linear, Hermitian operators . They are also associated with a vector space, but they are not state vectors.⁵² That in quantum mechanics every measurable quantity has an associated observable, represented by a Hermitian operator and not a variable is indeed very subtle! Thus, Hermitian (or self-adjoint) operators play a crucial role in measurement as observable-operators. Such operators are postulated to represent the observable quantities of a quantum system. The set of the operator’s eigenvalues represent the set of possible measurement outcomes the operator can produce. For each eigenvalue, there is a corresponding eigenstate (or eigenvector), which will be the state of the system after the measurement. Note that

1.

A Hermitian matrix can be unitarily diagonalized, generating an orthonormal basis of eigenvectors which spans the state space of the quantum system.

2.

The eigenvalues of a Hermitian matrix are real. The possible outcomes of a measurement are precisely the eigenvalues of the given observable-operator.

3.

The matrix representing the product of two linear operators is the product of the matrices representing the two factors.

The above implies that any state of a quantum system can be expressed as a linear combination of the eigenvectors of any Hermitian operator . Thus, any state of a quantum system is describable by a linear superposition of the eigenstates of an observable-operator. Classical mechanics deals with real variables, and quantum mechanics deals with operators whose eigenvalues are real. When the focus is on operators whose fundamental property is to bring about change, it is not surprising that quantum mechanics needs a separate postulate for measurement, while classical mechanics, which ideally views measurement as a passive activity, does not. In quantum mechanics, an observable-operator, inter alia, does two things: (1) it randomly collapses the wave function on which it operates to one of its (operator’s) eigenstate according to Born’s probabilistic rule; and (2) on collapse, it produces a real-valued eigenvalue corresponding to the eigenstate to which the wave function has collapsed.

An exception arises when two or more eigenstates of an observable belonging to the same eigenvalue arise. Then any linear superposition of such eigenstates will also be an eigenstate of the observable for the same eigenvalue. This means that if two or more states for which a measurement of the observable is certain to belong to the same eigenvalue, then any state formed by their superposition will also belong to the same eigenvalue. Since two eigenstates of an observable belonging to different eigenvalues are orthogonal, it follows that two states for which a measurement of the observable is certain to give two different results are orthogonal.

Note that while we cannot speak of an observable having a value for a particular state, we can speak of the probability of its having any specified value when one makes a measurement, and we can define an average value for the state. It is permissible for a quantum state to be simultaneously an eigenstate of two observables; the chances for such an existence are most favorable if the two observables commute and rather exceptional when they do not. When the observables do commute, there exist so many simultaneous eigenstates that they form a complete set.⁵³ The case when two observables commute is special in the sense that the observations are non-interfering or compatible in such a way that one can give a meaning to the two observations being made simultaneously and discuss the probability of any particular results being obtained. Indeed, one may view the two observations as a single observation of a more complicated type, the result of which is two real numbers rather than one. A natural generalization of this is that any two or more commuting observables may be counted as a single observable, a measurement of which produces two or more numbers. The states for which this measurement is certain to lead to one particular result are the simultaneous eigenstates.

After any measurement, a quantum system remains in its collapsed state unless altered. Till altered, any repetition of the measurement will only reproduce the first measurement. One may, of course, alter the post-measurement state of the system by measuring a different observable or measuring in a different basis or by applying a quantum (unitary) operation.

2.8.2 The Need for Observable-Operators

When coaxing information about a real measurable variable out of an abstract entity such as the wave function, some conversion operation must be done on that abstract entity to physically extract the information it stores. Such an operation on a wave function is what we have been referring to as an observable. In quantum mechanics, the term observable, by convention, also refers to mathematical operators. The task of an “observable-operator” is to extract from the wave function information that is physically measurable. In classical physics, one directly deals with observables which are themselves physically measurable variables. In quantum mechanics one needs to get used to the convention that observables are operators which act on the abstract wave function, which, while containing all the information about the system it represents, is not directly measurable. A quantum observable, as an operator, is the intermediary that produces a value for a related classical observable when it acts on a wave function.⁵⁴ The classical value the quantum observable-operator produces is one of its (operator’s) eigenvalues chosen randomly according to Born’s probabilistic rule.

In physics, the notion of space is intimately related to distribution of matter; two material objects cannot occupy the same space at the same time. The notion of time comes from change. Without change of some kind, there is no need for the notion of time. In mathematics, we have two basic entities: operands (nouns) and operators (verbs). For an operand (noun) to change its value, it needs to be operated on (verbed!). Operands and operators act together to bring about change. Operands have space-like property in that they can attain a value from a set of values depending upon the operator acting on it. Operators have time-like property. Operators may also show space-like properties as when describing the gradient of a distribution of operands. In natural languages, we have, over time, become used to certain commonly used nouns also being used as verbs and vice versa. Remember that mathematics is also a language for which quantum mechanics has provided the insight that what people have done with natural languages can also be done in mathematics—use an operator as an alias for an operand.

2.8.3 Remarks on Vector Spaces

In quantum mechanics, the state vector carries both classical and quantum information. The classical information is accessible to classical measurement, the quantum information is not. The state space of a composite quantum system is the tensor product of individual Hilbert spaces, e.g., given two spaces H₁ and H₂, the composite space H is given by $$ H = H_{1} \otimes H_{2} $$. If a state $$ \left| \psi \right\rangle $$ in a composite system in H cannot be written in the form $$ \left| \psi \right\rangle = \left| {\psi_{1} } \right\rangle \otimes \left| {\psi_{2} } \right\rangle $$ of its components, it is an entangled state. Note further that the wave function is essentially complex because its real and imaginary parts are coupled, i.e., not independent of each other.⁵⁵

Finally, when working with vectors in an n-dimensional vector space, it is extremely useful to have a set of n mutually orthogonal unit vectors and use them as the basis (orthonormal basis) to construct any vector in that space. Such a vector will necessarily be a linear combination of the basis vectors. In fact, the dimension of the vector space can be defined as the maximum number n of mutually orthogonal vectors that can be found in that space. An orthonormal basis is not unique, but n is. Any two bases can be related to each other for conversion purposes. For example, one may choose one basis for preparing a quantum system but measure it in another basis. Specifying an orthonormal basis is conceptually similar to choosing a Cartesian coordinate system in physical three-dimensional space.

2.9 Weirdness of Quantum Mechanics (In Summary)

In dealing with quantum mechanics, it is inadvisable to give free rein to intuition and common sense. View it as a mathematical framework for constructing physical theories and focus on the rules of the game as if trying to understand an alien world. Patience and diligence will be amply rewarded because what will be revealed is the finest set of scientific conjectures created by the human mind to date. In physics, as a deeper understanding of Nature evolved so did the mathematical complexity of the Laws of Nature that describe it. In quantum mechanics, mental visualization has gone out of reach. We understand it only in the abstract language of mathematics (it is both a language and a tool for reasoning); we see the beauty in it only if we have a deep understanding of how rational reasoning can be communicated in mathematics, precisely and without ambiguity. Quantum mechanics therefore requires extreme mental discipline and meticulous training. A good understanding of linear algebra and Fourier analysis is necessary. Common sense derived from classical mechanics should be shunned. Therefore, let us now look at those aspects of quantum mechanics that appear weird because of our initial training in classical mechanics where all our intuitions were built. Had we been initially trained in quantum mechanics, classical mechanics would have looked similarly weird. Just remember Feynman’s advice:

Do not keep saying to yourself, if you can possibly avoid it, ‘But how can it be like that?’ because you will get … into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that.⁵⁶

We do not have a bridging theory that firmly connects quantum mechanics with classical mechanics. The correspondence principle speculates how the continuous deterministic laws of classical physics may emerge, while at the quantum level the basic processes are discontinuous and have probabilities associated with them. The essential ideas are that (1) discontinuities are too small to be seen on a classical level, and (2) so many quantum processes occur simultaneously in any classical process, that the deviation of the actual result from the statistical average is negligible. However, to ensure a “smooth” transition from the quantum level to the classical level, it is necessary to limit severely both the possible spacings of the quantum states and the probabilities of quantum processes.

Niels Bohr in 1923 suggested that the results of classical mechanics may be obtained as a limiting case of quantum mechanics when the quantum numbers describing the system are large, meaning either some quantum numbers of the system are excited to a very large value, or the system is described by a large set of quantum numbers, or both. This heuristic view is Bohr’s correspondence principle, the essence of which, as Max Born notes, is the “obvious requirement that ordinary classical mechanics must hold to a high degree of approximation in the limiting case where the numbers of the stationary states, the so-called quantum numbers, are very large [] and the energy changes relatively little from place to place, in fact practically continuously.”⁵⁷ Finally, it was left to Born to give Schrödinger’s wave equation a probabilistic interpretation. The journey from classical mechanics to quantum mechanics—from Galileo–Newton to Schrödinger–Born —has been a long and a remarkably unexpected one.

Wave-particle duality came as a totally unexpected discovery. It is now experimentally verified that subatomic particles (matter/energy) behave as waves when they are moving (refraction) and as particles when they are created or when they hit something (photoelectric effect). Individual events are always particle-like while wave behavior manifests itself as a statistical pattern, i.e., interference. Thus, a quantum object can interfere with itself. One may reconcile wave-particle duality if we posit that it is not a property of any quantum object but a property of its interactions, say, with a measuring device. Because energy is quantized, and indivisible when energy transfer occurs, electromagnetic waves can take on particle-like properties. In particular, the sudden appearance of all energy at one point suggests that light is made up of particles. Yet, even when a single photon is present, it demonstrates interference properties, which suggests that light is a wave (see Chap. 8, Sect. 8.​1.​1). The two aspects are related by the fact that wave intensity determines the probability that all the energy appears at one point. Probability, wave-particle duality , and the indivisibility of quantum transfers (a quantum cannot be transferred in fractions) are intimately related in quantum mechanics.

As Henry Stapp notes, the physical world, according to quantum mechanics, is

not a structure built out of independently existing unanalyzable entities, but rather a web of relationships between elements whose meanings arise wholly from their relationships to the whole.⁵⁸

Thus, the Universe presents itself to human intelligence as webs (patterns) of relations or correlations. Individual entities are only idealizations, the result of correlations based on “inspired” hypotheses or conjectures which have survived intense rational refutations and “experimental verification.” Newtonian physics depicts and predicts events based upon observations and measurements we can make in the classical world; quantum physics depicts the world in conceptual terms that are impossible to visualize, and it does not predict events but predicts only the probabilities with which potentially possible events can occur. It appears impossible to make a complete mental picture of events in the quantum world. Werner Heisenberg wrote:

The mathematically formulated laws of quantum theory show clearly that our ordinary intuitive concepts cannot be unambiguously applied to the smallest particles. All the words or concepts we use to describe ordinary physical objects, such as position, velocity, color, size, and so on, become indefinite and problematic if we try to use them of elementary particles.⁵⁹

At the mental level, we feel we understand the large-scale world only when we have developed a mental picture of it. Newtonian mechanics pandered to that mental level by dealing with measurable and observable quantities such as position, velocity, etc. Quantum events cannot be pictured in our mind, and therefore it leaves us with the queasy feeling that it is unreal, further strengthened by the fact that quantum mechanics deals with not real numbers but complex numbers.

Newtonian physics tells us that it is possible, in principle, to predict exactly the future from the present (and likewise from the past) if we have enough information about the present (and likewise about the past). It is only the enormity of the task of gathering information that prevents us from doing so. Quantum mechanics acts spoilsport through Heisenberg’s uncertainty principle , since it says that even in principle it is not possible to know enough about the present to make a complete prediction about the future. In fact, quantum mechanics does not predict individual events at all; it concerns itself only with group behavior, that is, it states the statistical laws which govern collections of events. Probability is the major characteristic of quantum mechanics.

2.10 Interpretations of Quantum Mechanics

The brain is the only kind of object capable of understanding that the cosmos is even there, or why there are infinitely many prime numbers, or that apples fall because of the curvature of space-time, or that obeying its own inborn instincts can be morally wrong, or that it itself exists.⁶⁰—David Deutsch

Abstract mathematical statements, if they are to describe the world to a human mind, need an isomorphic interpretation that establishes a correspondence between mathematical symbols and the rules by which those symbols are aggregated on the one hand and entities and mental concepts about our Universe on the other. The discovered isomorphisms may be many. When that happens, we seek a more unified view of our Universe by drawing analogies among the isomorphisms. In the subsections below, we provide three well-known interpretations of quantum mechanics. The conceptual diversity of these interpretations is remarkable. In Chap. 12, a hypothesized subquantum level view of the quantum world is presented. Note that interpretations are about perceptions, not necessarily reality.

2.10.1 Copenhagen Interpretation

The Copenhagen interpretation , named after the city in Denmark, where Neils Bohr lived and worked, was provided by him and Werner Heisenberg about 1927. It is the most cited Interpretation. It extends the probabilistic interpretation of the wave function proposed by Max Born.⁶¹ As Jammer notes:

The Copenhagen view is not a single, clear-cut, unambiguously defined set of ideas but rather a common denominator for a variety of related viewpoints.⁶²

The interpretations’ interesting features include: The position of a particle is essentially meaningless; a measurement causes an instantaneous collapse of the wave function and the collapsed state is randomly picked to be one of the many possibilities allowed for by the system’s state vector; the fundamental objects handled by the equations of quantum mechanics are not actual particles that have an extrinsic reality but “probability waves ” that merely have the capability of becoming “real” when an observer makes a measurement. Even then, it does not explain entanglement; this observed “nonlocality” is a mathematical consequence of quantum theory.⁶³ In Bohr’s view,

… there is no quantum world. There is only abstract quantum physical description. It is wrong to think that the task of physics is to find out how nature is. Physics [only] concerns what we can say about nature.⁶⁴

This view is contrary to Einstein’s who believed that the job of physical theories is to “approximate as closely as possible to the truth of physical reality.”⁶⁵ David Mermin once dubbed the Copenhagen interpretation as the “shut up and calculate” interpretation.⁶⁶

2.10.2 Everett’s Many-World Interpretation

Hugh Everett III, in his 1956 doctoral dissertation⁶⁷ (journal version⁶⁸), proposed his “relative state interpretation,” now generally known as the many world interpretation (the name was coined by Bryce DeWitt in the late 1960s). This interpretation is perhaps the most bizarre and yet perhaps the simplest because it circumvents the measurement conundrum. Everett posits that when a quantum system is faced with a choice as in a measurement, the entire Universe splits into a number of universes equal to the number of collapse-choices available. When a universe splits, observers in it also split. Thus, there will be parallel copies of observers in parallel universes, each of whom will see the specific outcome that appears in his respective split universe.⁶⁹ Variants of the many-world interpretation are the multiverse interpretation, the many-histories interpretation, and the many-minds interpretation.

Physicists, including Bohr, at the time ridiculed Everett’s interpretation (he got his Ph.D. alright and the work was published). It was said that “Bohr gave him the brush-off when Everett visited him in Copenhagen.”⁷⁰ He was so discouraged by the ridicule that he left physics, became a defense analyst, and then a private contractor to the US defense industry which made him a multimillionaire.⁷¹ “Everett advised high-level officials in the Eisenhower and Kennedy administrations on the best methods for selecting hydrogen bomb targets and structuring the nuclear triad of bombers, submarines and missiles for optimal punch in a nuclear strike.”⁷² Everett is also renowned for his generalized Lagrange multiplier method.⁷³ After Everett’s death, his interpretation has gained widespread respect of physicists, especially of David Deutsch who believes:

The quantum theory of parallel universes is not the problem, it is the solution. It is not some troublesome, optional interpretation emerging from arcane theoretical considerations. It is the explanation, the only one that is tenable, of a remarkable and counter-intuitive reality.⁷⁴

2.10.3 Bohm’s Interpretation

In Bohm’s interpretation , ⁷⁵ which appeared in 1952 predating Everett’s, the whole universe is entangled, and one cannot isolate one part of the universe from the other. In Bohm’s view, the interaction between entangled particles is not mediated by any conventional field known to physics (such as the electromagnetic field), but by an all-pervasive field that is instantaneous. He showed that the non-local interactions can be described in terms of a very special anti-relativistic quantum information field (pilot-wave) that does not diminish with distance and that binds the whole universe together. This field is not physically measurable but manifests itself in terms of non-local correlations. The idea is not only interesting but entirely derivable from the Schrödinger equation . Consequently, in Bohm’s interpretation, e.g., the electron is a particle with well-defined position and momentum at any instant. However, the path an electron follows is guided by the interaction of its own pilot wave with the pilot waves of other entities in the universe.

In the Copenhagen interpretation , we are asked to believe in a cloud of probabilities as to the position and momentum of a particle even though the wave function evolves deterministically. To know anything about the particle, you must collapse the wave function by making a measurement. Bohm held a radically different view. He drew inspiration from de Broglie’s hypothesis of a “pilot wave” accompanying the particle. In Bohm’s view, every particle has an actual, definite location whether observed or not. Changes in the positions of the particles are given by another equation, known as the “pilot wave” equation (or “guiding equation”) derivable from Schrödinger’s equation. Hence, the theory is fully deterministic and non-locality is even more conspicuous. Indeed, the trajectory of any particle depends on what the rest of the Universe is doing. If the initial state of a system, and its wave function is known, the location of each particle can be calculated. The particle behaves like a surfer riding the pilot wave; the particle goes where the pilot wave, which acts like a wave, takes it. So, while the particle takes a deterministic path through just one slit (say, in the two-slit experiment), the pilot wave passes through both slits, and the result is as dictated by wave interference generated by the pilot wave. The key is the pilot wave.

Recent experiments by Mahler et al.⁷⁶ support Bohm’s interpretation. A major supporter of Bohm’s interpretation was John Bell after whom the Bell inequality mentioned in Chap. 1, Sect. 1.​5 is named. We shall meet this important inequality again in Chap. 4, Sect. 4.​5.​4.

2.11 From Galileo–Newton to Schrödinger–Born

From Galileo⁷⁷ onwards, physics has moved from one level of mathematical abstraction to another higher level, in the sense that mental visualization by means of interpreting those abstractions has become increasingly difficult. Quantum mechanics works at levels of abstraction where physicists have had to abandon mental visualizations. Mathematics is used to mechanically compute results. The best way to “understand” quantum mechanics appears to be in the language of mathematics; any translation of quantum mechanical concepts in any natural language only creates misunderstanding, confusion, ambiguity, and sometimes outright disbelief. This is because of the natural language’s inability to deal with many of those concepts.

Apart from the deep role played by mathematics in physics, inductive reasoning, conjectures and refutations play an immense role in developing physical theories. Largely influenced by Karl Popper, physicists have reconciled themselves to the idea that they can never know what the real world is like; they can only make conjectures, which are essentially free creations of the human mind, and try to incessantly refute those conjectures. If a conjecture falls, physicists try to amend the conjecture or look for a new one.⁷⁸ As Heisenberg has perceptively noted: “What we observe is not nature itself, but nature exposed to our method of questioning.”⁷⁹

Gary Zukav, a non-physicist, in his classic, The Dancing Wu Li Masters, said it rather eloquently:

Thus it was that the wave aspect of quantum mechanics developed. Just as waves have particle-like characteristics (Planck, Einstein), particles also have wave-like characteristics (de Broglie). In fact, particles can be understood in terms of standing waves (Schrödinger). Given initial conditions, a precise evolution of standing-wave patterns can be calculated via the Schrödinger wave equation . Squaring the amplitude of a matter wave (wave function) gives the probability of the state that corresponds to that wave (Born). Therefore, a sequence of probabilities can be calculated from initial conditions by using the Schrödinger wave equation and Born’s simple formula.⁸⁰

As Bohr once wrote, the break from classical mechanics to quantum mechanics entails “the necessity of a final renunciation of the classical ideal of causality and a radical revision of our attitude toward the problem of physical reality.”⁸¹ Quantum mechanics dares us to reexamine our instincts, common sense, and even experiences we have come to rely upon. The law of causality is no longer applied in quantum theory, and the law of conservation of matter is no longer true for the elementary particles.

2.12 Concluding Remarks

Since 1900 when Planck resorted to quantizing energy, a new and bizarre quantum world has emerged. In this world, quantum entities are both particles and waves, and they know whether they are being observed or not. The link between the Schrödinger equation and the classical wave equation is the de Broglie relation. The Schrödinger equation constitutes a fundamental law of nature (an axiom); the evolution of the wave function it describes is a complex function.

This bizarre world has prodigiously produced Nobel Laureates in Quantum Mechanics: They include: Max Karl Ernst Ludwig Planck (1918), Albert Einstein (1921), Niels Henrik David Bohr (1922), Prince Louis-Victor Pierre Raymond de Broglie (1929), Werner Karl Heisenberg (1932), Erwin Schrödinger (1933), Paul Adrien Maurice Dirac (1933), Sir George Paget Thomson (1937), Clinton Joseph Davisson (1937), Wolfgang Pauli (1945), Born [11], Gerardus ‘t Hooft (1999), Martinus J.G. Veltman (1999), David J. Wineland (2012), Serge Haroche (2012), François Englert (2013), and Peter W. Higgs (2013).

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Footnotes

1

Deutsch [21].

2

Hence, in our context, quantum computing is not about computational quantum physics, or quantum chemistry or modeling of quantum systems using classical computers.

3

The terms “classical mechanics” and “classical physics” refer to a non-quantum theory. It is possible to give separately, for classical and quantum theories, a relativistic and a non-relativistic formulation.

4

Maxwell [47].

5

Dyson [26].

6

Bohr [7].

7

See Wheeler [68].

8

Faye [33].

9

Young [69, 70].

10

Thomson [65] for this discovery J. J. Thomson received the 1906 Nobel Prize in physics. His son George Paget Thomson won the 1937 Nobel Prize in physics (shared with Clinton Davisson) for showing that a beam of electrons could also be diffracted and hence behave as waves too.

11

Planck [52]. In this paper, Planck provided a phenomenological fit to the blackbody radiation spectrum and introduced the constant h later named after him.

12

Notwithstanding, the Nobel Prize in Physics 1918 was awarded to Max Karl Ernst Ludwig Planck “in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta.”

13

Einstein [27]. This paper was fundamental to the development of quantum theory.

14

This was in his doctoral thesis (titled Recherches sur la théorie des quanta (Researches on the Quantum Theory) at the Sorbonne in Paris.

15

A. Einstein, Letter to P. Langevin, December 16, 1924. The quote appears in James [43], p. 311.

16

Davisson and Germer [19]. The first published experiments to confirm de Broglie’s theory. See also: Davisson [20].

17

See Thomson [66]. See also: Thomson [67].

18

In 1929, de Broglie was awarded the Nobel Prize in Physics “for his discovery of the wave nature of electrons.” In 1937, Sir George Paget Thomson and Clinton Joseph Davisson shared the Nobel Prize for physics “for their experimental discovery of the diffraction of electrons by crystals.”

19

In passing we note that the concept of force may be considered redundant. In principle, one can always express classical physics in terms of the positions, velocities, and accelerations of all the particles of the universe.

20

We now know from chaos theory that there are limitations to this; it appears in the form of “deterministic chaos.” It was first noticed in 1890 by Henri Poincaré (1854–1912). See Poincaré [53].

21

See, e.g., Bohm [4], Chap. 2.

22

Renkel [55].

23

Born won the 1954 Nobel Prize in physics “for his fundamental research in quantum mechanics, especially for his statistical interpretation of the wavefunction.” It was shared with Walther Bothe, who won it “for the coincidence method and his discoveries made therewith.” Born’s Nobel lecture, 11 December 1954 (see Born [11]) is highly recommended to the reader.

24

It may well be that Nature, at its core, is completely deterministic and the dynamics of the Universe is preordained.

25

See, e.g., Aspect et al. [3], Aspect [2]. In February 2017, even more stringent experimental evidence of entanglement was provided by Handsteiner et al. [35].

26

Given a set S and two binary operators * and + on S, we say that the operation * is left-distributive over + if, given any elements x, y, and z of S, x * (y + z) = (x * y) + (x * z) and right-distributive if (y + z) * x = (y * x) + (z * x), and simply distributive if both left and right hold. Note that when * is commutative, then all the three are logically equivalent.

27

Susskind and Friedman [64], pp. 35, 94.

28

Susskind and Friedman [64], p. 96.

29

Heisenberg [37]. See also: Born and Jordan [14], and Born et al. [13].

30

Nobelprize [51].

31

Heisenberg [40], p. 57.

32

Heisenberg [40], p. 75.

33

The original paper is Schrödinger [61]. This paper was preceded by Schrödinger [58–60]. All these papers are available at Schrödinger (Collected papers). See also: Nanni [49].

34

See, e.g., Cresser [18], especially Chaps. 2, 3, and 6.

35

Schrödinger [57].

36

Born [11].

37

Dirac [24].

38

Hawking [36].

39

Since Schrödinger’s wave equation is non-relativistic, it does not work at high energies. At high energies, particle physicists use the S Matrix (also called scattering matrix), an array of mathematical quantities that predicts the probabilities of all possible outcomes of a given experimental situation. E.g., two colliding particles may alter in speed and direction or even change into entirely new particles: The S-matrix for the collision gives the likelihood of each possibility. An S-matrix is expressed in terms of observable quantities. Complete knowledge of the S-matrix for all processes would amount to having complete understanding of all physical laws.

40

For example, de Broglie and Schrödinger had attempted to make the obvious analogy between matter waves of quantum mechanics and classical waves of Maxwellian electrodynamics without success, because such an analogy, inter alia, was incompatible with the intrinsic nonlocality of quantum phenomena.

41

Invented by Paul Dirac.

42

Nielsen and Chuang [50], see Chap. 2, Sect. 2.2. We have generally tried to stay close to their formal statements and notations to enable readers to easily transition to their excellent book.

43

A linear operator U whose inverse is its adjoint (conjugate transpose) is called unitary.

44

Born [12], p. 95.

45

Born [12], p. 96.

46

Born [12], p. 102.

47

Heisenberg [38]. See also: Heisenberg [39].

48

Heisenberg [38]. See also: Heisenberg [39].

49

For a proof, see Nielsen and Chuang [50], p. 89. See also Busch et al. [15]; Cowan [17].

50

Heisenberg [38], Hilgevoord and Uffink [42].

51

See, e.g., Lamb [45].

52

Susskind and Friedman [64], p. 52.

53

For a proof, see, e.g., Dirac [25], pp. 49–50. The converse is also true, i.e., if there are two observables such that their simultaneous eigenstates form a complete set, then the two observables commute.

54

In Chap. 2, Sect. 2.2, we noted the two-layer description of the world . In quantum mechanics, one may view observable-operators as partly doing the task of the second layer. What is unusual here is that an observable-operator initiates the random “collapse” of the wave function on which it acts.

55

Bohm [4], pp. 84–85.

56

Feynman [34], Chap. 6.

57

Born [11].

58

Stapp [63].

59

Heisenberg [41], p. 114. (Quoted in Zukav [70], p. 47.)

60

Deutsch [23].

61

Born [9, 10].

62

Jammer [43], p. 87. Quotation as reproduced in Al-Kahlili [1], p. 134.

63

Seife [62].

64

Al-Kahlili [1], p. 153.

65

See Footnote 64.

66

Mermin [48].

67

Everett [29].

68

Everett [30].

69

Byrne [16].

70

Al-Kahlili [1].

71

See Footnote 70.

72

Byrne [16].

73

Everett [31].

74

Deutsch [22], p. 51.

75

Bohm and Hiley [6], Chap. 3. The original paper is: Bohm [5].

76

Mahler et al. [46]. See also: Falk [32].

77

Galileo is the only scientist referred to in the scientific literature by his first name rather than his last.

78

Popper [54]. Great scientific theories have built into them the risk of making predictions of possible effects not yet observed. The matter wave theory of de Broglie is one of them. A theory which is not refutable by any conceivable event is non-scientific. Irrefutability is not a virtue of a theory (as people often think) but a vice. As for Popper, “the criterion of the scientific status of a theory is its falsifiability, or refutability, or testability.” Theories which fail to meet the criterion of falsifiability adopt the soothsayer’s practice of making their interpretations and prophesies sufficiently vague so that they are able to explain away anything that might have been a refutation of the theory had the theory and prophesies been more precise.

79

Heisenberg [40], p. 58.

80

Zukav [70], p. 129.

81

Bohr [8], p. 60.

© Springer Nature Singapore Pte Ltd. 2020

R. K. BeraThe Amazing World of Quantum ComputingUndergraduate Lecture Notes in Physicshttps://doi.org/10.1007/978-981-15-2471-4_3

3. Mathematical Elements Needed to Compute

Rajendra K. Bera1  

(1)

Acadinnet Education Services India, Bangalore, Karnataka, India

Rajendra K. Bera

Email: rajendrabera@yahoo.com

Young man, in mathematics you don’t understand things. You just get used to them.

—John von Neumann (as quoted in Zukav [5, p.208, footnote]).

Abstract

This chapter begins with an introduction to propositional calculus and first-order predicate calculus , goes through elements of linear algebra highlighting those elements that are essential in quantum computing (e.g., eigenvalues and eigenvectors, commutator and anti-commutator), and ends with a brief introduction of the Pauli matrices. The presentation assumes that the reader already knows linear algebra and has used it in classical physics.

3.1 Introduction

Do read von Neumann’s quote above, made in humor. Inventing mathematics and mathematical algorithms requires a high level of intelligence, applying already invented algorithms does not. By and large, if you just try to get used to the mathematics described in this chapter, especially the operators and what each operator takes in as input and gives out as output, you will do fine. Carefully note how the entire thing is symbolically represented and the meaning attached to it. It can be a tedious job, but if you patiently bear with it, the magic of quantum computing will begin to unfold in later chapters. So, let us get started with something that you are expected to be familiar with—linear algebra. But before that, a few general comments about the foundations of mathematics. In dealing with a language, especially mathematics, we talk about three distinct things:

The syntax, which comprises the rules of grammar that define properly formed sentences in the language.

The semantics, which is concerned with meaning (interpretation).

The pragmatics, which deals with what people do with language, that is, the practical aspects of getting things done or said.

3.1.1 Propositional Calculus (Propositional Logic)

Propositional calculus (also known as sentential calculus, propositional logic, or sentential logic) is a formal axiomatic system for reasoning about propositional formulas. It is about the simplest of logical calculus in mathematical use today. It is a part, indeed the foundation, of symbolic logic¹ where we reason about abstract formulas that represent statements. Propositional calculus (or reasoning) is built around our notions of the correct usage of the words if … then … (or implies), or, and, not. Propositional calculus has a vocabulary, rules for the formation of well-formed formulas (i.e., statements constructed according to a prescribed syntax) based on the vocabulary, and inference rules for deriving formulas from a given set of well-formed formulas. The inference rules are chosen such that if the formulas in the set represent true statements, then the derived formulas also represent true statements. When we argue, we instinctively use propositional calculus .

Propositional calculus is closely related to Boolean algebra.² However, in Boolean algebra we do not derive well-formed formulas from sets of well-formed formulas, rather we rewrite well-formed formulas into equivalent well-formed formulas given a certain set of algebraic identities.

In logic, the logical value and the deductive sequence of the formulas does not, in the least, depend upon the meanings that may be given to them. The meanings serve to make the formulas intelligible, to give them clarity, and to make their meaning obvious, but never to justify them. The meanings may be omitted without destroying the formal rigidity of the system.

Propositional calculus can be extended in several ways. For example, one may add rules sensitive to more fine-grained details of the sentences being used. When the atomic sentences of propositional calculus are split into term, variables, predicates, and quantifiers, they yield first-order predicate calculus (also known as first-order logic), which keeps all the rules of propositional calculus and adds some new ones.

3.1.2 First-Order Predicate Calculus (First Order Logic)

First-order predicate calculus is a theory in symbolic logic wherein predicate means to declare something about the subject of a proposition. Higher-order predicates talk about predicates. Like any logical theory, it comprises

A specification regarding the construction of syntactically correct statements (also known as well-formed formulas),

A set of axioms, where each axiom is a well-formed formula,

A set of inference rules, using which one can prove theorems from axioms or earlier proven theorems.

There are two types of axioms: the logical axioms, which embody general truths about proper reasoning involving quantified statements, and the axioms describing the subject matter at hand (e.g., axioms describing sets in set theory or axioms describing numbers in arithmetic). The number of axioms may be zero, non-zero but finite, or infinite.³ The set of inference rules is always finite.

By a well-formed formula, we mean that such a formula contains

Variables such as x, y, … which are placeholders for objects of the domain under consideration,

Object constants such as 0, 1, or the empty set ∅ which stand for fixed individual objects in the said domain,

Predicate constants such as < (meaning less than), ∈ (is in), = (equals), which stand for fixed relations between or properties of said objects; these are also known as first-order predicates to distinguish them from predicates that talk about predicates,

Function constants such as +, ×, which stand for fixed functions taking objects as arguments, and returning objects as values,

Logical connectives such as ∧ (and), ∨ (or), ⇒ (implies), ¬ (not), ∃ (there exists; existential quantifier), and ∀ (for all; universal quantifier).

The object, predicate, and function constants will typically depend on the specific domain we are talking about. With the tools of first-order logic, it is possible to formulate theories, either with explicit axioms or by rules of inference that can themselves be treated as logical calculi. Arithmetic is the best known of these; others include set theory. On the other hand, multi-valued logics, since they allow sentences to have values other than true and false, such as neither and both or a continuum of values (as in fuzzy logic), often require calculational devices quite distinct from propositional calculus .

3.2 Elements of Linear Algebra

The basic objects in the study of linear algebra are vector spaces, in particular the space Cn of all n-tuples of complex numbers (z₁, …, zn). The elements of a vector space are called vectors. Physicists sometimes refer to complex numbers as c-numbers. The most familiar notation for a state vector in describing a quantum system is $$ \left| \psi \right\rangle $$ (Physicists can be rather flippant in their choice of symbols other than ψ, e.g., +, −, ↑, ↓, etc.). The standard vector-matrix notation is used when the state vector is expressed in terms of its components. For example,

is the column vector with n-components, and

is the corresponding row vector with n-conjugate components.

Like ordinary vectors, state vectors are specified by a particular choice of basis vectors (eigenstates) and a particular set of complex numbers, corresponding to the amplitudes with which each eigenstate contributes to the complete state vector. Once the state vector $$ |\psi \rangle $$ of a quantum system is known, the expected value⁴ of any observable attribute of the system can be calculated since $$ |\psi \rangle $$ contains the complete information about the system. This is similar to descriptions of systems in classical physics in which the complete state of the system is known once the time-dependent functions for position and momentum are determined.

In what follows, it is advisable to pay attention to minute details of notation and syntax. Heed von Neumann’s observation and get used to them! Pay careful attention to all those mathematical properties which remain invariant under a given transformation, especially unitary transformations. Under unitary transformations state vectors only rotate, their lengths do not change (first invariant). We have generally stayed with the notations and manner of presentation of results provided in the excellent text book by Nielsen and Chuang.⁵ This will help readers in going back and forth between this book and that by Nielsen and Chuang. The notations have a certain elegance. It will serve you well to glance through Tables 3.1, 3.2 and 3.3 before going further and get familiar with the notations used in this book. Keep referring to the tables till you get a hang of the notation. In any subject, getting familiar with its symbolic system is a crucial part of grasping the knowledge it enshrines.

Table 3.1

Dirac notations

Bra-ket notations

Remarks

The symbols: $$ \left\langle { \cdot \left|~ {{\text{is}}\,{\text{called}}\,bra;} \right| \cdot } \right\rangle $$ is called ket; the dot inside the symbols is a placeholder for labels

A bra and a ket with the same label are Hermitian conjugates. For example, $$ \left| u \right\rangle = \left| u \right\rangle^{\dag } = \left| u \right\rangle^{{*\,{\text{T}}}} \equiv \left| u \right\rangle^{{{\text{T}}\,*}} $$

Superscripts *, T, and †, respectively, denote complex conjugate, transpose, Hermitian conjugate of the entity to which it is attached, e.g.,

is the column vector with n-components, $$ z_{1} , z_{2} , , z_{n} $$ and

is the corresponding row vector with n-conjugate components

Writing bras, kets, and linear operators next to each other implies matrix multiplication. Such combinations are interpreted using matrix multiplication

Note that matrices, depending on their place in an expression, are used as operands or operators

$$ \left| u \right\rangle $$ and $$ \left| v \right\rangle $$ denote, respectively, row vector u and column vector v

Their direction corresponds to the states of a dynamical system at a particular time. $$ \left| u \right\rangle $$ and $$ \left| v \right\rangle $$ must belong to the same vector space

Inner product

The bra-ket $$ \left\langle {u||v} \right\rangle \equiv \left\langle {u|v} \right\rangle = \left\langle {v|u} \right\rangle^{*} = z $$ denotes the inner product of two vectors and results in a complex number, z. It is a measure of how the two vectors tilt toward or overlap each other

For any vector, v

The norm (squared length) of v is $$ \left| v \right|^{2} = \left\langle {v|v} \right\rangle $$

v is a unit vector if $$ \left\langle {v|v} \right\rangle = 1 $$

u and v are orthogonal if $$ \left\langle {u|v} \right\rangle = 0 $$

Linearity in the second argument

$$ \left\langle {u|c_{1} v_{1} + c_{2} v_{2} } \right\rangle = c_{1} \left\langle {u|v_{1} } \right\rangle + c_{2} \left\langle {u|v_{2} } \right\rangle $$, where c₁ and c₂ are complex numbers

Antilinearity in the first argument

Outer product

The ket-bra $$ \left| u \right\rangle \left\langle v \right| = M $$ denotes the outer product of $$ \left| u \right\rangle $$ and $$ \left| v \right\rangle $$; it produces a transformation matrix, M

The matrix M converts $$ \left\langle {\left. u \right|} \right.\,{\text{to}}\,\left\langle {\left. v \right|} \right. $$ and $$ \left. {\left| v \right.} \right\rangle \,{\text{to}}\,\left. {\left| u \right.} \right\rangle $$: , where $$ \gamma = \left\langle {u|u} \right\rangle $$, and

$$ \left| u \right\rangle \left\langle {v||v} \right\rangle \equiv \left| u \right\rangle \left\langle {v|v} \right\rangle = \delta \left| u \right\rangle $$, where $$ \delta = \left\langle {v|v} \right\rangle $$

$$ \left\langle {u|M|v} \right\rangle $$ produces a number. It is the inner product between $$ \left| u \right\rangle $$ and $$ M\left| v \right\rangle $$. Equivalently, between M†$$ \left| u \right\rangle $$ and $$ \left| v \right\rangle $$

$$ \left\langle {\psi |M|\psi } \right\rangle $$ and $$ \left\langle {\psi |f(M)|\psi } \right\rangle $$ denote, respectively, the average value of an observable M and the average value of any function f(M) for a particular state $$ \left| \psi \right\rangle $$

Cauchy–Schwarz inequality

An important property of Hilbert space:

Where $$ \left. {\left| v \right.} \right\rangle $$ and $$ \left. {\left| w \right.} \right\rangle $$ are two vectors

Table 3.2

Important bra and ket operations

Notation

Description

Complex number, conjugate of complex number $$ z $$

Ket vector, related bra vector

Bra-ket (inner product)

$$ \left\langle {\psi ||\phi } \right\rangle \equiv \left\langle {\psi |\phi } \right\rangle = \left\langle {\phi |\psi } \right\rangle^{*} = z $$.

Inner product of two vectors $$ \left. {\left| \psi \right.} \right\rangle $$ and $$ \left. {\left| \phi \right.} \right\rangle $$ produces a complex number $$ z. $$

Ket-bra (outer product )

Outer product of two vectors $$ \left. {\left| \psi \right.} \right\rangle $$ and $$ \left. {\left| \phi \right.} \right\rangle $$ produces a complex matrix $$ M $$

Tensor product of $$ \left. {\left| \phi \right.} \right\rangle $$ and $$ \left. {\left| \psi \right.} \right\rangle $$

Matrix, complex conjugate of matrix, transpose of matrix, $$ A $$

Hermitian conjugate or adjoint of $$ A $$

Inner product between $$ \left. {\left| \psi \right.} \right\rangle $$ and $$ A\left. {\left| \phi \right.} \right\rangle ; $$ equivalently between $$ A^{\dag } \left. {\left| \psi \right.} \right\rangle $$ and $$ \left. {\left| \phi \right.} \right\rangle $$

Table 3.3

Operator/matrix types and their salient properties

Operator type/representation

Description

Diagonal representation of operator $$ A $$ (orthonormal decomposition)

on a vector space V

The vectors $$ \left. {\left| i \right.} \right\rangle $$ form an orthonormal set of eigenvectors for $$ A $$, with corresponding eigenvalues $$ \lambda_{i} $$. The $$ i $$-th diagonal element of $$ A $$ is $$ \lambda_{i} $$ and all non-diagonal elements of $$ A $$ are zero

Note the outer product representation of $$ A $$

Normal operator: if

A normal operator is always diagonalizable. Its eigenvalues can be complex

Hermitian operator : if

A Hermitian operator is also normal; its eigenvalues are always real. A normal matrix is Hermitian if and only if it has real eigenvalues

Spectral decomposition theorem implies that any normal operator $$ M $$ on a vector space $$ V $$ has an outer product representation as

$$ M $$ on a vector space $$ V $$ is diagonal with respect to some orthonormal basis for $$ V $$. Conversely, any diagonalizable operator is normal

Unitary operator: for such an operator $$ U, $$ and each of its matrix representation

It preserves inner products between vectors

It follows that $$ U^{ - 1} = U^{\dag } $$. If $$ U $$ is real, then $$ U^{ - 1} = U^{\text{T}} $$. Further, $$ U $$ is normal and has a spectral decomposition (it can be diagonalized). Since $$ U $$ has unit length, it can only rotate the vector it operates on. Unitarity is the only constraint required of quantum operators (also called gates)

Positive operator: a positive operator $$ A $$ is such that for any vector $$ \left. {\left| v \right.} \right\rangle $$, $$ \left\langle {v\left| A \right|v} \right\rangle $$ is a real, non-negative number. If $$ \left\langle {v\left| A \right|v} \right\rangle $$ is strictly greater than zero for all $$ \left. {\left| v \right.} \right\rangle \ne 0 $$, then A is positive definite

Any positive operator is Hermitian and therefore has the diagonal representation $$ \sum\nolimits_{i } {\lambda_{i} \left. {\left| i \right.} \right\rangle \left\langle {\left. i \right|} \right.} $$ with non-negative eigenvalues λi. Also, for any operator A, $$ A^{\dag } A $$ is positive. Positive operators are an important subclass of Hermitian operators

Trace: the trace of a matrix $$ A $$ (diagonal or otherwise) is

The trace is invariant under the unitary similarity transformation, $$ A \to UAU^{\dag } $$, i.e.,

Further,

Commutator and anti-commutator: given two operators $$ A $$ and $$ B $$, we have

Commutator: $$ \left[ {A, B} \right] AB {-} BA $$

Anti-commutator: $$ \left\{ {A, B} \right\} AB + BA $$

The simultaneous diagonalization theorem states that if $$ A $$ and $$ B $$ are Hermitian operators , then $$ \left[ {A, B} \right] = 0 $$ if and only if there exists an orthonormal basis such that both $$ A $$ and $$ B $$ are diagonal with respect to that basis

If $$ \left[ {A, B} \right] = 0 $$, the operators commute

If $$ \left\{ {A, B} \right\} = 0 $$, the operators anti-commute

Pauli matrices

$$ \sigma_{0} \equiv I \equiv \left[ {\begin{array}{*{20}c} 1 &amp; 0 \\ 0 &amp; 1 \\ \end{array} } \right] $$,

$$ \sigma_{1} \equiv \sigma_{x} \equiv X \equiv \left[ {\begin{array}{*{20}c} 0 &amp; 1 \\ 1 &amp; 0 \\ \end{array} } \right] $$,

$$ \sigma_{2} \equiv \sigma_{y} \equiv Y \equiv \left[ {\begin{array}{*{20}c} 0 &amp; { - i} \\ i &amp; 0 \\ \end{array} } \right] $$,

Pauli matrices (also, called σ-matrices) are Hermitian and unitary, and except for I, have trace zero. They are used to manipulate the state of single qubits

Any 2 × 2 matrix $$ M $$ or operator $$ M $$ can be expressed as

$$ M = \alpha I + \beta X + \gamma Y + \delta Z $$,

α, β, γ, δ are complex constants

The eigenvectors of a Hermitian operator form a unitary basis

3.2.1 Various Representations of a State Vector

A vector $$ \left. {\left| v \right.} \right\rangle $$ having $$ n $$ vector components $$ \left| {v_{1} } \right\rangle , \ldots ,\left| {v_{n} } \right\rangle $$ is generally written as the linear summation

where $$ a_{1} , \cdots , a_{n} $$ are $$ n $$ complex constants. It is also customary to represent $$ \left. {\left| v \right.} \right\rangle $$ in the following alternative matrix forms, if it is apparent from the context that $$ \left. {\left| v \right.} \right\rangle $$ has the components $$ \left. {\left| {v_{1} } \right.} \right\rangle , \cdots , \left. {\left| {v_{n} } \right.} \right\rangle $$:

Note that in this notation, the matrix representation of $$ \left. {\left| {v_{i} } \right.} \right\rangle $$ will have all $$ a_{k} = 0 $$ for $$ k = 1, \ldots ,n $$ except for $$ k = i $$, for which $$ a_{i} = 1 $$. For example, $$ \left. {\left| {v_{3} } \right.} \right\rangle $$ has the matrix representation

Note that when we use matrix notation to describe state transformations, the ordering of the basis vectors in the matrix representation must be settled a priori so that we can keep track of how each basis vector is transforming as it transforms. Finally, when the context is clear, the abstract index form is also used for an abstract linear transformation or a set of basis vectors. For example, $$ \left| i \right\rangle $$ may stand either for itself or for the basis set of which it is a member. (Warning: When the abstract index form is used, make sure that you know the context!)

3.2.2 Bases and Linear Independence

A set of vectors $$ \left. {\left| {v_{1} } \right.} \right\rangle, \ldots ,\left. {\left| {v_{n} } \right.} \right\rangle $$ is said to be a spanning set for a vector space V if any vector $$ \left. {\left| v \right.} \right\rangle $$ in V can be written as a linear combination

where, for the given $$ \left. {\left| v \right.} \right\rangle $$, the complex coefficients $$ a_{i} $$ are unique. Such a vector space V is said to have $$ n $$ dimensions, a fact symbolically stated by Cn.

An example of a spanning set for the vector space C² is the set

since any vector $$ \left. {\left| v \right.} \right\rangle \equiv \left[ {\begin{array}{*{20}c} {a_{1} } &amp; {a_{2} } \\ \end{array} } \right]^{\text{T}} $$ in C² can be written as the following linear combination

A vector space may have many different spanning sets. For example, a second spanning set for the vector space C² is the set

as once again, any vector $$ \left. {\left| v \right.} \right\rangle \equiv \left[ {\begin{array}{*{20}c} {a_{1} } &amp; {a_{2} } \\ \end{array} } \right]^{\text{T}} $$ in C² can be written as the linear combination

A set of non-zero vectors $$ \left. {\left| {v_{1} } \right.} \right\rangle , \ldots , \left. {\left| {v_{n} } \right.} \right\rangle $$ is said to be linearly dependent if there exists a set of complex numbers $$ a_{1} , \ldots , a_{n} $$ with $$ a_{i} \ne 0 $$ for at least one value of $$ i $$, such that

otherwise it is linearly independent. A linearly independent set is called a basis for V, and such a set always exists. The number of elements $$ n $$ in the basis is defined to be the dimension of V. Any two sets of linearly independent vectors, which span a vector space V, contain the same number $$ n $$ of elements.

3.3 Linear Operators and Matrices

A linear operator between vector spaces V and W, where $$ \left. {\left| {v_{1} } \right.} \right\rangle , \cdots , \left. {\left| {v_{m} } \right.} \right\rangle $$ is a basis for V and $$ \left. {\left| {w_{1} } \right.} \right\rangle , \cdots , \left. {\left| {w_{n} } \right.} \right\rangle $$ is a basis for W (note that m and n may be different), is defined to be any function A: V → W, which is linear in its inputs,⁶

A linear operator A is said to be defined on a vector space V if A is a linear operator from V to V. The identity operator IV on a vector space V is defined by the equation IV $$ \left. {\left| v \right. } \right\rangle \equiv \left. {\left| v \right. } \right\rangle $$ for all vectors $$ \left. {\left| v \right. } \right\rangle $$. If the context is clear, IV is often abbreviated to I. In addition, there is a zero-operator denoted by 0, which maps all vectors to the zero vector, i.e., 0 $$ \left. {\left| v \right.} \right\rangle \equiv 0 $$. Note that the ket notation for the zero vector is not used as, by convention, it is reserved for the zero vector $$ \left. {\left| 0 \right.} \right\rangle $$ in quantum computing where it means something entirely different.

Sometimes it is easier to see linear operators in terms of their equivalent matrix representation. The claim that the matrix A is a linear operator simply means that

is true as an equation where the operation is matrix multiplication of A by column vectors. The linear operator’s matrix representation is given by

for each j in the range 1, …, m and an $$ A_{ij} $$ is an element of A when represented in matrix form. The matrix representation of A is completely equivalent to the operator A. However, to make the connection between matrices and linear operators, we must specify a set of input and output basis states for the input and output vector spaces, i.e., V and W, respectively, of the linear operator A.

Three kinds of products between a pair of vectors $$ \left. {\left| v \right. } \right\rangle $$ and $$ \left. {\left| w \right. } \right\rangle $$ are defined: inner product , outer product , and tensor product.

3.3.1 Inner Product

The inner product of $$ \left| v \right\rangle $$ and $$ \left| w \right\rangle $$ in the same vector space is represented by $$ \left\langle {v|w} \right\rangle $$ (a short form of $$ \left\langle {v\left\| w \right.} \right\rangle $$). Let

or alternatively, in the abbreviated form as

written with respect to the same orthonormal basis (this is a good time to get used to the abbreviated representation of vector components represented by their indices i and j such as $$ \left. {\left| i \right. } \right\rangle $$ for $$ \left. {\left| {v_{i} } \right.} \right\rangle $$ and $$ \left. {\left| j \right.} \right\rangle $$ for $$ \left. {\left| {w_{j} } \right.} \right\rangle $$ as done here, if the context is clear) with respect to some orthonormal basis of which $$ \left. {\left| i \right.} \right\rangle $$ is a member (this means that $$ \left\langle {i |j} \right\rangle = \delta_{ij} , $$ where the Dirac delta function⁷ $$ \delta_{ij} = 1 $$ if i = j else $$ \delta_{ij} = 0 $$; this can always be done by adjusting the values of $$ a_{i} $$ and $$ b_{j} $$ depending on how the basis $$ \left. {\left| i \right. } \right\rangle $$ is constructed. Several construction methods exist, and a popular one is the Gram–Schmidt procedure.⁸) Then

Since we can multiply any state vector by a non-zero complex number without changing its physical interpretation, we can always normalize the state so that it has unit length to make it a unit vector or put it, so to say, in a normalized state.

Note also that the inner product remains unaffected if each of the two vectors in the product is multiplied by the factor $$ \exp \left( { - i\theta } \right) $$ (or its conjugate as applicable), where θ is real:

3.3.2 Outer Product

The outer product

results in a n × n matrix. It is a useful way of representing linear operators. Moreover, the expression $$ \left| w \right\rangle {{\langle v | {v^{\prime } } }} \rangle $$ can be freely given any one of two meanings: (1) to denote the result when the operator $$ \left. {\left| w \right.} \right\rangle \left\langle {\left. v \right|} \right. $$ acts on $$ {|v^{{\prime }} } \rangle $$ and (2) to denote the result of multiplying $$ \left. {\left| w \right. } \right\rangle $$ by the complex number $$ \left\langle {v |v^{{\prime }} } \right\rangle $$. Mathematicians usually try to construct such clever symbolic systems for ease of manipulation and economy of expression when equivalences exist. One also notices that

where the set of vectors represented by $$ \left. {\left| i \right.} \right\rangle $$ is an orthonormal basis (i.e., $$ \left\langle {i |j} \right\rangle = \delta_{ij} $$) so that $$ \left\langle {i |v} \right\rangle = a_{i} , $$ and I is a n × n unit matrix. This equation is known as the completeness relation . By complete we mean that any state vector in the chosen vector space can be represented as a weighted sum of just the $$ \left. {\left| i \right.} \right\rangle $$ vectors, e.g.,

For example, suppose A: V → W, is a linear operator, $$ \left. {\left| {v_{i} } \right.} \right\rangle $$ is an orthonormal basis for V, and $$ \left. {\left| {w_{j} } \right.} \right\rangle $$ an orthonormal basis for W. Assume

Now apply $$ \left\langle i \right| $$ to the above equations:

This allows us to extract any element of A with indices i, j by defining

Thus, revealing the matrix version of $$ A\left| a \right\rangle = \left| b \right\rangle $$:

3.3.3 Tensor Product

The tensor product of $$ \left. {\left| v \right.} \right\rangle $$ and $$ \left| w \right\rangle $$, represented by $$ \left. {\left| v \right.} \right\rangle \otimes \left. {\left| w \right.} \right\rangle $$ (alternatively, by $$ \left| v \right\rangle \left| w \right\rangle $$ or $$ \left. {\left| {v, w} \right.} \right\rangle $$ or $$ \left. {\left| {vw} \right.} \right\rangle $$), produces a matrix. The tensor product puts vector spaces together to form larger vector spaces. This simple method is crucial for constructing the Hilbert space for multi-qubit systems. Let A be a m × n matrix, and B a p × q matrix. Then the way vector spaces are put together is as follows:

Here, each element $$ A_{ij} B $$ represents a p × q submatrix whose entries are the entries of B multiplied by $$ A_{ij} $$. The fully expanded matrix form of A ⊗ B is a larger mp × nq matrix.

Example 1

Example 2

Given $$ X = \left[ {\begin{array}{*{20}c} 0 \,&amp;\, 1 \\ 1 \,&amp;\, 0 \\ \end{array} } \right] $$ and $$ Y = \left[ {\begin{array}{*{20}c} 0 \,&amp;\, { - i} \\ i \,&amp;\, 0 \\ \end{array} } \right], $$

Then

Finally, we have the useful notation $$ \left| {v^{ \otimes k} } \right\rangle $$, which stands for $$ \left. {\left| v \right.} \right\rangle $$ tensored with itself k times. For example,

By definition,

We shall shortly see that an analogous notation is used for operators (as they also have a matrix representation) on tensor product spaces.

By definition, the tensor product satisfies the following basic properties:

1.

For an arbitrary scalar z and elements $$ \left. {\left| v \right.} \right\rangle $$ of V and $$ \left. {\left| w \right.} \right\rangle $$ of W,

2.

For arbitrary $$ \left. {\left| {v_{1} } \right.} \right\rangle $$ and $$ \left. {\left| {v_{2} } \right.} \right\rangle $$ in V and $$ \left. {\left| w \right.} \right\rangle $$ in W,

3.

For arbitrary $$ \left. {\left| v \right.} \right\rangle $$ in V and $$ \left. {\left| {w_{1} } \right.} \right\rangle $$ and $$ \left. {\left| {w_{2} } \right.} \right\rangle $$ in W,

One may also show that

where the superscripts *, T, and †, respectively, denote complex conjugate, transpose, and Hermitian conjugate⁹ (or adjoint) operations on the entity they operate on.

Finally, we come to the nature of linear operators that act on the space $$ \left. {\left| V \right.} \right\rangle \otimes \left. {\left| W \right.} \right\rangle $$. By convention (and this is important), a matrix representation of a linear operator will imply that the representation is with respect to orthonormal input and output bases. Further, if the input and output spaces for a linear operator are the same, then the input and output bases are assumed to be the same, unless noted otherwise.

Suppose $$ \left. {\left| v \right.} \right\rangle $$ and $$ \left. {\left| w \right.} \right\rangle $$ are vectors in V and W, and A and B are linear operators on V and W, respectively. Then we define a linear operator $$ A \otimes B $$ on $$ V \otimes W $$ by the equation

The definition of $$ A \otimes B $$ is then extended to all elements of $$ V \otimes W $$ in the natural way to ensure linearity of A ⊗ B, that is,

One can show that $$ A \otimes B $$ so defined is a well-defined linear operator on $$ V \otimes W $$. This notion of the tensor product of two operators extends in the obvious way to the case where A: V → V′ and B: W → W′ map between different vector spaces. Indeed, an arbitrary linear operator C mapping $$ V \otimes W $$ to $$ V^{{\prime }} \otimes W^{{\prime }} $$ can be represented as a linear combination of tensor products of operators mapping V to V’ and W to W’,

where by definition

The inner products on the spaces V and W can be used to define a natural inner product on $$ V \otimes W $$. Define

It can be shown that the function so defined is a well-defined inner product . From this inner product , the inner product space $$ V \otimes W $$ inherits other familiar structures, such as notions of unitarity,¹⁰ adjoint, normality, and Hermiticity.

3.4 Eigenvalue, Eigenvector, Spectral Decomposition, Trace

The reader should read this section carefully and get conversant with the terms and results presented herein. They form the core of mathematical manipulations in quantum computing.

3.4.1 Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors of a matrix are among the most important notions relevant to linear transformations. They are vital to quantum mechanics. The eigenvalues λ₁, …, λn of a n × n matrix A are those values of λ for which the equation det (A − λI) = 0, where det is the determinant function for matrices, and I is the unit matrix. In its expanded form, it is a polynomial equation of degree n in λ. By the fundamental theorem of algebra,¹¹ we can factorize this polynomial so that det (A − λI) = 0 reduces to

where the complex numbers λ₁, λ₂, λ₃, …, λn, are the eigenvalues of A.

Example

Given

The roots of the quadratic (a − λ) (d − λ) – bc = 0 in λ are the eigenvalues of A.

Note that for λ to be an eigenvalue, the matrix A − λI must be singular and vice versa.

An eigenvector corresponding to an eigenvalue λi of a linear operator A on a vector space is a non-zero vector $$ \left. {\left| {v_{i} } \right.} \right\rangle $$ such that $$ A\left. {\left| {v_{i} } \right.} \right\rangle = \lambda_{i} \left. {\left| {v_{i} } \right.} \right\rangle $$. That is, an eigenvector of a linear transformation A is a non-zero complex vector $$ \left. {\left| {v_{i} } \right.} \right\rangle $$ which A sends to a multiple λi of itself (i.e., to $$ \lambda_{i} \left. {\left| {v_{i} } \right.} \right\rangle $$). For a given linear operator, eigenvectors are therefore special vectors whose directions are unaltered by the operator.

If the eigenvalues of two or more eigenvectors have the same value, say $$ \lambda^{{\prime }} $$ (i.e., an eigenvalue repeats itself), then the set of such eigenvectors form an eigenspace corresponding to the eigenvalue $$ \lambda^{{\prime }} $$. An eigenspace is a vector subspace of the vector space on which A acts. The dimension of an eigenspace is the number of eigenvectors in that eigenspace. When an eigenspace is more than one-dimensional, we say that it is degenerate. It turns out that eigenvalues and eigenvectors of a quantum operator play a fundamental role both in terms of the representation of a wave function on which the operator operates and the information that can be extracted from the wave function in a measurement of the quantum system the wave function represents (see Postulate 3 in Chap. 2, Sect. 2.​7.​3).

3.4.2 Diagonal Representation of an Operator or Orthonormal Decomposition

A diagonal representation for an operator $$ A $$ on a vector space $$ V $$ is a representation

where the vectors $$ \left. {\left| i \right.} \right\rangle $$ form an orthonormal set of eigenvectors for $$ A $$, with corresponding eigenvalues $$ \lambda_{i} $$. The $$ i $$-th diagonal element of $$ A $$ is $$ \lambda_{i} $$ and all non-diagonal elements of $$ A $$ are zero. An operator which has a diagonal representation is said to be diagonalizable. Diagonal representations are also known as orthonormal decompositions . A diagonal representation of $$ A $$ simplifies computations tremendously since one needs to deal with only $$ n $$ elements of $$ A $$ rather than $$ n^{2} $$ elements for a $$ {\textit{n}} \times {\textit{n}} $$ representation of $$ A $$. If all the eigenvalues of $$ A $$ are distinct, they can be used to index its eigenvectors.

3.4.3 Normal Operators and Spectral Decomposition

An operator A is said to be normal if $$ AA^{\dag } = A^{\dag } A $$. Thus, an operator which is Hermitian (i.e., $$ A = A^{\dag } = \left( {A^{\text{T}} } \right)^{*} = \left( {A^{*} } \right)^{\text{T}} $$) is also normal. Clearly, the (principal) diagonal elements of a Hermitian matrix must be always real. It can be shown that the eigenvalues of a Hermitian matrix are indeed real. A normal matrix is Hermitian if and only if it has real eigenvalues. A very important result regarding normal operators is the following spectral decomposition theorem (presented here without proof):

Any normal operator $$ M $$ on a vector space $$ V $$ is diagonal with respect to some orthonormal basis for $$ V $$. Conversely, any diagonalizable operator is normal.

The reader must commit this theorem to memory. What it means is that $$ M $$ can be written in terms of the outer product representation as

where $$ \lambda_{i} $$ are the eigenvalues of $$ M $$, the set of vectors $$ \left. {\left| i \right.} \right\rangle $$ is an orthonormal basis for V, and each $$ \left. {\left| i \right.} \right\rangle $$ is an eigenvector of $$ M $$ paired with its corresponding eigenvalue $$ \lambda_{i} $$.

3.4.4 Unitary Operators

A matrix $$ U $$ is said to be unitary if $$ U^{\dag } U = I $$. Hence, the inverse of a unitary matrix exists. If $$ U $$ is real, it is called an orthogonal matrix. For such a matrix $$ U^{\text{T}} = U^{ - 1} $$. Similarly, an operator $$ U $$ is unitary if $$ U^{\dag } U = I $$, hence it is invertible. One can show that an operator is unitary if and only if each of its matrix representations is unitary. Since $$ U^{\dag } U = I $$, $$ U $$ is normal and has a spectral decomposition (i.e., it can be diagonalized). Unitarity simply means the equality of two lengths. Any unitary transformation is therefore equivalent to a rotation and has a L² length that remains invariant to a rotation. An L² length has the form of a sum of squares. Rather surprisingly, the unitarity constraint is the only constraint required of quantum operators (also called gates). Thus, any unitary matrix specifies a valid quantum operator (gate) and hence their great importance in quantum computing. In short , unitary operators only rotate the vector they operate on.

Unitary operators preserve inner products between vectors. Let $$ \left. {\left| v \right.} \right\rangle $$ and $$ \left. {\left| w \right.} \right\rangle $$ be any two vectors. Then the inner product of $$ U\left. {\left| v \right.} \right\rangle $$ and $$ U\left. {\left| w \right.} \right\rangle $$ is

Another interesting property of a unitary operator is that if $$ \left. {\left| {v_{i} } \right.} \right\rangle $$ is any orthonormal basis set, and $$ \left. {\left| {w_{i} } \right.} \right\rangle \equiv U\left. {\left| {v_{i} } \right.} \right\rangle $$, then $$ \left. {\left| {w_{i} } \right.} \right\rangle $$ is also an orthonormal basis set, since unitary operators preserve inner products. Further, . Conversely, if $$ \left. {\left| {v_{i} } \right.} \right\rangle $$ and $$ \left. {\left| {w_{i} } \right.} \right\rangle $$ are two orthonormal bases, then the operator defined by $$ U \equiv \mathop \sum \nolimits_{i} \left. {\left| {w_{i} } \right.} \right\rangle \left\langle {\left. {v_{i} } \right|} \right. $$ is a unitary operator . Each eigenvalue of a unitary matrix has modulus 1 (i.e., it has the form $$ e^{i\theta } = \cos \theta + i\sin \theta $$ for some real $$ \theta $$). The tensor product of two unitary operators is unitary. Note that Hermitian and unitary matrices are normal.

3.4.5 Positive Operator

A positive operator A is defined as an operator such that for any vector $$ \left. {\left| v \right.} \right\rangle $$, $$ \left\langle {v\left| A \right|v} \right\rangle $$ is a real, non-negative number. If $$ \left\langle {v\left| A \right|v} \right\rangle $$ is strictly greater than zero for all $$ \left. {\left| v \right.} \right\rangle \ne 0 $$, then A is said to be positive definite. It can be shown that any positive operator is Hermitian and therefore has the diagonal representation $$ \mathop \sum \nolimits_{i } \lambda_{i} \left. {\left| i \right.} \right\rangle \left\langle {\left. i \right|} \right., $$ with non-negative eigenvalues λi. One may also show that for any operator A, $$ A^{{\text{\dag }}} A $$ is positive. Positive operators are a very important subclass of Hermitian operators .

3.4.6 Trace of a Matrix

Another important matrix function is the trace of a matrix , tr(A), defined as the sum of the diagonal elements of A (which need not be a diagonal matrix),

The trace is easily seen to be cyclic, that is, tr(AB) = tr(BA), and linear, i.e., tr(A + B) = tr(A) + tr(B), tr(zA) = z tr(A), where A and B are arbitrary matrices, and z is a complex number. It also follows from the cyclic property that the trace of a matrix is invariant (note the invariance) under the unitary similarity transformation, $$ A \to UAU^{\dag } $$, since

We define the trace of an operator A to be the trace of any matrix representation of A. The invariance of the trace under unitary similarity transformations ensures that the trace of an operator is well defined. It also means that if we choose a unitary transformation that diagonalizes A, then $$ {\text{tr}}\left( A \right) = \sum\nolimits_{i} {\lambda_{i} ,} $$ i.e., the trace of A is also the sum of its eigenvalues, λI.

An extremely important formula (presented here without proof) for evaluating $$ \text{tr} (A\left. {|\psi} \right\rangle \left\langle {\psi |} \right.) $$, where $$ \left| \psi \right\rangle $$ is a unit vector and A is an arbitrary operator, is

This result is extremely useful in evaluating the trace of an operator because it plays a crucial role in deciding the probability with which a quantum system will collapse to a particular state when that system is measured. If A is the identity operator, then $$ \text{tr} (\left. {|\psi } \right\rangle \left\langle {\psi |} \right. = \left\langle {\psi |\psi } \right\rangle $$.

3.4.7 Commutator and Anti-Commutator

The commutator between two operators A and B is defined as

If [A, B] = 0, i.e., AB = BA, then we say A commutes with B. Similarly, the anti-commutator of two operators is defined by

A is said to anti-commute with B if $$ \{ A,B\} = 0 $$. It so happens that many important properties of pairs of operators can be deduced from their commutator and anti-commutator. One such is the simultaneous diagonalization theorem, which states that if A and B are Hermitian operators , then [A, B] = 0 if and only if there exists an orthonormal basis such that both A and B are diagonal with respect to that basis. (We omit the proof.) Thus, A and B are simultaneously diagonalizable in this case. It turns out that non-zero commutators play an essential role in the famous Heisenberg’s uncertainty principle (see Chap. 2, Sect. 2.​7.​6) in quantum mechanics. Indeed, non-commuting operators lie at the heart of quantum mechanics.

3.4.8 Polar and Singular Value Decompositions

The polar and singular value decompositions are useful ways of breaking linear operators up into simpler parts. In particular, these decompositions allow us to break general linear operators up into products of unitary operators and positive operators . We present, without proof, the following results.

Polar decomposition theorem. Let A be a linear operator on a vector space V. Then there exists unitary U and positive operators J and K such that

where the unique positive operators J and K satisfying these equations are defined by

Moreover, if A is invertible then U is unique. (Here, we call the expression A = UJ the left polar decomposition and A = KU the right polar decomposition.)

Singular value decomposition. Let A be a square matrix. Then there exist unitary matrices U and V, and a diagonal matrix D with non-negative entries such that

The diagonal elements of D are called the singular values of A.

3.4.9 Completeness Relation

The usefulness of the outer product notation becomes evident in the completeness relation for orthonormal vectors. Let $$ \left. {\left| i \right.} \right\rangle $$ be any orthonormal basis set for the vector space V. In this space, an arbitrary vector $$ \left. {\left| v \right.} \right\rangle $$ can be written as $$ \left. {\left| v \right.} \right\rangle = \sum\nolimits_{i} {v_{i} \left. {\left| i \right.} \right\rangle ,} $$ where each $$ v_{i} $$ is a complex coefficient. Note that $$ \left\langle {i |v} \right\rangle = v_{i} . $$ Therefore,

Thus, we have the important relationship

Since this relationship is true for all $$ \left. {\left| v \right.} \right\rangle $$, it follows that

This equation is known as the completeness relation . By complete we mean that any state vector in the Hilbert space can be represented as a weighted sum of just the $$ \left. {\left| i \right.} \right\rangle $$ vectors. The completeness relation is a convenient means for representing any operator in the outer product notation. For example, suppose $$ A:V \to W $$ is a linear operator, $$ \left. {\left| {v_{i} } \right.} \right\rangle $$ is an orthonormal basis for V, and $$ \left. {\left| {w_{j} } \right.} \right\rangle $$ an orthonormal basis for W, then by using the completeness relation twice, we get

which is the outer product representation for A.

3.5 Cauchy–Schwarz Inequality

An important geometrical property of Hilbert spaces is provided by the Cauchy–Schwarz inequality ,

where $$ \left. {\left| v \right.} \right\rangle $$ and $$ \left. {\left| w \right.} \right\rangle $$ are two vectors.

To prove the inequality, let $$ \left. {\left| i \right.} \right\rangle $$ be an orthonormal basis for the vector space such that the first member of the basis $$ \left. {\left| i \right.} \right\rangle $$ is $$ \left. {\left| w \right.} \right\rangle /\sqrt {\left\langle {w |w} \right\rangle } $$. This can always be arranged, say, by using the Gram–Schmidt procedure mentioned in Sect. 3.3.1 (when discussing Inner product). Now by using the completeness relation $$ \sum\nolimits_{i} {\left. {\left| i \right.} \right\rangle \left\langle {\left. i \right|} \right. = I} $$ and dropping some non-negative terms, we get

which establishes the desired inequality. Equality occurs if and only if $$ \left. {\left| v \right.} \right\rangle $$ and $$ \left. {\left| w \right.} \right\rangle $$ are linearly related, that is, $$ \left. {\left| v \right.} \right\rangle = z\left. {\left| w \right.} \right\rangle $$ or $$ \left. {\left| w \right.} \right\rangle = z\left. {\left| v \right.} \right\rangle $$, for some scalar z.

3.6 Pauli Matrices

In quantum computing, four of the most frequently encountered 2 × 2 matrices are noted below along with their several notations prevalent in the literature:

These are known as the Pauli matrices (also, as the σ-matrices). Pauli matrices are Hermitian and unitary, and except for I, have trace zero. They play a central role in the state manipulation of single qubits . Some authors exclude I in the set of Pauli matrices (in this book it is included). One may easily verify the following commutation relations

The importance of the Pauli matrices is that any 2 × 2 matrix M or operator M can be expressed as

where α, β, γ, and δ are complex constants. The 2 × 2 Pauli matrices are closely related to quaternions,¹² which too can be written as 2×2 complex matrices of the form:

where z and w are complex numbers and a, b, c, and d are real numbers.

3.7 Concluding Remarks

As the reader might have realized from the earlier chapters that without mastering linear algebra and complex variables, one cannot hope to learn quantum computing. All data related to the wave function is represented in n-dimensional vectors. This way it is easy to summarize and manipulate data swiftly, with economy of representation, and keeping them within visual range. It is designed for efficient manipulation of data. Linear algebra is all about setting up and solving linear equations using a variety of transformations. In quantum mechanics, the most important transformation is the rotation of the state vector (wave function) in Hilbert space. The abilities to calculate eigenvalues and eigenvectors, to diagonalize matrices, and to find the trace of a matrix are priceless elements of linear algebra . Once you feel comfortable with these elements, dealing with quantum computing will become, if not a cakewalk, something approaching it. This chapter tells you what you must know. After reading this chapter, do keep a good text book on mathematics, e.g., Kreyszig,¹³ nearby for ready reference. Tables 3.1, 3.2 and 3.3 are provided for quick reference.

References

1.

E. Haaparanta, in The Development of Modern Logic (Oxford University Press, 2009)

2.

E. Kreyszig, in Advanced Engineering Mathematics, 10th edn (Wiley Plus, 2011)

3.

M.A. Nielsen, I.L. Chuang, in Quantum Computation and Quantum Information (Cambridge University Press, 2000). Errata at http://​www.​squint.​org/​qci/​

4.

N. Wheeler, in Simplified Production of Dirac Delta Function Identities (1997). https://​www.​reed.​edu/​physics/​faculty/​wheeler/​documents/​Miscellaneous%20​Math/​Delta%20​Functions/​Simplified%20​Dirac%20​Delta.​pdf

5.

G. Zukav, in The Dancing Wu Li Masters: An Overview of the New Physics (Rider, 1979)

Footnotes

1

Logic was first studied by the ancient Greeks, to provide a logical framework to do mathematics. This Aristotelian logic (developed about the fourth century B.C.) provided a list of 14 syllogisms (a form of argument that contains a major premise, a minor premise, and a conclusion.) that were meant to sum up the major ideas of logic. These syllogisms were examples of all the known ways to draw a conclusion (when such a conclusion was possible) from two given statements. Example: (Major premise) All mammals are warm-blooded. (Minor premise) All humans are mammals. Conclusion: Therefore, all humans are warm-blooded. Aristotle’s 14 syllogisms plus another 5 that were added by medieval logicians represented logic for about 2000 years. Thus, to check the validity of an argument in a natural language one showed that it has one of a number of standard forms, or by paraphrasing it into such a form. For development of modern logic, see Haaparanta [1].

2

In 1848, George Boole (1815–1864) invented symbols to facilitate the study of logic in much the same way that symbols are used in algebra. Moreover, Boole organized his study of what we today call symbolic logic along the lines of a deductive system. He selected undefined concepts and axioms and used these to build up the system of symbolic logic. Just as algebra is the study of the ways that numbers can be compared and operated upon, symbolic logic is the study of ways that statements can be compared and operated upon. The concept of “statement” is left undefined except to say that a statement can be either true or false but cannot be both true and false at the same time.

3

When an infinite set of axioms is allowed, it comes with the requirement that we can mechanically check if each statement is an axiom or not. In computer science, this is known as having a recursive set of axioms. An example is the set of Peano axioms in Peano arithmetic, which are: (1) zero is a number; (2) if a is a number, the successor of a is a number; (3) zero is not the successor of a number; (4) two numbers of which the successors are equal are themselves equal; and (5) if a set S of numbers contains zero and also the successor of every number in S, then every number is in S (This is an induction axiom, that is, an axiom scheme comprising infinitely many axioms.).

4

The expected value is something similar to the arithmetic mean. It is formally defined in Chap. 6, Sect. 6.​2.​2.

5

Nielsen and Chuang [3].

6

A: V → W means that A is a mapping from V to W, i.e., the input to A is V and the output of A is W. The space V is called the domain of A, and W the codomain of A. The range of A is the space Y = {y | y ∈ W and y = Ax for some x ∈ V}.

7

First introduced by Paul Dirac in 1926. The curious reader may read Wheeler [4] about the initial controversy that was raised by mathematicians regarding Dirac’s delta function. For example, John von Neumann dismissed the δ-function as a “fiction” and “wrote his monumental Mathematische Grundlagen der Quantenmechanik [Springer, 1932] largely to demonstrate that quantum mechanics can (with sufficient effort!) be formulated in such a way as to make no reference to such a fiction.” “Dirac’s first use of the δ-function occurred in a paper published in 1926, where δ(x − y) was intended to serve as a continuous analog of the Kronecker delta δmn, and thus to permit unified discussion of discrete and continuous spectra.” Dirac introduced it as a “convenient notation” in his influential 1930 book The Principles of Quantum Mechanics. He called it the “delta function” since he used it as a continuous analogue of the discrete Kronecker delta. The delta function is not a “function” but a “distribution.”

8

See, e.g., Nielsen and Chuang [3], p. 66.

9

A† ≡ (AT)* ≡ (A*)T is the Hermitian conjugate or adjoint of matrix A. An operator A whose adjoint is also A is known as a Hermitian or self-adjoint operator.

10

A n × n matrix U is unitary if U†U = UU† = I, where I is the unit matrix. This condition means that U is unitary if and only if it has an inverse which is equal to its conjugate transpose U†.

11

The theorem states that every non-zero single-variable polynomial of degree n with complex coefficients has exactly n complex roots if each root is counted up to its multiplicity.

12

Historical Note. The quaternions are members of a non-commutative division algebra, first invented by Sir William Rowan Hamilton in 1843. The idea struck him while walking along the Royal Canal in Dublin, Ireland, on his way to a meeting of the Irish Academy. He was so pleased with his discovery that he scratched the fundamental formula of quaternion algebra i² = j² = k² = ijk = −1, into the stone of the Brougham bridge. Hamilton called a quadruple with these rules of multiplication a quaternion. Clearly, i, j, and k are three square roots of −1. The quaternions are non-commutative, but they are associative and form a group called the quaternion group. Pauli’s reinvention of the system is the Pauli matrices in the context of quantum mechanics. See, e.g., Weisstein, Eric W., Quaternion, MathWorld—A Wolfram Web Resource, http://​mathworld.​wolfram.​com/​Quaternion.​html.

13

Kreyszig [2, Chaps. 7–12].

© Springer Nature Singapore Pte Ltd. 2020

R. K. BeraThe Amazing World of Quantum ComputingUndergraduate Lecture Notes in Physicshttps://doi.org/10.1007/978-981-15-2471-4_4

4. Some Mathematical Consequences of the Postulates

Rajendra K. Bera1  

(1)

Acadinnet Education Services India, Bangalore, Karnataka, India

Rajendra K. Bera

Email: rajendrabera@yahoo.com

The unreasonable effectiveness of mathematics in the natural sciences.

—Eugene P. Wigner (1960).

A concept is only as good as the theorems that it leads to!

—Gregory Chaitin.

Abstract

This chapter introduces the reader to some known non-classical constraints that Nature imposes on quantum mechanical systems. It highlights, what are now called the “No-go” theorems in quantum computing, their implications, and how they may be circumvented in the design of quantum algorithms. The aim is to get the reader acclimatized to the fact that developing quantum algorithms requires one to work with concepts like superposition, entanglement, and measurement in the presence of the No-go theorems in the alien world of quantum mechanics.

4.1 Introduction

From the postulates and the consequences that follow, we learn that in the Hilbert space, vectors represent states of a quantum system, observable-operators measure quantities, and the eigenvalues of observable-operators represent possible results of measurements of those quantities. Hermitian operators have only real eigenvalues whose corresponding eigenvectors form a unitary basis. In the context of quantum computing, we note that a qubit can have complex linear combinations of Boolean values. A qubit is a minimal physical quantum system and it is measurable by an observable-operator whose spectrum contains only two elements. Of this qubit, quantum mechanics says some amazing and unsuspected things. We mention three of them here: (1) the impossibility of cloning a qubit of unknown state, (2) the impossibility of deleting a qubit of unknown state using unitary operators , and (3) the impossibility of hiding information. They are the result of linearity and unitarity aspects of quantum mechanics.¹

The no-cloning and no-deleting impossibilities taken together (and excluding wave function collapse which is a non-unitary process) provide a quality of permanence of information in a Universe ruled by the Schrödinger equation . Creation of copies can be achieved by importing the information from some other part of the Universe if it exists and its state is known or can be known; deleting a copy can only be achieved by exporting the information out to some other part of the Universe. This is very different from the preservation of information by any reversible dynamics. The no-hiding theorem says that quantum information cannot be completely hidden in correlations.

4.2 No-Cloning Theorem

In 1982, Wootters and Zurek² proved that we cannot make an exact copy of an unknown quantum state. This implies that it is impossible to determine by any kind of measurement or other means the unknown state reliably. Of course, if you know the state, say, because you prepared it in that state, then obviously you can make as many copies as you like. The no cloning property follows from the unitary property of operators.

Theorem 7.2.1

(No-cloning theorem)

Suppose we are given a qubit of an unknown state $$ \left| \psi \right\rangle = \alpha \left| 0 \right\rangle + \beta \left| 1 \right\rangle , $$ where α and β are unknown and $$ \left| \alpha \right|^{2} + \left| \beta \right|^{2} = 1 $$. Then, given an arbitrary state $$ \left| \psi \right\rangle $$ of an unknown qubit and a blank state $$ \left| \Omega \right\rangle $$, there does not exist a transformation

where $$ \left| A \right\rangle $$ and $$ \left| {A_{\psi } } \right\rangle $$ are the initial and final states of the ancilla (it could be the corresponding states of the cloning machine).

Proof

Since a qubit in an orthogonal state such as $$ \left| 0 \right\rangle $$ or $$ \left| 1 \right\rangle $$ carry classical information it can be copied. Let there be a machine that copies a qubit in the orthogonal states. So, we have

If we send an unknown qubit through this cloning machine, then by linearity we have

Ideally, if cloning had been possible, we should have obtained the state

Since the states in the first and the second equation cannot be equal for arbitrary values of α, β, and some choice of $$ \left| {A_{\psi } } \right\rangle $$, it therefore follows that there is no allowed machine that satisfies

Hence, reliable cloning of an unknown qubit is impossible. This contrasts with classical computers which can easily make copies of any classical state.

Although it is impossible to make a copy of an unknown quantum state reliably, it is possible to use quantum gates to copy classical information encoded as $$ \left| 0 \right\rangle $$ or $$ \left| 1 \right\rangle $$, i.e., classical bit copying is possible. We can do this using the controlled-not gate. Let a qubit be in an unknown state $$ \left| \psi \right\rangle = \alpha \left| 0 \right\rangle + \beta \left| 1 \right\rangle $$ and another qubit in the state $$ \left| 0 \right\rangle $$. The input state of the two qubits may be written as

The action of the controlled-not gate to this input is to negate the second qubit when the first qubit is 1, and thus the output is simply $$ \alpha \left| {00} \right\rangle + \beta \left| {11} \right\rangle $$. Thus, in the case where $$ \left| \psi \right\rangle = \left| 0 \right\rangle $$ (i.e., α = 1, β = 0) or $$ \left| \psi \right\rangle = \left| 1 \right\rangle $$ (i.e., α = 0, β = 1) (classical bit states) the gate indeed copies. What is not permitted is qubit copying, i.e., the creation of the state $$ \left( {\alpha \left| 0 \right\rangle + \beta \left| 1 \right\rangle } \right) \otimes \left( {\alpha \left| 0 \right\rangle + \beta \left| 1 \right\rangle } \right) $$ from an unknown state $$ \left( {\alpha \left| 0 \right\rangle + \beta \left| 1 \right\rangle } \right) $$. However, it is possible to clone a known quantum state.

In the multi-qubit case, one can obtain n particles in an entangled state $$ \left( {\alpha \left| {00 \ldots 0} \right\rangle + \beta \left| {11 \ldots 1} \right\rangle } \right) $$ from an unknown state $$ \left( {\alpha \left| 0 \right\rangle + \beta \left| 1 \right\rangle } \right) $$. Each of these particles will behave in exactly the same way when measured with respect to the standard basis for quantum computation {$$ \left| {00 \ldots 0} \right\rangle, \left| {00 \ldots 01} \right\rangle, \ldots, \left| {11 \ldots 1} \right\rangle $$}, but not when measured with respect to other bases. However, it is not possible to create the n particle state $$ \left( {\alpha \left| 0 \right\rangle + \beta \left| 1 \right\rangle } \right) \otimes \cdots \otimes \left( {\alpha \left| 0 \right\rangle + \beta \left| 1 \right\rangle } \right) $$ from an unknown state $$ \left( {\alpha \left| 0 \right\rangle + \beta \left| 1 \right\rangle } \right) $$. In contrast, classical bits can be easily measured, copied, and moved around.

4.2.1 Consequences of the No-Cloning Theorem

There were three interesting repercussions of the no-cloning theorem. First, people initially thought that quantum error correction in quantum computers might be impossible if one was unable to copy qubits of unknown states. In Chap. 11, we will see that this perception is false. It is possible to design quantum error codes using entanglement, which enables the reconstruction of error-free quantum states. The existence of quantum error codes made quantum computers more viable than expected.

Second, the theorem resolved the problem posed by the EPR paradox (see Sect. 4.5.2) that entanglement implied the possibility of signals travelling faster than the speed of light. The no-cloning theorem and the EPR paradox together reveal in a subtle way that non-relativistic quantum mechanics is a consistent theory. For, if cloning was indeed possible, then EPR correlations could be used to communicate with superluminal speed, which would imply that, once special theory was accounted for, an effect could precede a cause. To understand this, note that by generating many clones, and then measuring them in different bases, someone, say, Bob could deduce unambiguously whether his member of an EPR pair is in a state of the basis $$ \left\{ {\left| 0 \right\rangle ,\left| 1 \right\rangle } \right\} $$ or of the basis $$ \left\{ {\left| + \right\rangle ,\left| - \right\rangle } \right\} $$. Alice would communicate instantaneously by forcing the EPR pair into one basis or the other through her choice of measurement axis. The no-cloning result prevents that.

Third, the no-cloning aroused a great deal of interest; the result made it possible to devise a fool-proof key distribution algorithm (see Chap. 1, Sect. 1.​4).

4.3 No-Deleting Theorem

Pati and Braunstein³ have provided an interesting result known as the no-deleting principle. Quantum deletion means the creation of a blank state from two copies by acting jointly on the two copies and the ancilla. Imagine that we have two qubits in some state $$ \left| \psi \right\rangle $$, and an ancilla in some initial state $$ \left| A \right\rangle $$. The no-deletion theorem states that there is no physical operation whose joint action can be represented as

where $$ \left|\Omega \right\rangle $$ is a blank state, and $$ \left| A \right\rangle $$ and $$ \left| {A^{{\prime }} } \right\rangle $$ correspond to the initial and final state of the ancilla. The final state of the ancilla must be independent of the input state because one must exclude hiding of unknown state in the ancilla.

Note that a distinction between the terms “erasure” and “deletion” is being made here. Erasure refers to getting rid of the last bit of information from a collection of unordered bits, whereas deletion refers to resetting the last bit to a standard bit from a collection of identical ordered bits.

Now for an unknown state $$ \left| \psi \right\rangle = \alpha \left| 0 \right\rangle + \beta \left| 1 \right\rangle $$, where α and β are unknown and $$ \left| \alpha \right|^{2} + \left| \beta \right|^{2} = 1 $$, by linearity of the deleting transformation, we have

which is a quadratic polynomial in α and β. However, if $$ \left| \psi \right\rangle \left| \psi \right\rangle \left| A \right\rangle \to \left| \psi \right\rangle \left|\Omega \right\rangle \left| {A^{{\prime }} } \right\rangle $$ is to hold ideally, we should have obtained $$ \left( {\alpha \left| 0 \right\rangle + \beta \left| 1 \right\rangle } \right)\left|\Omega \right\rangle \left| {A^{{\prime }} } \right\rangle $$.

The actual and ideal output states are in general different and hence one can conclude that there cannot be a general-purpose deletion machine. The only case that is not covered is the obvious one where to begin with $$ \left| A \right\rangle = \left|\Omega \right\rangle $$ and one simply swaps $$ \left| A \right\rangle $$ with the second register, i.e.,

But then the second copy of $$ \left| \psi \right\rangle $$ remains in existence (albeit in the ancilla now). The no-deleting principle states that the second copy of $$ \left| \psi \right\rangle $$ can never be “deleted” in the sense that $$ \left| \psi \right\rangle $$ can always be resurrected from $$ \left| {A^{{\prime }} } \right\rangle $$. However, if wave function collapse via a measurement is allowed, then deletion is possible. We simply perform a complete measurement on $$ \left| \psi \right\rangle $$ and rotate the post-measurement state to $$ \left| 0 \right\rangle $$ by a unitary transformation depending on the measurement outcome.

4.4 No-Hiding Theorem

The no-hiding theorem is another landmark result in quantum mechanics. It essentially says that quantum information cannot be completely hidden in correlations.⁴ The theorem was experimentally verified in 2010.⁵ The no-cloning theorem and the no-deleting theorem provide permanence to quantum information. They suggest that in the quantum world information can neither be created nor destroyed. But quantum systems are fragile; any interaction with the environment may lead to loss of information. It is this issue of information loss that the no-hiding theorem addresses. It says that “if any physical process leads to bleaching of quantum information from the original system, then it must reside in the rest of the universe with no information being hidden in the correlation between these two subsystems.”⁶ Thus, the “missing information” can be fully recovered.

4.5 EPR Paradox and Bell Inequalities

An amazing and puzzling result of composite systems is the existence of entangled states. Consider a 2-qubit system with the qubits labeled 1 and 2 with respective state vectors $$ \left| {\psi_{1} } \right\rangle = \alpha_{1} \left| 0 \right\rangle + \beta_{1} \left| 1 \right\rangle , $$ and $$ \left| {\psi_{2} } \right\rangle = \alpha_{2} \left| 0 \right\rangle + \beta_{2} \left| 1 \right\rangle . $$ Therefore, any state of the form $$ \left| {\psi_{1} } \right\rangle \otimes \left| {\psi_{2} } \right\rangle $$ in this composite 2-qubit system can be written as

where the basis vectors for this 2-qubit system are $$ \left| {00} \right\rangle ,\left| {{\text{01}}} \right\rangle {\text{,}}\left| {{\text{10}}} \right\rangle {\text{,}}\left| {{\text{11}}} \right\rangle $$. The probability of the 2-qubit system collapsing to one of its basis vectors on measurement is given by the square of the amplitude of the corresponding vector’s coefficient. For example, $$ \left| { \alpha_{1} \beta_{2} } \right|^{2} $$ gives the probability of finding qubit 1 in state $$ \left| 0 \right\rangle $$ and qubit 2 in state $$ \left| 1 \right\rangle $$ if the 2-qubit system is measured. However, in this composite system we notice something peculiar. For example, the 2-qubit state

called a Bell state , while perfectly legitimate in the 2-qubit system’s Hilbert space, has the remarkable property that there are no single qubit states $$ \left| v \right\rangle $$ and $$ \left| w \right\rangle $$ such that $$ \left| \psi \right\rangle = \left| v \right\rangle \otimes \left| w \right\rangle $$. That is, there are no values of $$ \alpha_{1} ,\beta_{1} ,\alpha_{2} ,\beta_{2} $$ that will yield $$ \left| \psi \right\rangle $$. Such a non-factorizable state of a composite system is called an entangled state.⁷

If we make a measurement on $$ \left| \psi \right \rangle = \left( {\left| {00} \right \rangle + \left| {11} \right \rangle} \right)/\sqrt 2 $$, there is equal (50%) probability of finding both qubits in state 0 or both in state 1. That is, a measurement on either qubit will instantly fix the state of the other qubit—their fates are interlinked—independent of any distance between them. A measurement on the other qubit will now inevitably produce the same measurement as on the first qubit . That is, the measurements are correlated. The state vector $$ \left| \psi \right\rangle $$, nonetheless, represents a perfectly legitimate quantum state. In fact, in quantum mechanics, any vector

where α, β, γ, and δ are complex constants in the 2-qubit Hilbert space, is a valid state for a composite system, irrespective of whether it is factorizable or not. In fact, for composite systems with n qubits , most of the states are non-factorizable; they are the norm rather than the exception! Entangled states play a crucial role in quantum computation and quantum information.

4.5.1 An Analogy for Factorizable States

Factorizable states can be explained as follows. Let Alice and Bob each possess a qubit . Suppose, now, that Alice and Bob prepare their qubits independently, where both operations may depend on the output of a classical random number generator, e.g., a dice. Therefore, the source of the correlations is a classical random number generator. The 2-qubit states which Alice and Bob can prepare in this way using their respective dice are called classically correlated or separable. All other 2-qubit states are called entangled. Separable states occur when the dice are not related to each other. If the dice are related (e.g., if one dice showed 2 the other would invariably show 5, etc.), the result would have been entangled states. Clearly, related dice function as a unit and can only acquire a group state. In a group state, it is impossible to measure one without learning something about the other. Their properties are inextricably linked. Likewise, quantum entanglement means that one cannot describe an entangled quantum system in terms of just local descriptions, one for each component.

Since an entangled state cannot be factorized, it is impossible to specify a pure state of its constituent components; we can only specify a group state. Interestingly, this group state is a pure state in the Hilbert space! That is, we know the correlation between measurement outcomes on qubits 1 and 2 but we cannot, even in principle, identify a pure state with each of the qubits 1 and 2 individually. Entanglement demonstrates a fundamental difference between classical and quantum physics.

4.5.2 Einstein, Podolsky, Rosen Pose a Paradox

The entanglement enigma was brought into focus in a famous paper⁸ in 1935 by Einstein, Podolsky, and Rosen (EPR). They argued that the strange behavior of entanglement meant that quantum mechanics was an incomplete theory, and that there must be “hidden variables” (or additional variables) which, when discovered and accounted for, would complete the theory. In other words, what the EPR group suggested was that each particle has some internal state, which completely determines the result of any measurement. It is just that this state is hidden from us. Hence, the best we can do is make probabilistic predictions. Theories based on such assumptions are known as hidden variable theories. The EPR paradox stated below is a variant of the original one and was put forward by David Bohm.⁹

Suppose that two spin-1/2 particles such as an electron and a positron are created by the decay of a single spin-zero particle at some central point and the two particles move outward in opposite directions. Conservation of angular momentum requires that the spins of the two particles add up to zero since that was the angular momentum of the initial central particle. In quantum mechanical terms, let $$ \left| Q \right\rangle $$ represent the combined zero-angular-momentum state of the two particles. Then

where e and p, respectively, refer to the electron and the positron. Note that the conservation of angular momentum permits the superposition of only two joint spin states: $$ \left| { \uparrow e} \right\rangle \left| { \downarrow p} \right\rangle $$ and $$ \left| { \downarrow e} \right\rangle \left| { \uparrow p} \right\rangle $$. The two particles are born entangled! What is implied here is that when, say, the spin of the electron is measured in any arbitrary direction, the positron must immediately spin in the opposite direction no matter where in the Universe each is located.

The original paper by the EPR group produced a historic debate between Albert Einstein and Niels Bohr. Bohr argued that quantum mechanics was complete, and that Einstein’s problems arose because he tried to interpret the theory too literally, e.g., the EPR view cannot explain measurement results with respect to a different basis. The core issue was non-local behavior: whether a change in a quantum particle can result in an instantaneous effect on another distant particle.

Today, most physicists prefer the Copenhagen interpretation (see Chap. 2, Sect. 2.​10.​1). It says there is no deeper reality, no hidden variables—the world is simply probabilistic. We are not ignorant about quantum objects, it is just that there is nothing further to be known. Quantum correlations (entanglements) are real. In fact, so real that even when the two particles are separated by galactic distances, communication between the particles is unnecessary to explain the correlations. That is why entanglement is a non-local phenomenon. Further, note that although after a measurement, both qubits have the same state when

it does not imply a cloning operation, since no well-defined state can be attributed to a subsystem of an entangled state. The term cloning refers to a process whose result is a separable state with identical forms, such as $$ \left| \psi \right\rangle = \left| v \right\rangle \otimes \left| v \right\rangle $$. Quantum mechanical experiments have shown that there is no faster-than-light signal involved between entangled quantum entities (see Chap. 1, Sect. 1.​5). Rather than communicating, entangled pairs share the same existence, a joint destiny if you like, so that changes occur instantly and jointly. Further, any number of quantum entities can be entangled. Thus, entanglement allows a quantum computer to manipulate all of its qubits at the same time.

All his life, Einstein was skeptical of quantum theory even though some of his ideas were fundamental to it, e.g., his 1905 concept of the “photon”—the quantum of the electromagnetic field out of which the idea of wave-particle duality was developed by Louis de Broglie in 1924 in his Ph.D. thesis. The concept of the “boson” was partly Einstein’s along with Satyendra Nath Bose, as were many other ideas central to quantum theory. Yet he had great aversion to the probabilistic aspect of the theory. In 1926, in reply to a letter from Max Born, he wrote:

Quantum mechanics is very impressive. But an inner voice tells me that it is not yet the real thing. The theory produces a good deal but hardly brings us closer to the secret of the Old One. I am at all events convinced that He does not play dice.¹⁰

So “How reliable is our inner voice?” Notwithstanding Einstein’s reservations, the Nobel Prize in Physics for 1954 was divided equally between Max Born “for his fundamental research in quantum mechanics, especially for his statistical interpretation of the wavefunction” and Walther Bothe “for the coincidence method and his discoveries made therewith.”¹¹

4.5.3 What Does Hidden Variable Theory Mean?

A hidden variable theory is a rule for converting a unitary transformation into a classical probabilistic transformation, i.e., it is a function that takes as input an n × n unitary matrix U together with a quantum state

and produces an n × n stochastic matrix S, such that given as input the probability vector obtained from measuring $$ \left| \psi \right\rangle $$ in the standard basis, S produces as output the probability vector obtained from measuring $$ U\left| \psi \right\rangle $$ in the standard basis. Only then can we say that a hidden-variable theory reproduces the predictions of quantum mechanics.¹²

A stochastic matrix is a non-negative matrix in which every column sums to 1. Mathematically, if

then we must have

where

Such a stochastic matrix, $$ S $$, always exists, e.g.,

4.5.4 Bell Inequality

In a remarkable paper in 1964, John Bell pointed out that for certain experiments classical hidden variable theories made different predictions from quantum mechanics—the measurement correlations in the Bell state are stronger than could ever exist between classical systems.¹³ In fact, he published a theorem, which quantified just how much more strongly quantum particles were correlated than would be classically expected, even if hidden variables were taken into account. This made it possible to test whether quantum mechanics could be accounted for by hidden variables.

Suppose Charlie prepares two particles and can prepare as many copies as he wants. He sends one particle to Alice and another to Bob. Assume that Alice has measurement apparatuses AQ and AR, and Bob has apparatuses AS and AT, and that the values each apparatus can output is either +1 or −1. Let us further assume that Alice and Bob carry out measurements on their respective particles simultaneously at a predetermined time, selecting randomly one of the two measurement apparatuses in their possession. Simultaneity of measurement and random choice ensures that measurements made by Alice and Bob are independent of each other. For convenience, let the outputs of AQ, AR, AS, and AT be, respectively, Q, R, S, and T.

Consider now the expression QS + RS + RT − QT. In particular, note that

Since R, Q = ±1, then either (Q + R) S = 0 or (R − Q) T = 0. In either case QS + RS + RT – QT = ±2.

Suppose next that p(q, r, s, t) is the probability that, before measurements are performed, the system is in a state Q = q, R = r, S = s, and T = t. Note that these probabilities may depend on how Charlie prepares the qubits , and on experimental noise. Let E(·) denote the mean value of a quantity. Then,

Alternatively,

We thus have the inequality

This particular form is known as the CHSH inequality after its discoverers Clauser, Horne, Shimony, and Holt.¹⁴ It is one of a larger class of inequalities generically known as Bell inequalities , since John Bell found the first of such inequalities.

By repeating the experiment many times (that is why Charlie should be able to make, as many identical particle pairs as required), Alice and Bob can measure each quantity on the left-hand side of the Bell inequality and determine if the inequality is satisfied or not. Note that the result is not specific to quantum mechanics—no axioms or results of quantum mechanics were invoked. From a common-sense point of view, we would expect the inequality to be true. So, let us look at a quantum system. Suppose Charlie prepares 2-qubits in the state

and passes one qubit to Alice and the other to Bob. Both measure their respective qubit at the appointed time. Say, their measurements¹⁵ reveal the following:

where X and Z are the Pauli operators (representing observable quantities) operating on the qubit denoted by their subscript (1 denotes the qubit held by Alice and 2 the qubit held by Bob). One may easily verify that the average values of these observable-operators are

Thus,

This certainly violates the Bell inequality ! Thus, to determine if a state is entangled, Alice and Bob can do correlation experiments. For example, they can test Bell’s inequalities. These inequalities are fulfilled for all classically correlated states but not for entangled states. Entanglement is a physically observable phenomenon; entangled particles are now routinely produced in experiments.

In 1982, Aspect et al.¹⁶ provided experimental evidence of entanglement. In February 2017, even more stringent experimental evidence of entanglement was provided by Handsteiner et al.¹⁷ Thus, when two or more particles are entangled, a measurement on any one particle or a combined measurement on a subgroup of particles will cause a “collapse” to occur instantly on the remaining particles no matter where they are in the Universe. A group of entangled particles thus have a distributed existence yet function as a single unit. Thus, quantum theory is correct and local hidden variable theories are wrong. Quantum physics has never been the same again.

4.5.5 An Intriguing Question

This section is adapted from Aaranson.¹⁸

Consider the wave function $$ \left| \psi \right\rangle $$ = (3/5) $$ \left| X \right\rangle $$ + (4/5) $$ \left| Y \right\rangle $$, where $$ \left| X \right\rangle $$ represents the pretty girl and $$ \left| Y \right\rangle $$ represents the old woman of Fig. 1.​1 in Chap. 1. Say, on measurement, this will collapse to $$ \left| X \right\rangle $$ with probability 9/16 or to $$ \left| Y \right\rangle $$ with probability 16/25. Suppose when your eyes spotted the picture your brain went into the superposed state $$ \left| \psi \right\rangle $$, which your mind then interpreted (measured), resulting in the collapse of the wave function to $$ \left| \psi \right\rangle $$, the pretty girl. Now imagine that your brain subsequently underwent a unitary operation such that the wave function changed to $$ \left| \psi \right\rangle $$ = (4/5) $$ \left| X \right\rangle $$ + (3/5) $$ \left| Y \right\rangle $$, which means that your mind now has a probability of 16/25 of seeing $$ \left| X \right\rangle $$. Interesting! But conditioned on seeing the pretty girl earlier, what is the probability that you will see her again at the later time?

Amazingly, in quantum mechanics, this turns out to be a meaningless question. (Did your course on probability theory prepare you for this?) The measurement postulate of quantum mechanics provides you with the probability of getting a certain outcome if you make a measurement at a certain time. Nothing more, nothing less. It does not give you multiple-time or transition probabilities. For example, it does not tell you the probability of an electron being found at point y at time t + 1 given that had you measured the electron at time t (which you didn’t), it would have been at point x, because at time t it wasn’t anywhere (it was in a state of superposition!). On the other hand, if you did measure it at time t, then the entire sequence from time t to t + 1 would be a completely different experiment.

A philosophical question: “Till you observe a quantum mechanical system, what do you know of its past history?” “What is the meaning of history for a quantum mechanical system?” “Does history last only for 10⁻⁴³ s?” “If history is a questionable notion, what does it mean to make a prediction?” Indeed, quantum mechanics is silent about what, if anything, is likely to be true in the absence of observation. Furthermore, it is essentially statistical. The probabilities built into the state function are fundamental. That is, the probabilities do not arise from ignorance of fine details as happens in classical statistical mechanics.

4.5.6 Returning to the Bell Inequality

The Bell results provided the first deep insight that quantum mechanics allows information processing beyond what is possible in the classical world. It drew attention to the importance of correlations between separated quantum systems which have interacted (directly or indirectly) in the past, but which no longer influence one another. Therefore, there can be no “easy” explanation of the entangled correlations. The only kind of hidden variables not ruled out by the Bell tests are “non-local,” meaning that the hidden variables would be able to act instantaneously across a distance no matter how long the distance. Thus, quantum mechanics cannot be explained by any local hidden variable theory.

Bell inequalities are relations satisfied by the average values of product of random variables that are correlated classically (their correlations arise from the fluctuations of some common cause in the past). Entangled pairs indeed behave non-classically and in the way predicted by quantum mechanics. The Bell inequalities are indeed compelling examples of an essential difference between quantum and classical physics.

What does it all mean? There are two “obvious” assumptions implicit in the derivation of the Bell inequality : (1) that measurement values (of physical properties) such as Q, R, S, and T exist independent of observation (also known as the assumption of realism), and (2) that measurements made by Alice and Bob are independent of each other’s (also known as the assumption of locality¹⁹). Together the two assumptions are sometimes known as the assumptions of local realism. Clearly, either or both assumptions—locality and realism—must be dropped to reconcile with experimental evidence. To my knowledge, there is no conclusive answer as to which must be dropped.²⁰ The Bell inequality emphatically states that the world is not locally realistic.

Henry Stapp has noted that

… “Bell’s Theorem” … is widely known not only among physicists, but also to philosophers, journalists, mystics, novelists, and poets. … assumptions that not only can be stated in entirely nontechnical terms but are so compelling that the establishment of their falsity has been called, not frivolously, “the most profound discovery of science.”²¹

4.5.7 Would Newton Have Approved of Entanglement?

It is inconceivable that inanimate Matter should, without the Mediation of something else, which is not material, operate upon, and affect other matter without mutual Contact … That Gravity should be innate, inherent and essential to Matter, so that one body may act upon another at a distance thro’ a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it. Gravity must be caused by an Agent acting constantly according to certain laws; but whether this Agent be material or immaterial, I have left to the Consideration of my readers.—Isaac Newton, Letters to Bentley, 1692/3

Newton certainly would not have believed in entanglement! As we have noted earlier, it did not sit well with Einstein, Podolsky, and Rosen either who tried to use the apparent absurdity of the predicted effects of entanglement to prove that quantum mechanics gave an incomplete description of physical reality. Experiments since the 1980s have verified time and again that entanglement is indeed a reality in Nature. However, we don’t know how it works. Nature is far stranger than man-made fiction.

4.6 Superposition and Indeterminacy

Superpositions have no classical interpretation. The non-classical nature of superposition of states and measurement outcomes are striking features of the postulates of quantum mechanics . Unlike a classical system in a superposition of two states, say s₁ and s₂, when measured will produce s₁ + s₂; a quantum system similarly superimposed in the wave function, when measured will randomly produce, according to the probabilities defined by Postulate 3 (see Chap. 2, Sect. 2.​7.​3) an output which is either s₁ or s₂, and not some intermediate result derived from s₁ and s₂. As Dirac states:

The intermediate character of the state formed by superposition thus expresses itself through the probability of a particular result for an observation being intermediate between the corresponding probabilities for the original states, not through the result itself being intermediate between the corresponding results for the original states.²²

The source of this essential departure from classical mechanics is that the quantum evolution of the system and its observation are governed by two different postulates. Because of this, superposition cannot be explained in terms of familiar physical concepts that require a system in superposed states to be in some vague way intermediate between those of the individual states comprising a given superposed state; the meaning of superposition in quantum mechanics is of an essentially different nature from any occurring in classical mechanics. Postulates 2 and 3 jointly demand indeterminacy in the results of observations in order to be capable of a sensible physical interpretation.

4.7 Mathematical Consequences

The remarkable conceptual differences between Euclid’s axiomatization of geometry and of quantum mechanics are indeed breathtaking. Postulates or axioms are unprovable beliefs. In Euclid’s case, the postulates appear intuitively self-evident; in quantum mechanics, they do not. To understand quantum mechanics, we need to suspend most of our intuitions about the world and what we believe reality to be. The reader is forced to deal with quantum mechanics by strictly adhering to its postulates rather than seek cues from the classical world. He needs to carefully understand the meaning of the state of a quantum system, and what it means to observe such a system. Mathematical reasoning is the only sure guide to navigating within the Hilbert space, and in noting the important fact that in making the transition to a wave equation , the role of physical variables is taken over by “observable-operators.”

A most intriguing aspect of quantum mechanics is the “collapse” of the wave function when a quantum system is observed or measured. Max Born’s insight that the wave function should be interpreted in terms of probabilities (Postulate 3) is nothing short of dramatic. For example, if we measure the location of an electron, the probability of finding it in a given region depends on the intensity of its wave function there. This statistical interpretation of the wave function suggested that a fundamental randomness was built into the laws of Nature. Einstein never accepted such an interpretation (most of us accept because a professor of physics or a text book says so!). In a letter to Born he made the famous remark: “I can’t believe that God plays dice.”²³ Einstein was not alone; neither did Erwin Schrödinger nor Louis de Broglie.

The second unusual aspect is Heisenberg’s Uncertainty Principle ²⁴ which categorically limits the accuracy with which a pair of canonically conjugate variables can be measured at the same time no matter how accurate the instruments used. For the moving electron, the canonically conjugate pairs are: (1) momentum and position, and (2) energy and time. In the extreme case, absolute precision of one variable in a pair would entail absolute imprecision regarding the other. This is in sharp contrast to classical physics where “if we know the present exactly, we can calculate the future.” In quantum mechanics, it is not the conclusion that is wrong but the premise. Thus, one cannot calculate the precise future motion of a particle, but only a range of its possible motion. (However, the probability associated with each motion, and the distribution among many particles following these motions, can be calculated exactly from Schrödinger’s wave equation .) If one cannot know the precise position and momentum of a particle at a given instant, its future cannot be determined.

The third unusual aspect is entanglement, a consequence of Postulates 1, 2, and 4, which mandates that quantum systems evolve in Hilbert space, and that this space be composed of, in terms of the spaces of its subsystems, not as Cartesian products but as tensor products. A group of entangled particles react to changes simultaneously and instantly no matter where they are in the Universe. This seems to contradict the very core of Einstein’s special theory of relativity which postulates that nothing can travel faster than the speed of light. Indeed, it was this discrepancy that led Einstein, Podolsky, and Rosen (EPR) to question the veracity of non-relativistic quantum theory by posing the EPR paradox. They hinted that unaccounted “hidden variables” exist which would explain the outcome of measurements, and that non-local, “action at a distance” was mythical. In short, quantum theory was incomplete. The EPR paper was important because it was carefully argued and, as it turned out, the fallacy was hard to find. In 1964, Bell brilliantly ferreted out the fallacy in terms of the Bell inequalities that could be tested experimentally. Experiments by Aspect, Dalibart, and Roger reported in 1982 and later experiments as noted in Sect. (4.​5.​4) have verified that reality is indeed non-local, and that nature does provide for entanglement. There is no counterpart of entanglement in classical physics. It is intriguing that two amazing results of quantum mechanics—Heisenberg’s uncertainty principle and Bell inequality —have roots outside of quantum mechanics.

The fourth unusual aspect is that no matter how delicately a quantum system is measured, it inevitably affects it, and usually does so in a dramatic manner, unless the system is already in the observable eigenstate. We have no idea how a quantum system knows that it is being measured, i.e., when it is evolving according to Postulate 2 and when it is collapsing according to Postulate 3. Indeed, one cannot “deduce” the measurement rule (Postulate 3) as a complicated instance of the Schrödinger equation (Postulate 2). The measurement rule is simply a different procedure; it contains all the non-determinism found in the theory. Both postulates are needed for all the remarkable agreements that quantum theory has so far provided with observational facts.²⁵ The measured output is the joint product of “system” and “apparatus,” i.e., the complete experimental setup; in classical physics, it is the product of the “apparatus.”

The fifth unusual aspect is the unexpected prohibitions imposed by the “No cloning” and the “No deleting” theorems. Together they provide a quality of permanence of information that is very different from the preservation of information by any reversible dynamics.

The sixth unusual aspect is complementarity or wave-particle duality, the central mystery of quantum mechanics (see Chap. 2, Sect. 2.​2.​2).

Notwithstanding these unusual aspects, within a few years, even though no one understood what quantum mechanics means, it was used to explain very successfully a large number of unexplained measurements, including the spectra of complicated atoms and various chemical reactions. While physicists have now developed an abiding faith in quantum mechanics, the question remains: “What does it all mean?” The honest answer is, “We do not know!” But because of the striking success of the theory in predicting physical phenomena, most physicists have gone ahead and worked out many exciting applications of the theory including hitherto unsolved problems of nuclear physics rather than dwell on the ontological²⁶ implications of the equations. In a way that was good. Quantum mechanics was instrumental in predicting anti-matter, understanding radioactivity (hence, nuclear power), accounting for materials such as semiconductors, explaining superconductivity, describing interactions such as those between light and matter (leading to the invention of the laser), and of radio waves and nuclei (leading to magnetic resonance imaging), and the invention of the electron microscope. Many successes of quantum mechanics have come from its extension, quantum field theory, which forms the foundation of elementary particle physics, including neutrino oscillations, search for the Higgs boson, and super-symmetry.²⁷

Quantum physics, which comprises the most precisely tested theories in the history of science, has revolutionized twentieth-century science and society. Its applications in industry, especially in electronics- and photonics-based technologies, have been phenomenal. They include the development of the transistor, which led to Nobel Prizes in Physics (1956) for William B. Shockley, John Bardeen, and Walter Houser Brattain; the development of the laser, which led to Nobel Prizes in Physics (1964) for Charles Hard Townes, Nicolay Gennadiyevich Basov, and Aleksandr Mikhailovich Prokhorov; development of the microchip, which led to Nobel Prizes in Physics (2000) for Zhores I. Alferov, Herbert Kroemer, and Jack S. Kilby; and the invention of efficient blue light-emitting diodes which led to Nobel Prizes in Physics (2014) for Isamu Akasaki, Hiroshi Amano, and Shuji Nakamura.

4.8 Concluding Remarks

In a paper delivered at the Solvay Congress of 1927, Heisenberg and Born had said:

We regard quantum mechanics as a complete theory for which the fundamental physical and mathematical hypotheses are no longer susceptible of modification.²⁸

This was quite a bold statement considering that in 1931 Gödel²⁹ would come out with a remarkable paper about axiomatic systems, possibly the most important paper in the whole of mathematics that would formally establish the following:

Gödel’s first theorem: In any axiomatic system sufficiently strong to allow one to do basic arithmetic, one can construct a statement that either can be neither proven nor disproven within that system (that is, the system is incomplete), or can be both proven and disproven within that system (that is, the system is inconsistent).

Gödel’s second theorem: A sufficiently strong consistent system cannot prove its own consistency.

Despite its success, quantum mechanics because of Postulates 2 and 3 gives rise to a perplexing question: “Do unobserved particles possess physical properties that exist independent of observation?” Physicists are still struggling to provide an answer. Quantum non-locality remains an enigma wrapped in a mystery. For some recent developments on the subject, see the book edited by Vaidman.³⁰ A bridging theory that would firmly connect quantum mechanics with classical mechanics remains elusive. Bohr’s speculative correspondence principle is about all we have (see Chap. 2, Sect. 2.​9). Even Bohr was pessimistic:

The repeatedly expressed hopes of avoiding the essentially statistical character of quantum mechanical description by the assumption of some causal mechanism underlying the atomic phenomena and hitherto inaccessible to observation would indeed seem to be as vain as any project of doing justice to the increased profundity of the picture of the world achieved by the general theory of relativity by means of the ordinary conceptions of absolute space and time. Above all such hopes would seem to rest upon an underestimate of the fundamental differences between the laws with which we are concerned in atomic physics and the every day experiences which are comprehended so completely by the ideas of classical physics.³¹

Recently, Peter Renkel with remarkable insight has hypothesized a bridge. He “starts from the generalization of a point-like object and naturally arrives at the quantum state vector of quantum systems in the complex valued Hilbert space, its time evolution and quantum representation of a measurement apparatus of any size. … [He shows] that a measurement apparatus is a special case of a general quantum object. [Finally, he provides an] example of a measurement apparatus of an intermediate size ….”³² This is a paper worth studying carefully.

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A. Aspect, in Testing Bell’s Inequalities, (1991), pp. 415–425, http://​inspirehep.​net/​record/​1406213/​files/​C91-01-26_​415-426.​pdf (For a video, see http://​cds.​cern.​ch/​record/​423022) The text presented is very close to the one that Aspect prepared for the special issue of Europhysics News; A. Aspect, Testing Bell’s inequalities. Europhys. News 22, 73–75 (1991), on John Bell and Quantum Mechanics (issue of April 1991)

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A. Aspect, J. Dalibard, G. Roger, Experimental test of Bell’s inequalities using time-varying analyzers. Phys. Rev. Lett. 49(25), 1804–1807 (1982). http://​www.​drchinese.​com/​David/​Aspect.​pdf

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S.L. Braunstein, A.K. Pati, Quantum information cannot be completely hidden in correlations: implications for the black-hole information paradox. arXiv:​gr-qc/​0603046v1, http://​arxiv.​org/​pdf/​gr-qc/​0603046v1.​pdf, 13 Mar 2006

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A. Einstein, B. Podolsky, N. Rosen, Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 41, 777–780. http://​www.​drchinese.​com/​David/​EPR.​pdf Known as the EPR paper, it claimed that QM was an incomplete theory. After Einstein died in 1955, John Bell and others would prove him wrong

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A. Fine, The Einstein-Podolsky-Rosen argument in quantum theory, in Stanford Encyclopedia of Philosophy. http://​plato.​stanford.​edu/​entries/​qt-epr/​, 10 May 2004

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K. Gödel, Über formal unentseheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik 38, 173–198 (1931) (On Formally Undecidable Propositions of Principia Mathematica and Related Systems I.) (Visit http://​jacqkrol.​x10.​mx/​assets/​articles/​godel-1931.​pdf for an English translation by B. Meltzer.)

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J. Handsteiner et al., Cosmic Bell test: measurement settings from milky way stars. Phys. Rev. Lett. 118, 060401 (2017). https://​journals.​aps.​org/​prl/​pdf/​10.​1103/​PhysRevLett.​118.​060401

18.

W. Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik Zeitschr, (The actual content of quantum theoretical kinematics and mechanics), Physics 43(3–4), 172–198 (1927). Translation available at J.A. Wheeler, W.H. Zurek, in Quantum Theory and Measurement (Princeton University Press, N.J., 1983), pp. 62–84

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B. Hensen et al., Loophole-free Bell inequality violation using electron spins separated by 1.3 km. Nature 526, 682–686 (2015). Preprint at https://​arxiv.​org/​abs/​1508.​05949

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M.-N. Luo, H.-R. Li, H. Lai, X. Wang, Unified quantum no-go theorems and transforming of quantum states in a restricted set. arXiv:​1701.​04166v2 [quant-ph]. https://​arxiv.​org/​pdf/​1701.​04166.​pdf, 19 March 2017

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Z. Merali, Toughest test yet for quantum ‘spookiness’. Nature 525, 14–15 (2015) http://​www.​nature.​com/​polopoly_​fs/​1.​18255!/​menu/​main/​topColumns/​topLeftColumn/​pdf/​nature.​2015.​18255.​pdf

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N.D. Mermin, Hidden variables and the two theorems of John Bell. Rev. Mod. Phys. 65(3), 803–815 (1993). https://​upload.​wikimedia.​org/​wikipedia/​commons/​6/​68/​Variables_​ocultas_​y_​los_​teoremas_​de_​Bell.​pdf Erratum: Hidden variables and the two theorems of John Bell. Rev. Mod. Phys. 65, 803 (1993) 89, 049901—Published 28 December 2017. “In the last sentence of the first complete paragraph on page 814 the word “noncontextuality” should be ‘contextuality.’” https://​journals.​aps.​org/​rmp/​abstract/​10.​1103/​RevModPhys.​89.​049901

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News (20070227), in A Hidden Twist in the Black Hole Information Paradox (University of York, 27 February 2007). https://​www.​york.​ac.​uk/​news-and-events/​news/​2007/​blackhole/​

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J. Ortigoso, Twelve years before the quantum no-cloning theorem, arXiv:​1707.​06910v2 [physics.hist-ph], https://​arxiv.​org/​pdf/​1707.​06910.​pdf, 22 February 2018

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A. Pais, in ‘Subtle is the Lord …’: The Science and Life of Albert Einstein (Clarendon Press, Oxford, 1982)

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J.L. Park, The concept of transition in quantum mechanics. Found. Phys. 1(1), 23–33 (1970)

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A.K. Pati, S.L. Braunstein, Impossibility of deleting an unknown quantum state. Nature 404, 164–165 (2000). http://​www-users.​cs.​york.​ac.​uk/​~schmuel/​papers/​pb00.​pdf

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R. Penrose, in The Emperor’s New Mind (Vintage, 1990)

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A. Peres, How the no-cloning theorem got its name. arXiv:​quant-ph/​0205076v1, http://​arxiv.​org/​PS_​cache/​quant-ph/​pdf/​0205/​0205076v1.​pdf. (Peres notes that Asher reported the title of Wooters and Zurek’s paper on no-cloning was contributed by John Wheeler.), 14 May 2002

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P. Renkel, Building a bridge between classical and quantum mechanics. arXiv:​1701.​04698v2 [physics.gen-ph], https://​arxiv.​org/​pdf/​1701.​04698.​pdf, 06 Nov 2017

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J.R. Samal, A.K. Pati, A. Kumar, Experimental test of quantum no-hiding theorem. arXiv:​1004.​5073v1 [quant-ph], http://​arxiv.​org/​pdf/​1004.​5073v1.​pdf, 28 Apr 2010

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J.R. Samal, A.K. Pati, A. Kumar, Experimental test of the quantum no-hiding theorem. Phys. Rev. Lett. 106, 080401 (2011)

33.

E. Schrödinger, Die gegenwärtige Situation in der Quantenmechanik. Die Naturwissenschaften 23, 807–812, 823–828, 844–849 (1935); in English translation in Quantum Theory and Measurement, eds. by J.A. Wheeler, W.H. Zurek (Princeton University Press, 1983)

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H. Stapp, Are superluminal connections necessary? Nuovo Cimento 40B, 191–204 (1977). http://​www-physics.​lbl.​gov/​~stapp/​NCimento.​pdf

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C. Tresser, The simplest Bell’s theorem, with or without locality, arXiv:​quant-ph/​0501030v4, 14 Jun 2005 (revised 1 Feb 2008), available at http://​arxiv.​org/​PS_​cache/​quant-ph/​pdf/​0501/​0501030v4.​pdf

36.

L. Vaidman (ed.), Quantum Nonlocality (MDPI Books, June 2019). https://​www.​mdpi.​com/​books/​pdfdownload/​book/​1340

37.

W.K. Wootters, W.H. Zurek, A single quantum cannot be cloned. Nature 299, 802–803 (1982). http://​puhep1.​princeton.​edu/​~mcdonald/​examples/​QM/​woottersnature29​980282.​pdf

Footnotes

1

The three examples are members of a group of theorems colloquially known as the No-Go theorems. More examples can be found in Luo et al. [20].

2

Wooters and Zurek [37]. See also: Peres [29]. As Asher reports, the title of the paper was contributed by John Wheeler. Historical note: Ortigoso [24] reports that the no-cloning theorem had already been stated by Park [26]. Wootters and Zurek, and independently, Dieks [12] rediscovered the theorem.

3

Pati and Braunstein [27].

4

Braunstein and Pati [9, 10]; News (20070227) [23].

5

Samal et al. [31, 32].

6

Samal et al. [31, 32].

7

The term “entanglement” is a translation of the German “Verschränktheit” coined by Schrödinger [33].

8

Einstein et al. [14]. Podolsky and Rosen were postdoctoral research associates of Einstein at the Institute for Advanced Study. See also: Fine [15]. The problem in a simple form was first raised by Einstein in 1928 at the 5th Solvay Conference. See also: Bohr [6], which has Niels Bohr’s response to EPR.

9

See, e.g., Bohm [5]. Bohm discusses some of his ideas concerning hidden variables.

10

Source of quote: Pais [25, p. 443]; as reproduced in Penrose [28, p. 361]. See also: Born [8], p. 91. (In the literature the actual wording varies in translation from German to English.)

11

The Nobel Prize in Physics 1954. NobelPrize.org. Nobel Media AB 2018. https://​www.​nobelprize.​org/​prizes/​physics/​1954/​summary/​.

12

See, e.g., Aaronson [1].

13

Bell [4]. The mathematical proof is brilliant and relatively straightforward.

14

Clauser et al. [11].

15

Note that in quantum mechanics, physical variables take the form of “operators.”

16

Aspect et al. [3]. This paper provides experimental evidence that over-ruled Einstein’s objections described in his EPR paper. See also Aspect [2].

17

Handsteiner et al. [17]. See also: Hensen et al. [19]; and Merali [21].

18

Aaranson [1].

19

If two systems are far enough apart, the measurement of one system does not directly affect the reality that pertains to the unmeasured system, i.e., the behavior of a physical system is determined solely by the forces and influences that arise in the immediate vicinity. Quantum entanglement leads to non-local of “ghostly-action-at-a-distance” effects.

20

See, e.g., Tresser [35].

21

Quote as reproduced in Mermin [22]. See also: Stapp [34].

22

Dirac [13, p. 13].

23

See Born [8], p. 91.

24

Heisenberg [18].

25

Penrose [28, pp. 323–4].

26

Ontological: Of or relating to essence or the nature of being. In the context of knowledge sharing, it is a description (like a formal specification of a program) of the concepts and relationships that can exist for an agent or a community of agents; an explicit specification of a conceptualization.

27

Neutrino oscillation: Neutrino oscillation is a quantum mechanical phenomenon whereby a neutrino created with a specific lepton flavor (electron, muon, or tau) changes to a different flavor.

Higgs boson: The existence of the particle was announced in 2012.

Super-symmetry: For every type of boson, there exists a corresponding type of fermion, and vice versa.

28

As quoted at: Quantum Mechanics, 1925–1927: Triumph of the Copenhagen Interpretation , http://​www.​aip.​org/​history/​heisenberg/​p09.​htm.

29

Gödel [16].

30

Vaidman [36].

31

Bohr [7].

32

Renkel [30].

© Springer Nature Singapore Pte Ltd. 2020

R. K. BeraThe Amazing World of Quantum ComputingUndergraduate Lecture Notes in Physicshttps://doi.org/10.1007/978-981-15-2471-4_5

5. Waves and Fourier Analyses

Rajendra K. Bera1  

(1)

Acadinnet Education Services India, Bangalore, Karnataka, India

Rajendra K. Bera

Email: rajendrabera@yahoo.com

“Waves are the voices of tides. Tides are life,” murmured Niko. “They bring new food for shore creatures, and take ships out to sea. They are the ocean’s pulse, and our own heartbeat.”

―Tamora Pierce, Sandry’s Book

Abstract

This chapter is meant to refresh the reader about waves and Fourier analysis and explain their importance in quantum mechanics. The presentation assumes that the reader already knows about their importance in classical mechanics.

5.1 Introduction

Wave phenomena appear in many contexts throughout physics and are related to oscillating systems. Wave oscillations may appear not only as time-oscillations at one place but propagate in space as well. The weirdness of quantum mechanics is that a quantum object shows both particle-like and wave-like behavior (complementarity; see Chap. 2, Sect. 2.​2.​2). A knowledge of waves and Fourier analysis is therefore central to quantum computing.

5.2 Waves

Waves are disturbances that undergo cyclic changes and are often modeled as sinusoidal functions of time (t) or some other variable. The amplitude of a wave is the measure of the magnitude of the maximum disturbance a wave undergoes in a wave cycle. Waves have crests (highs) and troughs (lows). Consider, e.g., the function

and the wave represented by

where $$ s\left( t \right) $$ is a disturbance that varies periodically with time t, A is the wave’s amplitude, f is the wave’s frequency, and θ is a constant, and mod is the modulo operation. The initial phase (at $$ t = 0 $$) of this sinusoid is the initial angle $$ \varphi \left( 0 \right) = \theta $$, which is commonly referred to as just phase. The instantaneous phase at time t is $$ 2ft + \theta $$ expressed in radians. Note the following nomenclature about waves (see Fig. 5.1):

The amplitude (A) of a wave is the maximum disturbance a wave undergoes in a cycle.

The wavelength (λ) of a wave is the length of one complete wave.

The frequency ($$ f = 1/T = \omega /2\pi $$) of a wave is the number of oscillations per unit time.

The period ($$ T = 1/f $$) of a wave is the time that it takes for one complete oscillation.

The velocity ($$ v = \lambda f = \omega /k $$) of a wave is the distance travelled per unit time by the wave.

The angular frequency ($$ \omega = 2\pi f $$) is the rate of change of phase with time.

The wave number $$ (k = 2\pi /\lambda ) $$ of a wave is the rate of change of phase with distance.

Fig. 5.1

a Wave nomenclature. Source https://​www.​miniphysics.​com/​properties-of-waves.​html, b (Left, top) The two waves of like frequency reinforce each other when in phase. (Left, bottom) The two waves cancel each other when they are maximally out of phase. (Right) Wave interference leading to beats. The red and blue waves are constantly in sections of constructive and destructive interference. Beats occur when two waves are similar in frequency, but slightly off.

Source Yap, J. AP Physics 1 Exam Review—Beats. AP Physics, 14 September 2015. http://​blog.​omninox.​org/​ap-physics-1-question-review-1/​

Points on a wave which are a whole number of wavelengths apart are said to be in phase.

Points which are an odd number of half wavelengths apart are said to be in anti-phase.

In the literature, wavelength, phase, time, frequency, and period are expressed in units of distance (e.g., meter), radians, seconds, cycles per second (Hertz), and seconds per cycle, respectively. The amplitude is measured in units depending on the type of wave. When the frequency of an oscillation is time invariant, then one often uses time instead of angle to express instantaneous phase. Thus, the Earth’s rotation is measured in hours instead of in radians, and the interval between two time zones is a good example of phase shifts. Other measures of phase shifts in use are distance and fraction of the wavelength.

The wave velocity v is determined by the properties of the medium in which the wave is travelling and is independent of the other parameters. Certain kinds of waves, e.g., electromagnetic waves, do not need a medium to travel or to even exist. In any case, v can be calculated from the measurements of the frequency f and wavelength λ from the relationship $$ v = f\lambda $$.

Waves may be electromagnetic (light, radio frequency, etc.), acoustic (sound), probability (quantum mechanics), etc. When the phase difference between two waves is zero (i.e., the waves are whole number of wavelengths apart), the waves are said to be in phase with each other, otherwise they are out of phase. When the phase difference is 180° (π radians) (i.e., the waves are an odd number of half wavelengths apart), then the two waves are said to be in anti-phase. Waves can be superposed on one another and the result will depend on the amplitudes, frequencies and phase of the participating waves (see Fig. 5.1 for some simple examples). Phase coherence is the quality of a wave to display self-defined phase relationship in different regions of its domain of definition.

In physics, all systems which oscillate harmonically are quantized in terms of their energy, whether these systems are material oscillators, sound waves, or electromagnetic waves. Since we see diverse systems interacting with each other, it follows that the quantization of any one type of harmonic oscillator will require a similar quantization of all other types. Reflection, refraction, diffraction, and interference are characteristic behaviors of all types of wave (see Fig. 5.2).

Fig. 5.2

Characteristic behavior of waves

Reflection occurs when a wave bounces from the surface of an obstacle. None of the properties of a wave are changed by reflection. The wavelength, frequency, period, and speed are the same before and after reflection. The only change is in the direction in which the wave is travelling—the incident angle equals the reflected angle.

Refraction is the change in direction of a wave due to a change in its speed. This is most commonly seen when a wave passes from one medium to another. Refraction of light is the most commonly seen example, but any type of wave can refract when it interacts with a medium, for example, when sound waves pass from one medium into another or when water waves move into water of a different depth.

Diffraction refers to various phenomena associated with the bending of waves when they interact with obstacles in their path. Diffraction is most obvious when the wavelength is greater than the obstacle’s size. As wavelength increases, the degree of diffraction increases. It occurs with any type of wave, including sound waves, water waves, and electromagnetic waves such as visible light, X-rays, and radio waves. The complex patterns resulting from the intensity of a diffracted wave are a result of interference between different parts of a wave that travelled to the observer by different paths. A wavelength does not change on diffraction.

Interference occurs when two (or more) waves are superposed on one another. The result is a composite wave. There can be interference in time and in space. Interference in time occurs when two wave sources, say of sound, emit slightly different frequencies and if we listen to both at the same time, then we hear a rising and falling of the sound, a phenomenon called beats, that is, the rising sound happens when the crests of the two waves come together (constructive interference) and the falling sound happens when the crest of one wave and the trough of another come together (destructive interference). Interference in space involves wave patterns which result when waves are confined within a given volume and reflect back and forth, say, from a wall.

Interference is the test for wave motion. In quantum mechanics, the waves of particular interest are those that give the probability amplitude of finding a particle at a given place—the so-called matter waves. Their frequency is proportional to the energy and their wave number is proportional to the momentum of the particle. Complex algebra amazingly allows us to calculate the striking interference effects seen in quantum phenomena.

5.2.1 The Wave Equation

The one-, two-, three-dimensional equations for a wave travelling in the x, (x, y), and (x, y, z) Cartesian space are, respectively,

where v is the wave speed and $$ \psi $$ represents the variable that is changing as the wave passes. The equations describe, respectively, e.g., a wave on a stretched string (1-dimension), a wave on a stretched membrane (2-dimensions), and a spherical wave from a blast in space (3-dimensions).

5.2.2 Travelling Waves

A wave moving in space is called a travelling wave . A travelling wave which is confined to one plane in space and varies sinusoidally in both space and time can be expressed as linear combinations of

as may be verified by direct substitution in the wave equation . Here A and B are wave amplitudes. It is sometimes convenient (as in quantum mechanics) to use the following complex form¹ (where $$ i = \sqrt { - 1} $$, and not an indexing variable)

In the case of classical waves, either the real or imaginary part is chosen since the wave must be real, but in quantum mechanics, such as for the wave function of a free particle, the complex form is usually preferred.

5.2.3 Standing or Stationary Waves

A standing wave (or a stationary wave) is a wave that is not travelling, i.e., it remains in a constant position. Standing waves can occur for a variety of reasons, e.g., the medium in which the wave is travelling is moving in the opposite direction to the wave, or in a stationary medium it may be due to the interference of two waves travelling in opposite directions.

5.2.4 Wave Packets

Since the travelling wave solution

is valid for any values of the wave parameters (A, λ, ω, etc.), and the wave equation is linear, any linear superposition is also a solution, that is

(Here, i is an index and not $$ \sqrt { - 1} $$.) A group of waves of different wavelengths can produce an interference pattern which localizes the group over a fuzzily observed short length $$ \triangle x $$ through constructive interference and destructive interference elsewhere (see Fig. 5.3). This group, outlined by an “envelope” called a wave packet , travels as a unit. Since $$ \triangle x $$ cannot be definitively measured, its associated wave number too becomes fuzzy. A wave packet solution to the wave equation must contain a range of frequencies. The narrower the wave packet , the greater the range of frequencies required for the fast transient behavior. This requirement can be stated as a kind of uncertainty principle for classical waves:

Fig. 5.3

Forming of wave packets.

Adapted from: Hyperphysics [4], Uncertainty principle, http://​hyperphysics.​phy-astr.​gsu.​edu/​hbase/​uncer.​html

The actual numbers involved obviously depend upon the definition of the wave packet width but creating narrow widths inherently requires a large frequency bandwidth. Depending on the evolution equation, the wave packet’s envelope may remain constant (non-dispersive) or it may change (dispersive) while propagating. There are two velocities involved in a wave packet : (1) the phase velocity, the speed at which a wave crest moves, and (2) the group velocity at which the wave “packet” moves. The reader may have begun to suspect that this fuzziness surrounding wave packets may have some connection with Heisenberg’s uncertainty principle (see Sect. 5.4). There indeed is. The wave packet’s group velocity is identified with the velocity of a particle and the packet region contains all the information related to all the mechanical properties, such as, energy and momentum, normally associated with a particle, and finally the relevant wave equation is the Schrödinger’s wave equation . The wave packets generated with this equation are probability waves , which describe the probability that a particle or particles in a particular state will be measured to have a given position and momentum, where amazingly the momentum is directly related to a wave number. See Sect. 5.4 for some mathematical details about wave packets .

5.2.5 Probability Waves

In quantum mechanics, the vibrating object is an abstract thing. It is the amplitude of a probability function that gives the probability of finding a quantum system in a given configuration. This amplitude function can vary in space and time and satisfies the linear Schrödinger equation . By some interesting transformations, it turns out that what we call frequency of the probability amplitude is equal, in the classical idea, to energy. Therefore, what we understand as frequency in wave theory applies to energy in quantum mechanics. In particular, a quantum system can always be represented as a linear superposition of states of definite energy. The energy of each state is a characteristic of the system and so is the pattern of amplitude which determines the probability of finding a system in different states. The general motion of a quantum system can be described by giving the amplitude of each of these different energy states.²

5.3 Fourier Analysis

Resolving a wave into a superposition of its Fourier components in linear systems is now a well-established mathematical technique. It is widely used in solving problems both in classical and quantum physics. Inter alia, Fourier analysis allows us to estimate the period from a discrete set of values sampled at a fixed rate.

The finite, or discrete, Fourier transform of a complex vector y with n elements $$ y_{j} $$ is another complex vector Y, also with n elements $$ Y_{k} $$:

where $$ \Omega $$ is a complex n-th root of unity given by³

In matrix notation, the above Fourier transform can be expressed as

where the elements of the matrix F are given by

It can be shown that

which then allows us to invert the Fourier transform:

Hence

It is obvious that $$ \frac{F}{\sqrt n } $$ is unitary. A direct application of the Fourier transform

requires n multiplications and n additions for each of the n components of Y for a total of $$ 2n^{2} $$ floating-point operations in addition to the operations required to generate the powers of $$ \Omega $$. Smart calculation strategies, generally known as Fast Fourier transform (FFT) algorithms, are able to do the calculations in $$ O\left( {n\,{ \log }_{2} \,n} \right) $$ floating-point operations instead of in $$ O\left( {n^{2} } \right) $$ operations.⁴ The importance of Fourier analysis in quantum computing became evident once it was realized that quantum computers could perform a type of Fourier transform much faster than classical computers (see Chap. 10, Sect. 10.​6). This enabled many important quantum algorithms to be developed.

5.4 Wave Packets in Some Detail

An ordinary plane wave of definite wavelength λ is spread over all space. But it is possible to construct a wave packet by combining waves of different wavelengths, with phases and amplitudes so chosen that they interfere constructively over a small region of space, outside of which they produce an amplitude that reduces to zero rapidly as a result of destructive interference. In this manner, the wave packet can describe the motion of a pulse. Consider, e.g., the following function $$ \psi \left( x \right) $$:

Here, at $$ x = x_{0} $$, the argument of the exponential is zero for all k; hence, all contributions to the integral coming from different values of k add up in phase, and the result is large. On the other hand, as $$ x - x_{0} $$ becomes large, the exponential becomes a rapidly oscillating function of k, and its integral tends to cancel out. Thus, $$ \psi $$ is a function that is large only near $$ x = x_{0} $$, while farther away it is decreasingly small because the contributions of different k interfere destructively. Thus, $$ \psi $$ has the form of a wave packet .

Consider the example⁵ where

represents a Gaussian function. We then get

By adding and subtracting the following term

in the integrand, we have

Note that a Gaussian function in k space has led to a Gaussian function in x space. It turns out that the Gaussian function is the only one that has this peculiar symmetry in x and k space. Also note that the resulting packet has a maximum at x = x₀ and clearly becomes negligible for large values of (x − x₀).

5.4.1 Group and Phase Velocities

The motion of a wave packet is caused by the change of phase of all the wavelengths comprising it. That is,

In general, not only the position of the center of the wave packet but also its shape may change with time. The general case is rather complicated to handle. So, let us consider only how the packet as a whole (i.e., the overall shape of the wave’s amplitude, known as the envelope of the wave) moves by finding the position of the maximum of the packet. This means that for each instant t, there will be one point where waves of different k do not tend to interfere destructively. This will happen wherever the phase of the exponential $$ \varphi = k\left( {x - x_{0} } \right) - \omega \left( k \right)t $$ has an extremum. At this point, there will be a range of k where all waves have nearly the same phase and therefore will interfere constructively. This extremum is given by $$ \partial \varphi /\partial k = 0 $$, i.e.,

This means that the maximum of the wave packet moves through space with the group velocity

On the other hand, the phase velocity, $$ V_{p} = \lambda f = \frac{\omega }{k}, $$ which is the speed with which a point of constant phase moves when ω and k are defined. It appears rather amazing that the group velocity is the derivative of ω with respect to k and the phase velocity is $$ \frac{\omega }{k} $$. Generally, the phase velocity has little physical significance.⁶ The group velocity is of greater significance, say, for energy transport by the pulse.

The function $$ \omega \left( k \right) $$ is known as the dispersion relation. When ω is directly proportional to k, the group velocity equals the phase velocity (this is the case for electromagnetic waves). Otherwise, the envelope of the wave will become distorted as it propagates.

5.5 Concluding Remarks

Once Joseph Fourier (1768–1830) proved that any continuous function could be mimicked by an infinite sum of sine and cosine waves, it became an important tool in mathematics.⁷ The decomposing of a periodic function into its constituent Fourier components is called Fourier analysis . The Fourier transform works like a prism—you feed in a function and it separates the components of that function into its Fourier components. Those components when added together will reconstruct the wave. The role of waves in quantum mechanics is fundamental. In quantum computing, we deal with the abstract probability waves and, in particular, in managing the interference patterns of such waves where the focus is generally on creating wave packets .

References

1.

D. Bohm, in Quantum Theory (Prentice-Hall, New Jersey, 1951)

2.

R. Feynman, R. Leighton, M. Sands, in The Feynman Lectures on Physics, vol. 1, Chap. 49 (Addison-Wesley, 1963)

3.

J. Fourier, in The Analytical Theory of Heat. Translated with notes from the French edition of 1822 translated by Alexander Freeman (The University Press, Cambridge, 1878)

4.

Hyperphysics (n.d), in Uncertainty principle, Hyperphysics. http://​hyperphysics.​phy-astr.​gsu.​edu/​hbase/​uncer.​html

5.

C. Moler, Numerical Computing with MATLAB, Chapter 8, Fourier Analysis. MathWorks (2004). https://​www.​mathworks.​com/​moler/​chapters.​html or https://​www.​mathworks.​com/​content/​dam/​mathworks/​mathworks-dot-com/​moler/​fourier.​pdf

6.

W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, in Numerical Recipes: The Art of Scientific Computing, 3rd edn (Cambridge University Press, 2007). https://​e-maxx.​ru/​bookz/​files/​numerical_​recipes.​pdf

Footnotes

1

The formula $$ e^{i\theta } = \cos + i\,\sin \,\theta $$, known as Euler’s identity, is an amazing result that relates an exponential function to harmonic functions! Such is the non-intuitive world of complex number theory.

2

Feynman et al. [2. V. 1, Chap. 49, p. 624].

3

See, e.g., Moler [5].

4

See, e.g., Press et al. [6].

5

See, e.g., Bohm [1, pp. 62–63].

6

The phase velocity of electromagnetic radiation may, under certain circumstances, exceed the speed of light in a vacuum. However, this does not indicate that superluminal transfer of information or energy is possible. Such transfers are dependent on the group velocity whose speed never exceeds than that of light.

7

Fourier [3].

© Springer Nature Singapore Pte Ltd. 2020

R. K. BeraThe Amazing World of Quantum ComputingUndergraduate Lecture Notes in Physicshttps://doi.org/10.1007/978-981-15-2471-4_6

6. Getting a Hang of Measurement

Rajendra K. Bera1  

(1)

Acadinnet Education Services India, Bangalore, Karnataka, India

Rajendra K. Bera

Email: rajendrabera@yahoo.com

Karmanye vadhikaraste Ma Phaleshu Kadachana,

Ma Karmaphalaheturbhurma Te Sangostvakarmani.

(You have the right to work but not to its fruits. Let not the fruits be your motive, nor let your attachment be to inaction.)

—Bhagavad Gita, Chap. 2, Verse: 47

Abstract

This is a critical chapter in the book. Since the unusual role and importance of measurement in quantum mechanics is such as to require a separate postulate, it is discussed in some detail. The chapter concludes by revisiting Heisenberg’s uncertainty principle . The aim is to let the reader know that apart from unitary operators , measurement of a quantum system is an important tool in manipulating the evolution of the wave function by appropriately “collapsing” it under certain circumstances.

6.1 Introduction

Recall Fig. 1.​1 from Chap. 1. The superimposed picture of two women (wife and my mother-in-law) is a classical object. It captured only certain aspects of a quantum mechanical system. Of interest here is that a viewer at any instant would see only one of the women providing a fleeting sense of the collapse of the wave function to one of its eigenstates when a quantum mechanical system is measured (or observed). The difference is that a quantum system collapses irreversibly and remains in the collapsed state, if left alone or remeasured in an identical way while Fig. 1.​1 collapses only in the viewer’s mind and not in a physical sense. Thus, any observation of the picture is independent of all other observations. Further, more than one viewer can observe the picture concurrently without affecting or being affected by any other viewer. Hence, multiple observations of a single copy of the picture can be pooled to determine all of its possible views. In a quantum mechanical system, such pooling of information about the state of a single copy of a system is impossible.

Before reading further, the reader may wish to quickly revisit Sects. 2.​7.​3 and 2.​8 of Chap. 2 and refresh his memory on the material presented there on quantum measurements.

6.2 Measurement of Quantum Systems

Any measurement on a quantum system requires the system to interact with a suitable classical measuring device which, in general, will disrupt the unitary evolution of the system culminating in a thermodynamically irreversible process in which wave-particle duality (see Chap. 2, Sect. 2.​2.​2) will play an important role. For example, depending on the experiment, the behavior of an electron may appear to be particle-like or wave-like in different stages of an experiment. The wave aspect, especially phase relations, will play a central role in determining interference. Quantum measurement is governed by the following postulate, which we repeat from Chap. 2, Sect. 2.​7.​3:

Postulate 3: Quantum measurements are described by a collection $$ \left\{ {M_{m} } \right\} $$ of measurement operators. These are operators, which act on the state space of the system being measured. The index m refers to the measurement outcomes that may occur in the experiment. If the state of the quantum system is $$ \left| \psi \right\rangle $$ just before the measurement, then the probability that result m occurs is given by

and the state of the system after the measurement is

where the measurement operators satisfy the completeness condition

The completeness condition expresses the requirement that the respective probabilities associated with each state of $$ \left| \psi \right\rangle $$ must sum to one:

In this postulate, we come face-to-face with the probabilistic (pure chance) aspect of quantum mechanics.¹ It embodies the following: (1) the indivisibility of the quantum of action and (2) the unpredictability and uncontrollability of its consequences in each individual case. The postulate predicts the probabilities of certain results provided the measurement is made in a certain way. The measurement is a thermodynamically irreversible process. It cannot be undone. Till a measurement is made, the observed system develops deterministically according to the Schrödinger equation . Quantum mechanics is completely silent about the details of the measurement process and the source of randomness. It predicts only average behavior of an ensemble of quantum events; it says nothing about an individual event. Interestingly, the predicted probabilities follow deterministic laws in the same way that macroscopic events follow deterministic laws. That is, if enough information is available about the initial conditions of an experiment, we can calculate precisely what the probability will be for a certain result to occur. The postulate also makes clear that in quantum mechanics there are two distinct entities—the observed system and the observing system. The observing system is the environment (including humans making the measurement), which surrounds the observed system. The observed system cannot be observed until it interacts with the observing system and the observing system makes a detection. Even then, what we can observe is the observed system’s effect on the measuring device.

Postulate 3 is an information extraction postulate. Measurement of a quantum system is essentially an invasive process unlike in classical mechanics where the invasiveness can, in principle, be made arbitrarily small. In classical physics, energy, position, and velocity are directly accessible to observation; in quantum mechanics, they not only no longer appear as fundamental, but also are replaced by a state vector, which cannot be directly observed. The Bell inequality (see Chap. 4, Sect. 4.​5.​4) tells us that any attempt to reformulate quantum mechanics in a mathematically equivalent way that would give it a structure more like classical physics is doomed to failure. The Bell inequality is a compelling example of an essential difference between the classical and the quantum worlds.²

Embedded within the quantum measurement process is a decision process of which we are presently ignorant. Each measurement produces a probabilistic outcome chosen from a spectrum of values and the chosen outcome apparently has no physical cause.

In quantum computing, the objects of measurement are qubits . Most often they are measured in the computational basis $$ \left\{ {\left| 0 \right\rangle , \left| 1 \right\rangle } \right\}. $$ This is a measurement on a qubit with two outcomes defined by the two measurement operators $$ M_{0} = \left| 0 \right\rangle \left\langle 0 \right| $$ and $$ M_{1} = \left| 1 \right\rangle \left\langle 1 \right| $$. It is easily verified that they satisfy the completeness equation. Given that $$ \left| 0 \right\rangle = \left[ {\begin{array}{*{20}c} 1 &amp; 0 \\ \end{array} } \right]^{\text{T}} $$ and $$ \left| 1 \right\rangle = \left[ {\begin{array}{*{20}c} 0 &amp; 1 \\ \end{array} } \right]^{\text{T}} $$, it follows that

It is now easily verified that each measurement operator is Hermitian, and that

Thus, the completeness relation

is also obeyed.

Let $$ \left| \psi \right\rangle = a\left| 0 \right\rangle + b\left| 1 \right\rangle $$. Then the respective probability that the measurement will yield 0 or 1 is

Notice that we can only determine the complex coefficients, such as a and b describing the wave function $$ \left| \psi \right\rangle $$, in distinction to the states, up to an arbitrary phase factor only.

The state of the system after measurement will be, respectively, fixed at

Quite importantly, only certain sets of measurements can be made at any one time, and arbitrary quantum states cannot be measured with arbitrary accuracy. No matter how delicately done, the very first measurement will forever alter the state of the system. Recall that a quantum system can be described with many different, but related bases. Sometimes, it is easier to work with one basis than another just as in classical mechanics one coordinate system may be easier to work with than another.

6.2.1 Cascaded Measurements Are Single Measurements

Suppose $$ \left\{ {L_{l} } \right\} $$ and $$ \left\{ {M_{m} } \right\} $$ are two sets of measurement operators, then it can be shown that a measurement defined by the measurement operators $$ \left\{ {L_{l} } \right\} $$ followed by a measurement defined by the measurement operators $$ \left\{ {M_{m} } \right\} $$ is equivalent to a single measurement made by the measurement operator $$ \left\{ {N_{lm} } \right\} $$ where $$ N_{lm} \equiv M_{m} L_{l} . $$

6.2.2 Projective Measurements; Observable-Operators

In quantum mechanics, the measurement process affects the state of a system in a non-deterministic, but statistically predictable way. That is, given an ensemble of identical quantum systems (and hence describable by the same wave function), a measurement made on each such system may leave each of them in a different state with the unusual property that the ensemble now collectively exhibits a statistically predictable menagerie³ of states.

There is a special class of measurements known as projective measurements , which comes under Postulate 3. Such a measurement operator M is Hermitian and acts on the state space of the system being observed. M, traditionally called an observable, has a spectral decomposition,

where $$ P_{m } $$ is the projector onto the eigenspace of M with real eigenvalue $$ \lambda_{m} $$. The possible outcomes of the measurement correspond to the eigenvalues, $$ \lambda_{m} $$. Upon measuring the state $$ \left| \psi \right\rangle $$, the probability of getting a result corresponding to the index m is given by

Given that outcome m occurred, the state of the quantum system immediately after the measurement becomes

That projective measurement is a special case of Postulate 3 is easily seen. Suppose the measurement operators in Postulate 3, in addition to satisfying the completeness relation $$ \mathop \sum \nolimits_{m} M_{m}^{\dag } M_{m} = I, $$ also satisfy the conditions that $$ M_{m} $$ are orthogonal projectors, i.e., Hermitian, and $$ M_{m} M_{m'} = \delta_{mm'} M_{m} $$. Then with these additional restrictions, Postulate 3 reduces to a projective measurement just defined. It has some nice properties, e.g., the average value $$ \left\langle M \right\rangle $$ of the measurements is given by

This is a crucial result you must memorize; the average value of the observable M is

From here, we can derive the standard deviation Δ(M) associated with the observations of M,

The standard deviation (for a Gaussian distribution) is a measure of the spread of the observations of M about the average. This formulation of measurement and standard deviation in terms of observables gives rise to an elegant version of the Heisenberg uncertainty principle (see Sect. 6.3).

In general, a measurement will alter a quantum system’s state in an uncontrollable and unpredictable way that is consistent with Heisenberg’s uncertainty principle and the probabilistic outcome enunciated by Postulate 3. In principle, observable-operators exist for all possible measurable quantities. Thus, there is an observable-operator corresponding to each of the quantities position, momentum, angular momentum, energy, electron spin in a given direction, etc., e.g., the following mappings from quantity to observable-operator are well known (see Chap. 2, Sect. 2.​5.​1):

In quantum mechanics, a system’s state may be expressed as a weighted sum of the eigenvectors of any observable-operator.⁴ If one makes a measurement of some quantity, the only possible result is an eigenvalue of the observable-operator associated with that quantity. Once a measurement is made, the state of the system changes according to Postulate 3. This change manifests abruptly as a discontinuous “collapse of the wave function.” To which part of the wave function it collapses is a matter of chance. We are ignorant of the collapse mechanism and the instant of collapse.

In classical mechanics, the observable is a variable whose value, in principle, can be measured without disturbing the measured system, i.e., the observable can send out information carrying signals regarding its value that can be passively detected by the measurement apparatus, which then changes its state during measurement in direct correlation with the value of the observable. In quantum mechanics, the observable is an operator that can act on a system described by a superposition of the observable-operator’s eigenvectors. The task of the observable-operator is to act on the system and pick a unit vector $$ \left| {s_{i} } \right\rangle , i = 1, \ldots , n, $$ from among the unit vectors comprising the system’s wave function

i.e., the observable-operator acts as a filter. It is the measurement postulate that uniquely separates classical mechanics from quantum mechanics.

Of course, when we use operators we need to worry about the sequence in which they are operated. For example, consider the one-dimensional case where the position is given by x and the momentum by p. Then we have

Clearly, $$ xp\psi \ne px\psi . $$ Since this is true for any state vector $$ \psi $$, we say that x and p have a non-zero commutator, i.e.,

Similarly, we can show that time t and energy E also have a non-zero commutator.

Non-zero commutators enable an alternative means of expressing Heisenberg’s uncertainty principle : You cannot make simultaneous measurements of any two quantities with arbitrary precision if the commutator of their corresponding observable-operators is non-zero; with operators that commute you can. For example, in three dimensions, the components x, y, and z, of r, commute; therefore, their precise simultaneous measurement is possible; we can always precisely locate, say, the position of an electron or a photon.⁵ When dealing with a non-zero commutator, Heisenberg’s uncertainty principle forces us to choose which of the two quantities we want to measure more accurately. This, in a sense, means that making a measurement amounts to saying that we create certain properties, because we choose to measure those properties. In fact, it also raises the question, “Did a particle with the measured position or momentum exist before the measurement was made or did we create the particle by the measurement?” Indeed, John Wheeler once wrote:

May the universe in some strange sense be “brought into being” by the participation of those who participate? … The vital act is the act of participation. “Participator” is the incontrovertible new concept given by quantum mechanics. It strikes down the term “observer” of classical theory, the man who stands safely behind the thick glass wall and watches what goes on without taking part. It can’t be done, quantum mechanics says.⁶

The Galilean–Newtonian notion that we can observe, measure, and speculate about the world without changing it (absolute objectivity) and thereby proceed to know the “absolute truth” is emphatically negated by quantum mechanics. We are an integral part of the universe. There is no part of the universe that is ever really separate from the rest of the universe. Quantum mechanics does not claim to seek absolute truth; it only claims to correlate experience correctly.⁷

6.2.3 Distinguishing Quantum States

If it is known beforehand that the system to be measured can be in $$ s\left| {\psi_{i} } \right\rangle $$ with i = 1, …, n and that the states $$ \left| {\psi_{i} } \right\rangle $$ are orthonormal, then it is possible to reliably distinguish the orthonormal states by using the collection of measurement operators given by $$ M_{i} \equiv \left| {\psi_{i} } \right\rangle \left\langle {\psi_{i} } \right| $$, one for each index i, and an additional measurement operator $$ M_{0} $$ defined as the positive square root of the positive operator $$ I - \sum\nolimits_{i \ne 0} {\left| {\psi_{i} } \right\rangle \left\langle {\psi_{i} } \right|} $$. Clearly, these operators satisfy the completeness condition. Therefore, if the state of the quantum system is $$ \left| {\psi_{i} } \right\rangle $$, then

and the state of the system will be measured with certainty.

On the other hand, if the states $$ \left| {\psi_{i} } \right\rangle $$ are not orthonormal, then it can be shown that there is no quantum measurement capable of distinguishing the states. For example, there is no process allowed by quantum mechanics that will reliably distinguish between the states $$ \left| {\psi_{1} } \right\rangle = \left| 0 \right\rangle $$ and $$ \left| {\psi_{2} } \right\rangle = (\left| 0 \right\rangle + \left| 1 \right\rangle )/\sqrt 2 $$. The reason is that each vector will have a component parallel to the other. For example, $$ \left| {\psi_{2} } \right\rangle $$ can be decomposed into a non-zero component parallel to $$ \left| {\psi_{1} } \right\rangle $$ and a component orthogonal to $$ \left| {\psi_{1} } \right\rangle $$. Now if a measurement is made and the outcome is $$ \left| {\psi_{1} } \right\rangle $$, we cannot be completely sure that it was $$ \left| {\psi_{1} } \right\rangle $$ that was measured and not the $$ \left| {\psi_{1} } \right\rangle $$-component of $$ \left| {\psi_{2} } \right\rangle $$. So, with finite probability we will err in distinguishing between non-orthogonal states. As an aside, we mention that if it was possible to distinguish between arbitrary quantum states, it would imply that communication faster than light, using quantum entanglement, would be possible.

6.2.4 When Measurement Basis States Differ from Computational Basis States

When the basis states for both computation and measurement are the same, say, states $$ \left| {\psi_{1} } \right\rangle $$ and $$ \left| {\psi_{2} } \right\rangle $$ for a qubit , one can express its state as a linear superposition $$ \alpha \left| {\psi_{1} } \right\rangle + \beta \left| {\psi_{2} } \right\rangle $$. If the basis states are orthonormal, a measurement will output $$ \left| {\psi_{1} } \right\rangle $$ with probability $$ \left| {\alpha^{2} } \right| $$ and $$ \left| {\psi_{2} } \right\rangle $$ with probability $$ \left| {\beta^{2} } \right| $$. However, ambiguities will arise if the computational and measurement basis states differ. For example, suppose the computational basis states are $$ \left| 0 \right\rangle $$ and $$ \left| 1 \right\rangle $$, and the measurement basis states are $$ \left| + \right\rangle \equiv \left( {\left| 0 \right\rangle + \left| 1 \right\rangle } \right)/\sqrt 2 $$ and $$ \left| - \right\rangle \equiv \left( {\left| 0 \right\rangle - \left| 1 \right\rangle } \right)/\sqrt 2 $$, then $$ \left| \psi \right\rangle = \alpha \left| 0 \right\rangle + \beta \left| 1 \right\rangle $$ can be re-expressed in terms of $$ \left| + \right\rangle $$ and $$ \left| - \right\rangle $$ as⁸

If $$ \left| \psi \right\rangle $$ is now measured using the $$ \left| + \right\rangle $$, $$ \left| - \right\rangle $$ basis, the result will be $$ \left| + \right\rangle $$ with probability $$ \left| {\alpha + \beta } \right|^{2} /2 $$ and $$ \left| - \right\rangle $$ with probability $$ \left| {\alpha - \beta } \right|^{2} /2 $$. Thus, if $$ \beta = 0, $$ then

Any measurement on $$ \left| \psi \right\rangle $$ will collapse it to either state $$ \left| + \right\rangle $$ or state $$ \left| - \right\rangle $$ with equal probability! This also tells us that clever measurement schemes can be used to change the state of a quantum system in a controlled manner. In this case, the system was changed from state $$ \left| 0 \right\rangle $$ to a superposition of states $$ \left| 0 \right\rangle $$ and $$ \left| 1 \right\rangle $$. Note that when a qubit is measured, the measurement changes the state of the qubit to one of the basis states associated with the measurement system, i.e., to an eigenstate of the observable-operator.

6.2.5 Positive Operator-Valued Measure (POVM) Measurements

Postulate 3 involves two elements. First, it prescribes a rule for describing the measurement statistics, i.e., the respective probabilities of the different possible measurement outcomes. Second, it provides a rule for describing the post-measurement state of the system. Quite often, our main interest is in knowing the probabilities of the respective measurement outcomes, e.g., as in the case of an experiment where the system is measured only once upon the conclusion of an experiment. In such cases, the POVM formalism is well adapted to the analysis of the measurements. The POVM formalism is a simple consequence of the general description of measurements introduced in Postulate 3. The theory of POVM is both elegant and widely used.

Suppose a measurement described by measurement operators $$ M_{m} $$ is performed upon a quantum system in the state $$ \left| \psi \right\rangle $$. Then the probability of outcome labeled m is given by

Now define $$ E_{m} \equiv M_{m}^{\dag } M_{m} , $$ then from Postulate 3 and elementary algebra, $$ E_{m} $$ is a positive operator such that $$ \mathop \sum \nolimits_{m} E_{m} = I $$ and $$ p\left( m \right) = \langle \psi \left| E_{m} \right| \psi \rangle . $$ The operator set $$ \left\{ {E_{m} } \right\} $$, known as a POVM , is therefore sufficient to determine the probabilities of the different measurement outcomes. The POVMs are best viewed as a special case of the general measurement formalism, providing the simplest means by which one can study general measurement statistics, without the necessity for knowing the post-measurement state. They are a mathematical convenience that sometimes gives extra insight into quantum measurements.

A POVM application

If given a qubit and the information that it is in one of two states $$ \left| {\psi_{1} } \right\rangle = \left| 0 \right\rangle $$ or $$ \left| {\psi_{2} } \right\rangle = \left( {\left| 0 \right\rangle + \left| 1 \right\rangle } \right)/\sqrt 2 $$, we know that we cannot determine with perfect reliability the state of the qubit . However, it is possible to perform a measurement which distinguishes the state some of the time, but never makes an error of misidentification. This can be done by considering a POVM containing three elements,

It is easily verified that these are positive operators that satisfy the completeness relation $$ \sum\nolimits_{m} {E_{m} = I} $$ and therefore form a legitimate POVM .⁹

To show that it does not misidentify either $$ \left| {\psi_{1} } \right\rangle $$ or $$ \left| {\psi_{2} } \right\rangle $$, assume that the given state is $$ \left| {\psi_{1} } \right\rangle = \left| 0 \right\rangle $$. Perform the measurement using the POVM $$ \left\{ {E_{1} , E_{2} , E_{3} } \right\}. $$ There is zero probability that the result $$ E_{1} $$ will be observed since $$ E_{1} $$ has been cleverly chosen to ensure that $$ \left\langle {\psi_{1} } \right|E_{1 } \left| {\psi_{1} } \right\rangle = 0 $$ Therefore, if the measurement is $$ E_{1} $$, one can safely conclude that the state of the qubit must have been $$ \left| {\psi_{2} } \right\rangle $$. A similar line of reasoning shows that if the measurement outcome is $$ E_{2 } $$ then the qubit’s state must have been $$ \left| {\psi_{1} } \right\rangle $$. Finally, if the measurement is $$ E_{3} $$, nothing can be inferred about the state the qubit was in. Thus, one cannot err in identifying the state of the given qubit , but the price paid is that sometimes the state of the qubit cannot be identified at all.

6.2.6 The Effect of Phase on Measurement

Depending on the context, the term “phase has several meanings in quantum mechanics. (Phase was briefly mentioned in Chap. 2, Sect. 2.​7.​3 when discussing Postulate 3.) Here, we discuss two of them—global phase factor and relative phase factor .

Global phase factor . It turns out that the states $$ \left| \psi \right\rangle $$ and $$ e^{i\theta } \left| \psi \right\rangle $$, where θ is a real number, produce identical measurement statistics. To see this, suppose $$ M_{m} $$ is a measurement operator associated with some quantum measurement, and note that the respective probabilities that outcome labeled m will occur are

Thus, from a measurement point of view the two states are identical and we say that the state $$ e^{i\theta } \left| \psi \right\rangle $$ is equal to $$ \left| \psi \right\rangle $$, up to a global phase factor $$ e^{i\theta } $$. Thus, global phase factors are irrelevant to the observed properties of physical systems. This gives us the freedom to choose the global phase factor . Note that we have an invariant property or symmetry here. In physics, symmetry implies a corresponding conservation law, and for this one it turns out to be the conservation of electric charge.¹⁰

Relative phase factor . To understand the relative phase factor , consider the states $$ \left( {\left| 0 \right\rangle + \left| 1 \right\rangle } \right)/\sqrt 2 $$ and $$ \left( {\left| 0 \right\rangle - \left| 1 \right\rangle } \right)/\sqrt 2 $$. In the two states, the magnitude of the amplitude of $$ \left| 1 \right\rangle $$ differs only in sign. More generally, two amplitudes a and b differ by a relative phase if there is a real θ such that a = exp(iθ) b. Even more generally, two states are said to differ by a relative phase in some basis if each of the amplitudes in that basis is related by such a phase factor. For example, in the two states noted above, they are the same up to a relative phase shift because the $$ \left| 0 \right\rangle $$ amplitudes are identical (a relative phase factor of exp(i0) = 1), and the $$ \left| 1 \right\rangle $$ amplitudes differ only by a relative phase factor of exp(iπ) = −1. The difference between the relative phase factors and global phase factors is that for relative phase the phase factors may vary from amplitude to amplitude, which makes the relative phase a basis-dependent concept unlike the global phase. As a result, states which differ only by relative phases in some basis give rise to physically observable differences in measurement statistics, and it is no longer possible to regard these states as being physically equivalent.

6.2.7 Can Every Observable Be Measured?

Dirac in his influential text notes:

The question now presents itself – Can every observable be measured? The answer theoretically is yes. In practice it may be very awkward, or perhaps even beyond the ingenuity of the experimenter, to devise an apparatus which could measure some particular observable, but the theory always allows one to imagine that the measurement can be made.¹¹

Note that even though on measurement the wave function collapses so that only a single result is delivered, it is possible to measure certain joint properties of two or more qubits in a measurement. The business of measurement in quantum mechanics is so eerie that Erwin Schrödinger said:

Had I known that we were not going to get rid of this damned quantum jumping, I never would have involved myself in this business!¹²

6.2.8 Measurement with Photons and Electrons

In quantum mechanics, it is an observed fact that a wave of length λ is always associated with a momentum $$ p = h/\lambda $$ and a wave with frequency v is always associated with an energy $$ E = hv $$, where h is the Planck’s constant. In classical mechanics, a particle is something that has simultaneously a definite position and a definite momentum. Such is not the case in quantum mechanics.

It is impossible to localize a photon’s position within a region Δx, which is smaller than the wavelength of the photon except at the moment when it disappears by absorption. Therefore, the concept of a particle is of no use in interpreting the result of any other measurement. The situation is quite different for an electron for which we can say that immediately after it has been found at a given spot another observation will disclose the same electron at the same point. So, in this sense, the light quantum shows lesser resemblance to a classical particle than does an electron, but its definite momentum still makes it worthwhile to regard it as one sometimes. “Although there is a distribution of momenta, and therefore of energies, the relation between the energy and momentum for a light quantum, $$ E = pc $$, where c is the speed of light, remains exact and definite. This important fact follows from the de Broglie relations, plus the relation $$ \omega = kc $$, which holds for an electromagnetic wave in free space.”¹³ Here, $$ k = 2\pi /\lambda $$, and ω is the angular frequency.

6.2.9 Whither Causality?

The two different types of evolution of a quantum system (1) the unitary evolution as described by the Schrödinger wave equation (linear, reversible, lossless in information), and (2) measurement or projection (nonlinear, irreversible, loss in information as postulated by Max Born), which leads to the non-intuitive, probabilistic “collapse” of the wave function to a sub-state in an entirely random, causeless, and thus unpredictable manner. Subsequent measurement of the same variable will not affect the measured output if the system has not been disturbed in between. (This is physics without causality.) Of quantum measurements, only statistical statements can be made. The non-unitary nature of wave function “collapse” remains unsettling to minds that are firmly rooted in classical physics.¹⁴ Notwithstanding, the theory has been supported by numerous experimental tests to great precision.

One interesting view about the superposed state is that it appears so. It is a definite vector but not aligned without preferred coordinates. In fact, such a superposed state is frequently created by a unitary gate (e.g., the Hadamard gate) from an input that is in a classical state. Note that all unitary gates only rotate the state vector. Since a unitary transformation does not lose information or change the entropy of the transformed state, the superposed state must still be quite definite. Therefore, the wave function collapse is a serious issue as it introduces a fundamentally random element in quantum mechanics as well as irreversibility, which prevents backward temporal propagation of influences (or retrocause) even though unitary evolutions inherently permit backward influences to propagate. That the event is random, and hence causeless, breaks the dependency connection or causal chain of events. Shoup¹⁵ has argued that quantum measurement is not a collapse but a three-way unitary interaction that is not random and that does not preclude retrocausation. Quantum measurement, as postulated, is an interaction between the quantum system and the measurement apparatus. The entity not included is the environment, and each should be described as a fully general quantum system.

6.3 Heisenberg’s Uncertainty Principle (Revisited)

We now revisit Heisenberg’s uncertainty principle of 1927,¹⁶ one of the pillars of quantum mechanics that was at the center of the famous Bohr–Einstein (EPR) debate. The principle originates from a general property of waves that says (see Chap. 5, Sect. 5.​4) that a wave packet solution to any wave equation must contain a range of frequencies. The narrower the wave packet , the greater the range of frequencies required for the fast transient behavior. This requirement can be stated as a kind of uncertainty principle for classical waves:

where $$ \Delta t $$ is the range of time needed for a wave packet to pass a given point, $$ \Delta \omega $$ is the range of angular frequencies in this packet, $$ \Delta x $$ is the width of the packet, and $$ \Delta k $$ is the indefiniteness in the wave number that is related to the finite width of the wave packet . The actual numbers involved depend upon the definition of the wave packet width but creating narrow widths inherently requires a large frequency bandwidth. This relationship is not restricted to quantum theory and therefore is much easier to understand from the viewpoint of classical physics. In fact, Heisenberg’s original derivation of the principle dealt directly with this property of the wave equation solutions. He showed that the Fourier transform of a localized Gaussian position wave function is a localized Gaussian momentum-space wave function, with the momentum width of the latter Gaussian proportional to the reciprocal of the position width of the former Gaussian.¹⁷ This well-known property of Gaussian distributions under Fourier transformations has many analogs in classical physics. For example, electrical engineers know that a fast electrical pulse can be represented in the time-domain as a set of time-varying voltages, or in the frequency-domain as a continuous set of Fourier components (i.e., as a set of voltages varying continuously as a function of frequency). These two representations have exactly the Bohr–Heisenberg complementary relationship and exhibit their own “uncertainty principle.” The localization of a fast pulse in the time domain (i.e., making it extremely short in duration) requires a corresponding de-localization in the frequency domain, since the Fourier frequency spectrum of the pulse must include a broader range of frequencies, including very high ones. Conversely, one can increase the localization of the pulse in the frequency domain by passing the pulse through a “band pass filter,” which eliminates undesired Fourier components that do not fall within the frequency “window” of the filter. This kind of frequency localization leads to a corresponding broadening of the pulse in the time domain.

When we impose on this general property of waves the quantum mechanical de Broglie relationship between momentum and wave number, $$ p = h/\lambda = \hbar k, $$ where k is the wave number (a definite momentum implies a definite wave number) we get $$ \Delta p\,\Delta x \approx \hbar $$. It is the linkage between intensity and probability along with the linkage between wavelength (and hence wave number) and momentum that leads to the uncertainty principle. See Table 6.1 for an illustration.

Table 6.1

Illustration of Heisenberg’s uncertainty principle

The variables are position and momentum. Captured from animation ina. The Heisenberg uncertainty principle. Contributed by Sarah Woods, and Kris Baumgartner (UC Davis). Chemistry, LibreTexts, 11 September 2018

Remarks

Heisenberg’s uncertainty principle is illustrated here for the non-commuting observable-operators position and momentum. Note the greater spread in the momentum wave packet is accompanied by a smaller spread in the position wave packet and vice versa. A similar relationship exists between non-commuting observable-operators, energy, and time

Heisenberg’s uncertainty principle

aAt https://​chem.​libretexts.​org/​Textbook_​Maps/​Physical_​and_​Theoretical_​Chemistry_​Textbook_​Maps/​Map%3A_​Physical_​Chemistry_​(McQuarrie_​and_​Simon)/​01%3A_​The_​Dawn_​of_​the_​Quantum_​Theory/​1.​9%3A_​The_​Heisenberg_​Uncertainty_​Principle

It is important to note that there are two velocities involved in a wave packet : (1) the phase velocity, the speed at which a wave crest moves, and (2) the group velocity, the speed at which the wave “packet” moves (see Chap. 5, Sect. 5.​4.​1). In quantum mechanics, it is the group velocity that is identified with the velocity of a particle and the packet region displays all the mechanical properties, such as energy and momentum, one normally associates with a particle.

Like the uncertainty principle between position and momentum of a quantum particle, there is an energy–time uncertainty relation that can be derived by starting with $$ \Delta \omega \,\Delta t \approx 1 $$, where $$ \Delta t $$ is the range of time needed for a wave packet to pass a given point and $$ \Delta \omega $$ is the range of angular frequencies in this packet. The de Broglie relation, $$ \hbar \Delta \omega = \Delta E $$ leads to $$ \Delta E\,\Delta t \approx \hbar $$, where $$ \Delta E $$ is the range of indeterminacy in the energy, and $$ \Delta t $$ is the range of indeterminacy during the time at which the electron passes a given point.

Heisenberg’s uncertainty principle essentially says that given a pair of complementary observable-operators such as “position” and “momentum” we can never expect to measure both precisely because they have a non-zero commutator. For example, a precise knowledge of the position (or momentum) implies that all possible outcomes of measuring the momentum (or position) are equally probable. In classical physics, a moving particle always has a well-defined position and momentum, but in quantum mechanics even in principle it does not. That is, in the quantum world, it is not possible to know enough about the present to make a complete prediction about the future. Likewise, for energy and time. If the energy of the system is measured to accuracy $$ \Delta E $$, then the time to which this measurement refers must have a minimum uncertainty given by $$ \Delta E\,\Delta t \ge \hbar /2 $$. More generally, if $$ \Delta q $$ is the error in the measurement of any coordinate and $$ \Delta p $$ is the error in its canonically conjugate momentum, then $$ \Delta p\,\Delta q \ge \hbar /2 $$. The measurement limits imposed by the uncertainty principle cannot be overcome by refining measurement technology; it is a limit imposed by Nature as we penetrate into the subatomic world. The classical and quantum particles are entirely different entities.

Max Born summed the situation nicely:

[I]f we can never actually determine more than one of the two properties (possession of a definite position and of a definite momentum), and if when one is determined we can make no assertion at all about the other property for the same moment, so far as our experiment goes, then we are not justified in concluding that the “thing” under examination can actually be described as a particle in the usual sense of the term.¹⁸

The uncertainty principle provides a stringent limitation on the applicability of the concepts of classical determinism. This is because in any actual measurement process, there is always a stage where the uncontrollable and unpredictable transfer of an individual quantum intervenes. The modern and correct interpretation of Heisenberg’s uncertainty principle is in terms of the standard deviation and the commutator operator (see Chap. 3, Sect. 3.​4.​7) rather than the earlier misconceived notion that the uncertainty principle is about measuring a quantity C to some “accuracy” Δ(C) which then causes D to be “disturbed” by an amount Δ(D) in a way that some sort of inequality is satisfied.

Mathematically, given a large number of identical quantum systems each in state $$ \left| \psi \right\rangle $$, suppose we measure C on some of those systems, and D in others, where C and D are represented by two observable-operators, then the standard deviation $$ \Delta \left( C \right) $$ of the C results times the standard deviation $$ \Delta \left( D \right) $$ of the D results will satisfy the inequality¹⁹

It is, of course, true that measurements in quantum mechanics cause disturbance to the system being measured, but that is most emphatically not the content of the uncertainty principle. The deep significance of the uncertainty principle is that we cannot observe something without changing it. The independent observer, watching from the sidelines without influencing the observed phenomenon, simply does not exist. When dealing with commutator operators, we must necessarily choose, and the choice will alter the state of the universe. Heisenberg’s uncertainty principle throws up one obvious conclusion: The wave function determines at least two related probabilities for the principle to hold. Of course, as we have noted earlier, it also determines many more probabilities, in fact, the probabilities of all possible physical measurements.

Thus, indeterminism is built into the very structure of matter where certain pairs of variables (such as position and momentum) cannot even exist with simultaneously and perfectly defined values. Does this mean that quantum mechanics is incomplete in the sense that there are hidden variables unaccounted for in quantum mechanics that can actually determine what these quantities are at all times, but they are such that, in practice, we cannot predict or control them with complete precision? This was answered affirmatively by John Bell.²⁰ In passing we note that any contradiction of the uncertainty principle at any point would make the entire wave-particle duality untenable. It also implies that one cannot design an experiment in any way that would violate the principle.

The measurement postulate indicates that there is a subquantum world, not necessarily explained by string theory. We discuss such a subquantum world in Chap. 12.

6.4 Concluding Remarks

Heisenberg’s uncertainty principle and measurement are intimately related in quantum mechanics. Such is not the case in classical mechanics. A “wave packet ” shows how either the momentum or position of a particle can be precisely calculated, but not both simultaneously. The structure of wave packet holds the key. The waves constituting a wave packet define an average wavelength through an interference pattern; larger the number of waves comprising the wave packet , the more precisely we can measure the position of the particle and less precisely its momentum because more wavelengths of varying momenta are added. The converse is true when the wave packet comprises smaller number of waves.

References

1.

J.S. Bell, On the Einstein-Podolsky-Rosen Paradox. Physics 1, 195–200 (1964). http://​inspirehep.​net/​record/​31657/​files/​vol1p195-200_​001.​pdf. Reprinted in J.S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, Cambridge, 1987)

2.

D. Bohm, in Quantum Theory (Prentice-Hall, New Jersey, 1951)

3.

M. Born, in Atomic Physics, 8th edn (Courier Corporation, 2013)

4.

K.A. Brading, Which symmetry? Noether, Weyl, and conservation of electric charge. Stud. Hist. Philos. Sci. Part B Stud. Hist. Philos. Mod. Phys. 33(1), 3–22(20). http://​www.​nd.​edu/​~kbrading/​Research/​WhichSymmetryStu​diesJuly01.​pdf

5.

D.C. Cassidy, in Quantum Mechanics (1925–1927): The Gamma-Ray Microscope (The American Institute of Physics, 2018). https://​history.​aip.​org/​exhibits/​heisenberg/​p08b.​htm

6.

P.A.M. Dirac, in The Principles of Quantum Mechanics (Clarendon Press, Oxford, 1947)

7.

J. Gribbin, in Search of Schrödinger’s Cat. Updated Edition (1983)

8.

W. Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Zeitschrift für Physik 43, 172–198 (1927). https://​people.​isy.​liu.​se/​jalar/​kurser/​QF/​references/​Heisenberg1927.​pdf English translation in (Wheeler and Zurek, 1983), pp. 62–84. Another English translation (The Actual Content of Quantum Theoretical Kinematics and Mechanics), available as NASA TM 77379 (Dec 1983). https://​ia600500.​us.​archive.​org/​10/​items/​nasa_​techdoc_​19840008978/​19840008978.​pdf

9.

C. Misner, K.S. Thorne, J.A. Wheeler, in Gravitation (Freeman, New York, 1973)

10.

M.A. Nielsen, I.L. Chuang, in Quantum Computation and Quantum Information (Cambridge University Press, 2000). Errata at http://​www.​squint.​org/​qci/​

11.

E. Noether, Invariante Variationsprobleme, Nachr. d. König. Gesellsch. d. Wiss. zu Göttingen, Math-phys. Klasse, 235–257 (1918). http://​www.​physics.​ucla.​edu/​~cwp/​articles/​noether.​trans/​german/​emmy235.​html; English translation: M.A. Travel, Transp. Theory Stat. Phys. 1(3) 197, 183–207. Available as arXiv:​physics/​0503066v1, http://​arxiv.​org/​PS_​cache/​physics/​pdf/​0503/​0503066v1.​pdf, 8 Mar 2005

12.

A. Peres, in How to differentiate between non-orthogonal states. Phys. Lett. A 128, 19 (1988)

13.

R. Shoup, Physics without causality—theory and evidence. AIP Conf. Proc. 863, 169 (2006), California. http://​dx.​doi.​org/​10.​1063/​1.​2388754, http://​citeseerx.​ist.​psu.​edu/​viewdoc/​download?​doi=​10.​1.​1.​102.​4800&​rep=​rep1&​type=​pdf, http://​www.​boundaryinstitut​e.​org/​bi/​articles/​Physics_​without_​Causality.​pdf

14.

G. Zukav, in The Dancing Wu Li Masters: An Overview of the New Physics (Rider & Co., 1979)

Footnotes

1

Albert Einstein had tremendous reservations about this pure chance aspect of quantum mechanics. It led to the famous EPR paradox (see Chap. 4, Sect. 4.​5.​2).

2

Nielsen and Chuang [10, p. 96].

3

Menagerie: a collection of usually wild or exotic animals, or the place where they are exhibited. The term is used here in a metaphorical sense.

4

There is a bit of trick involved if more than one eigenvector has the same eigenvalue.

5

An eye adapted to the dark can detect a single photon.

6

Misner et al. [9, p. 1273]. Also quoted in Zukav [14, Chap. 1].

7

Zukav [14, Chap. 1].

8

Quantum physicists happily use such unusual notations as $$ \left| + \right\rangle $$, $$ \left| - \right\rangle $$, $$ \left| \uparrow \right\rangle $$, $$ \left| \downarrow \right\rangle $$, $$ \left| \to \right\rangle $$, $$ \left| \nwarrow \right\rangle $$, $$ \left| \nearrow \right\rangle $$, etc. It is a matter of getting used to them. Their meanings are usually obvious from the context.

9

The example is from Peres [12]. See also: Nielsen and Chuang [10, p. 92].

10

See, e.g., Brading [4]. The conservation of electric charge follows from Noether’s first theorem provided in Nöther [11].

11

Dirac [6, p. 37].

12

As quoted in Gribbin [7].

13

Bohm [2, p. 95].

14

Shoup [13].

15

See Footnote 14.

16

Heisenberg [8]. Heisenberg had originally and heuristically explained it in terms of optical microscopes. It is no longer a part of our current understanding of the uncertainty principle. See, e.g., Cassidy [5].

17

See, e.g., Bohm [2, pp. 62–63]. See also Chap. 5, Sect. 5.​4.​1.

18

Born [3, p. 97].

19

For a proof, see Nielsen and Chuang [10, pp. 89, 609].

20

Bell [1].

© Springer Nature Singapore Pte Ltd. 2020

R. K. BeraThe Amazing World of Quantum ComputingUndergraduate Lecture Notes in Physicshttps://doi.org/10.1007/978-981-15-2471-4_7

7. Quantum Gates

Rajendra K. Bera1  

(1)

Acadinnet Education Services India, Bangalore, Karnataka, India

Rajendra K. Bera

Email: rajendrabera@yahoo.com

First I shall do some experiments before I proceed farther, because my intention is to cite experience first and then with reasoning show why such experience is bound to operate in such a way. And this is the true rule by which those who speculate about the effects of nature must proceed.

—Leonardo Da Vinci, C. 1513 (Quotation as cited in Fritjof Capra, The Science of Leonardo, Doubleday, New York, 2007.)

Abstract

This chapter introduces the fundamentals of quantum gates (operators) and describes generally used gates in designing quantum algorithms.

7.1 Introduction

In Chap. 3, we set down the basic mathematical elements needed to study linear operators suited for designing quantum algorithms. Note that for both the state vectors and linear operators, we shall deal with have vector-matrix representations. They can thus be multiplied together in various ways. In particular, the associative and distributive axioms of multiplication will always hold, but the commutative axiom of multiplication will not hold. This is a good time to quickly glance at Tables 3.​1 to 3.​3 in Chap. 3 before proceeding further and to remember that notational conventions are extremely important and must be meticulously followed.

Note that certain linear operators correspond to certain classical dynamical variables, e.g., the coordinates of particles, components of velocity, momentum and angular momentum, and functions of these quantities. The striking difference about the use of these variables in quantum mechanics in contrast to their use in classical mechanics is that they are now subject to an algebra which bars the commutative axiom of multiplication. Despite this difference, since dynamical variables (observables) of quantum mechanics still share certain properties with their classical counterparts that in quantum mechanics we continue to use the same letter to denote a dynamical variable and the corresponding linear operator as if they were the same thing without getting confused. Confusion can be avoided by remembering that in quantum mechanics, the observable-operators are required to be Hermitian to ensure that their eigenvalues are real.

Unlike in classical mechanics where an observable A always has a value for the state of the system, it is generally not so in quantum mechanics due to Postulate 3 and the non-commutability of certain observables (Heisenberg’s uncertainty principle ). Instead, we can access the average value $$ \left\langle \psi \right|A \left| \psi \right\rangle $$ of an observable A as well as the average value $$ \left\langle \psi \right|f\left( A \right)\left| \psi \right\rangle $$ of any function $$ f\left( A \right) $$. We can also determine the probability of an observable having any specified value for the state, i.e., the probability of this specified value being obtained when one makes a measurement of the observable.¹ When we are dealing with two or more commuting observables, they may be counted as a single observable, and the result of a measurement then consists of two or more numbers. The states for which this measurement is certain to lead to one particular outcome are the simultaneous eigenstates.²

Just as for classical computers, there are universal sets of gates; likewise, there exist universal primitive quantum state transformations, called quantum gates , for quantum computation (see Sect. 7.5). Given enough qubits , it is possible to mimic a universal Turing machine, but to do so requires ingenuity, because, while most classical gates are irreversible, quantum gates (and consequently all quantum computations) are necessarily reversible. The required ingenuity came from Rolf Landauer,³ Charles Bennett,⁴ Tommaso Toffoli,⁵ Edward Fredkin,⁶ and others whose contributions showed that reversible computation on a Turing machine was indeed possible and computationally economical. Thus, Turing machines are those quantum computers whose dynamics ensure that they remain in a computational basis state at the end of each step, given that they start in one. To ensure unitarity, it is necessary and sufficient that the mapping from one state to another be bijective. In fact, the universal quantum computer can perfectly simulate any Turing machine and can simulate with arbitrary precision any quantum computer or simulator.⁷

In principle, any quantum particle can function as a gate. Take, e.g., an orbiting electron inside an atom.⁸ The electron can exist in either the so-called ground or excited states (respectively, labeled, say, $$ \left| 0 \right\rangle $$ and $$ \left| 1 \right\rangle $$; each state will be associated with a distinct eigenvalue of the appropriate Hamiltonian operator in the Schrödinger equation ). By shining light on the atom, with appropriate energy and for an appropriate length of time, it is possible to move the electron from state $$ \left| 0 \right\rangle $$ to state $$ \left| 1 \right\rangle $$ and vice versa. Further, by reducing the time we shine the light, the electron initially in the state $$ \left| 0 \right\rangle $$ can be moved half-way between $$ \left| 0 \right\rangle $$ and $$ \left| 1 \right\rangle $$ into, say, the $$ \left( {1/\surd 2} \right) \left( {\left| 0 \right\rangle + \left| 1 \right\rangle } \right) $$ state. More generally, by adding energy to the electron in a controlled manner it can also be put in an arbitrary superposed state $$ \left| \psi \right\rangle = a\left| 0 \right\rangle + b\left| 1 \right\rangle $$, where a and b are complex numbers. Additionally, the state of the electron can be altered such that $$ \left| a \right| $$ and $$ \left| b \right| $$ remain unchanged, that is, the electron’s state can be manipulated without affecting the probabilities associated with measurement. The electron’s state is a vector in a two-dimensional complex vector space.

While one can prepare a quantum particle in a specific state, $$ \left| \psi \right\rangle = a\left| 0 \right\rangle + b\left| 1 \right\rangle $$, it is not possible for another, ignorant of the particle’s state, to determine a and b since any measurement on the particle will collapse its state to either $$ \left| 0 \right\rangle $$ or $$ \left| 1 \right\rangle $$. Measurement on a quantum system (even if it is a one-particle system) elicits only a fraction of the total information that describes it and the rest is lost forever. Even if one had infinite number of identical copies of the system, it would still be impossible to determine a and b. Thus, it is impossible to clone an unknown quantum system (see Chap. 4, Sect. 4.​2). It is intriguing that Nature does keep track of all the continuous parameters describing the state, like a and b but keeps most of that information hidden from observers. Developers of quantum algorithms must always bear this fact in mind. Nature hides information.

7.2 Operators (A Summary)

We quickly recall (from Chap. 3, Sect. 3.​4) some important operators in quantum mechanics and hence in quantum computing. Table 7.1 provides a convenient summary for quick recall.

Table 7.1

Various types of operators used in quantum computing. See also: Chap. 3, Table 3.​3

Operator type, symbol

Operator definition

Remarks

Normal operator, A

A is diagonalizable with respect to some orthonormal basis for V. V is the vector space on which A operates. Conversely, any diagonalizable operator is normal

Hermitian operator , A

The diagonal elements of A are always real. A normal operator is Hermitian if and only if it has real eigenvalues

Positive operators, A

$$ \left\langle v \right|A\left| v \right\rangle $$ is real and non-negative for every vector $$ \left| v \right\rangle $$

A is Hermitian and has a diagonal representation with non-negative eigenvalues. This is an important sub-class of Hermitian operators

If $$ \left\langle v \right|A\left| v \right\rangle &gt; 0 $$ for all $$ \left| v \right\rangle \ne 0 $$, then A is positive definite

Unitary operators , U

U is invertible and $$ U^{ - 1} = U^{\dag } $$; U is diagonalizable. U only rotates the vector it operates on; it does not change the length of the vector. Unitary operators preserve inner products between vectors. If U is real, then $$ U^{ - 1} = U^{\text{T}} $$

Normal operators. An operator A is normal if $$ AA^{\dag } = A^{\dag } A $$. Any normal operator A on a vector space V is diagonal with respect to some orthonormal basis for V. Conversely, any diagonalizable operator is normal. This means that A can be written in the outer product form as $$ A = \mathop \sum\nolimits_{i} \lambda_{i} \left| i \right\rangle \left\langle i \right| $$, where $$ \lambda_{i} $$ are the eigenvalues of A, each $$ \left| i \right\rangle $$ is an eigenvector of A paired with its corresponding eigenvalue $$ \lambda_{i} $$, and the set of vectors $$ \left| i \right\rangle $$ is an orthonormal basis for V.

Hermitian operators . An operator A is Hermitian if $$ A = A^{\dag } = \left( {A^{T} } \right)^{*} = \left( {A^{*} } \right)^{\text{T}} $$. The diagonal elements of a Hermitian operator are always real. A normal operator is Hermitian if and only if it has real eigenvalues.

Positive operators. A positive operator A is defined as an operator such that for any vector $$ \left| v \right\rangle $$, $$ \left\langle v \right|A\left| v \right\rangle $$ is a real, non-negative number. If $$ \left\langle v \right|A\left| v \right\rangle $$ is strictly greater than zero for all $$ \left| v \right\rangle \ne 0 $$, then A is said to be positive definite. Any positive operator is Hermitian and therefore has the diagonal representation, $$ \mathop \sum\nolimits_{i} \lambda_{i} \left| i \right\rangle \left\langle i \right| $$, with non-negative eigenvalues $$ \lambda_{i} $$. One may also show that for any operator A, $$ A^{\dag } A $$ is positive. Positive operators are an important subclass of Hermitian operators .

Unitary operators. An operator U is unitary if $$ U^{\dag } U = I = UU^{\dag } $$. By definition, U is invertible and implies that $$ U^{ - 1} = U^{\dag } $$. Further, U is normal and hence has a spectral decomposition (i.e., it can be diagonalized). Any unitary transformation is equivalent to a rotation and has a L² length that remains invariant to a rotation. Unitary operators only rotate the vector they operate on. Unitary operators preserve inner products between vectors. If U is real, then $$ U^{ - 1} = U^{\text{T}} $$.

7.3 The Qubit

A single qubit is the simplest quantum system we can think of. It is a quantum system with a two-dimensional Hilbert space, capable of existing in a superposition of Boolean states and of being entangled with the states of other qubits . Mathematically, a qubit is a vector $$ \left| \psi \right\rangle = a\left| 0 \right\rangle + b\left| 1 \right\rangle $$, parameterized by two complex numbers a and b, satisfying $$ \left| a \right|^{2} + \left| b \right|^{2} = 1 $$. This implies that $$ \left| \psi \right\rangle $$ is a unit vector, i.e., the norm of $$ \left| \psi \right\rangle $$ is

Operations on a qubit must preserve this norm. Unitary operators do so. The condition $$ \left\langle {\psi } \mathrel{\left | {\vphantom {\psi \psi }} \right. \kern-0pt} {\psi } \right\rangle = 1 $$ is often known as the normalization condition for state vectors. Note that by convention the matrix representation of $$ \left| 0 \right\rangle \equiv \left[ {\begin{array}{*{20}c} 1 &amp; 0 \\ \end{array} } \right]^{\text{T}} $$ and $$ \left| 1 \right\rangle \equiv \left[ {\begin{array}{*{20}c} 0 &amp; 1 \\ \end{array} } \right]^{\text{T}} $$ and they are called the computational basis states. They form an orthonormal basis for this Hilbert space.

A string of n qubits exists in Hilbert spaces of dimensionality of up to $$ 2^{n} $$ and hence the string can have $$ 2^{n} $$ possible permutations of qubit states, say, $$ s_{1} , s_{2} , \ldots , s_{N,} $$ where $$ N = 2^{n} $$. The general state vector can be written as the unit vector (and hence with unit norm) as

The Hilbert space H (not to be confused here with the Hadamard operator that acts on a qubit ) is a complex vector space possessing a scalar product operation $$ \left\langle\cdot|\cdot\right\rangle $$, whose value is a complex number and which satisfies the following algebraic properties:

In the above properties, z is a complex number, the symbols $$ \psi $$ and $$ \phi $$ are interchangeable, and the arbitrary vector $$ \left| \chi \right\rangle $$ belongs to the same vector space as that of $$ \left| \psi \right\rangle $$ and $$ \left| \phi \right\rangle $$. Two of the above noted properties are particularly important:

Moreover, once the norm is known, the scalar product can be defined in terms of it, so linear transformations that preserve the norm (length) must also preserve the scalar product.

If A is some linear operator, then $$ \left\langle \phi \right|A\left| \psi \right\rangle $$ represents both scalar products: $$ \left\langle \phi \right| $$ with $$ A\left| \psi \right\rangle $$, as well as $$ \left\langle \phi \right|A $$ with $$ \left| \psi \right\rangle $$. Thus $$ \left\langle \phi \right|A\left| \psi \right\rangle = \left\langle \psi \right| A^{\dag } \left| \phi \right\rangle $$. Note that the wave function $$ \psi $$ is complex and that the phase of $$ \psi $$ (up to an overall constant multiplying factor) is an essential ingredient of the Schrödinger evolution. Note further that it is the phase alone that gives the wave function its “wave-like” character (see Sects. 7.3.1 and 7.3.2).

7.3.1 Global Phase Factor

Consider the state $$ {\text{e}}^{i\gamma } \left| \psi \right\rangle $$, where $$ \left| \psi \right\rangle $$ is a state vector and γ is a real number. We say that the state $$ {\text{e}}^{i\gamma } \left| \psi \right\rangle $$ is equal to $$ \left| \psi \right\rangle $$, up to a global phase factor $$ {\text{e}}^{i\gamma } $$. This means that the measurement statistics predicted for these two states are identical. That is, if $$ M_{m} $$ is a measurement operator, then

Thus, global phase factors are irrelevant to the observed properties of physical systems. This is an important invariant property or symmetry of Nature, and as noted in Chap. 6, Sect. 6.​2.​6, this symmetry implies, of all things, the conservation of electric charge!

7.3.2 Relative Phase Factor

To understand relative phase, consider the states $$ a\left| 0 \right\rangle + b\left| 1 \right\rangle $$ and $$ c\left| 0 \right\rangle + d\left| 1 \right\rangle $$. We say that two complex amplitudes, such as, a and c for $$ \left| 0 \right\rangle $$ differ by a relative phase factor if there is a real $$ \delta $$ (relative phase shift) such that $$ a = {\text{e}}^{i\delta } c $$. Likewise, we say that complex amplitudes b and d for $$ \left| 1 \right\rangle $$ differ by a relative phase factor if there is a real $$ \varepsilon $$ such that $$ b = {\text{e}}^{i\varepsilon } d $$. Note that $$ \delta $$ and $$ \varepsilon $$ depend on the basis being used to represent the states.

Example

In the two states (1/√2) ($$ \left |0 \right\rangle + \left |1 \right\rangle $$) and (1/√2) ($$ \left |0 \right\rangle - \left |1 \right\rangle $$), the amplitudes of $$ \left |0 \right\rangle $$ are the same in the two states, hence they have a relative phase factor of 1, while the amplitudes of $$ \left |1 \right\rangle $$ differ by a relative phase factor of −1.

The difference between the relative phase factors and global phase factors is that for relative phase the phase factors may vary from amplitude to amplitude, which makes the relative phase a basis-dependent concept unlike the global phase. As a result, states, which differ only by relative phases in some basis, give rise to physically observable differences in measurement statistics, and it is no longer possible to regard these states as being physically equivalent.

7.3.3 Unitary Operators

A matrix or an operator U is unitary if $$ U^{\dag } U = I = UU^{\dag } $$. An operator is unitary if and only if each of its matrix representations is unitary. U is invertible, normal, and has a spectral decomposition (i.e., it can be diagonalized), and preserves inner products between vectors. Each eigenvalue of a unitary matrix has modulus 1 (i.e., it has the form $$ {\text{e}}^{i\theta } $$ for some real $$ \theta $$). The tensor product of two unitary operators is unitary. Unitarity means equality of two lengths. Any unitary transformation means a rotation, and a L² length that remains invariant to a rotation. A L² length has the form of a sum of squares.

Simple unitary operations on qubits are called quantum “logic gates.” Such gates acting on a single qubit are described by 2 × 2 matrices. For example, if a qubit evolves as

then after time t we say that the operation, or “gate”

has been applied to the qubit , where $$ \theta = \omega t $$. This can also be written as

The unitarity constraint is the only constraint placed on quantum gates . Thus, any unitary matrix specifies a valid quantum gate ! Further, it can be shown that any arbitrary 2 × 2 unitary matrix may be decomposed as

where α, β, γ, and δ are real-valued. Here, the second matrix is just an ordinary rotation. The first and third matrices represent rotations in a different plane. This decomposition may be used to give an exact prescription for performing the action of an arbitrary single qubit quantum logic gate.

Quantum computing uses unitary operators to change the state of discrete state vectors when that change is being made under Postulate 2. Hence, such a change can be reversed, if desired, by the corresponding inverse unitary operator (which by definition exists) if applied before any measurement is made on the quantum system. Reversible operators (gates) only move states around. Since they have the same number of inputs and outputs and have a one-to-one mapping between input vectors and output vectors, input and output are uniquely retrievable from each other. In a thermodynamic sense, since no information is lost, energy is conserved. Every reversible gate can be described by a permutation.⁹ Finally, under Postulate 2, non-unitary operations (e.g., irreversible operations) are inadmissible. Thus, quantum gates do not allow (1) feedback loops from one part of the system to another, (2) fan-in operations (that is, bitwise OR of inputs), and (3) fan-out operations (because it would violate the no-cloning theorem).

7.3.4 Hermitian Operators

In quantum mechanics, Hermitian operators are postulated to represent the observable quantities of a quantum system. The set of the operator’s eigenvalues represent the set of possible measurement outcomes the operator can produce. For each eigenvalue, there is a corresponding eigenstate (or eigenvector), which will be the state of the system after the measurement. As we noted in Chap. 2, Sect. 2.​8.​1:

1.

A Hermitian matrix can be unitarily diagonalized, generating an orthonormal basis of eigenvectors which spans the state space of the quantum system.

2.

The eigenvalues of a Hermitian matrix are real. The possible outcomes of a measurement are precisely the eigenvalues of the given observable.

3.

The matrix representing the product of two linear operators is the product of the matrices representing the two factors.

The above implies that any state of a quantum system can be expressed as a linear combination of the eigenvectors of any Hermitian operator . Equivalently, it means that any quantum state can be expressed as a linear superposition of the eigenstates of an observable-operator. In quantum mechanics, an observable-operator, inter alia, does two things: (1) it randomly collapses the wave function on which it operates to one of its (operator’s) eigenstate, and (2) it produces a real-valued eigenvalue corresponding to the eigenstate to which the wave function has collapsed, as the measurement result. Quantum computing uses measurement operators to change the state of discrete state vectors when that change is being made under Postulate 3. This change, in general, is irreversible.

7.4 Important Qubit Gates

We now look at some qubit gates (transformations) which are central to quantum computing. The reader should commit these gates to memory and become familiar with their use. In particular, the 1-qubit Pauli gates and the Hadamard gate, and the 2-qubit Cnot gate. In addition, we describe some important 3-qubit gates. All unitary gates are described by unitary matrices. In matrix form, the basis states are lexicographically ordered, e.g., 3-qubit gates are ordered according to $$ \left |000 \right\rangle $$, $$ \left |001 \right\rangle $$, $$ \left |010 \right\rangle $$, $$ \left |011 \right\rangle $$, $$ \left |100 \right\rangle $$, $$ \left |101 \right\rangle $$, $$ \left |110 \right\rangle $$, and $$ \left |111 \right\rangle $$.

7.4.1 Pauli Gates and Other 1-Qubit Gates

In quantum computing, Pauli gates (also called sigma- or σ-matrices) play a central role in the manipulation of single qubits . They are represented by 2 × 2 matrices as shown in Table 7.2 along with their alternative labels (names). Except for I, all have trace zero. Some authors do not include I in the set of Pauli matrices (gates). We have included I. Any 2 × 2 matrix M can be expressed as the linear sum M = α I + β X + γ Y + δ Z, where α, β, γ, and δ are complex constants. Thus, there are infinitely many 2 × 2 unitary matrices, and therefore infinitely many single qubit gates, but each is expressible as linear combinations of the matrices, I, X, Y, and Z. This contrasts with classical information theory, where only two logic gates are possible for a single bit, namely the identity and the logical NOT operation.

Table 7.2

Pauli gates

Operation

Operator labels

Operator definition (matrix form)

Operator definition (bra-ket form)

Post operation qubit state

Identity

Negation

ZX

Phase shift

Note that $$ \left| 0 \right\rangle = \left[ {\begin{array}{*{20}c} 1 \\ 0 \\ \end{array} } \right] $$ and $$ \left| 1 \right\rangle = \left[ {\begin{array}{*{20}c} 0 \\ 1 \\ \end{array} } \right] $$

We present three frequently used 1-qubit gates that appear in quantum algorithms. These are the Hadamard gate, the phase gate, and the π/8 gate, as shown in Table 7.3.

Table 7.3

Some 1-qubit gates

Name

Symbol

Operation

Gate (bra-ket form)

Gate (matrix form)

Hadamard

H

Phase gate

S

π/8 gate

T

Note The π/8 gate is called so even though π/4 appears in its definition. This is because, up to an unimportant global phase, it is equal to a gate which has exp(±iπ/8) appearing on its diagonals: $$ T = {\text{e}}^{i\pi /8} \left[ {\begin{array}{*{20}c} {{\text{e}}^{ - i\pi /8} } &amp; 0 \\ 0 &amp; {{\text{e}}^{i\pi /8} } \\ \end{array} } \right] $$. Note that $$ H = \left( {X + Z} \right)/\sqrt 2 $$ and S = T² (Nielsen and Chuang [15, p. 174])

The Hadamard gate, H, is frequently used in quantum computations as the first step in creating input data at the start of computations. It is a special case of the more general Fourier transform. When it operates on a qubit in state $$ \left |0 \right\rangle $$ or $$ \left |1 \right\rangle $$, it transforms that state, respectively, into the superposed state H$$ \left |0 \right\rangle $$ = ($$ \left |0 \right\rangle $$ + $$ \left |1 \right\rangle $$)/√2 ≡ $$ \left |+ \right\rangle $$ and H$$ \left |1 \right\rangle $$ = ($$ \left |0 \right\rangle $$ − $$ \left |1 \right\rangle $$)/√2 ≡ $$ \left |- \right\rangle $$.¹⁰ That is, H rotates the state of a qubit such that the result may also be viewed as just a change in basis ($$ \left |0 \right\rangle $$, $$ \left |1 \right\rangle $$) → ($$ \left |+ \right\rangle $$, $$ \left |- \right\rangle $$). The result is that if the qubit is measured in the ($$ \left |0 \right\rangle $$, $$ \left |1 \right\rangle $$) basis, after being operated by the Hadamard gate, we will measure $$ \left |0 \right\rangle $$ or $$ \left |1 \right\rangle $$ with equal probability, but if it is measured using the ($$ \left |+ \right\rangle $$, $$ \left |- \right\rangle $$) basis we will measure $$ \left |+ \right\rangle $$ if the qubit’s earlier state was $$ \left |0 \right\rangle $$ and $$ \left |- \right\rangle $$ if the qubit’s earlier state was $$ \left |1 \right\rangle $$. Note that H is its own inverse, or H H ≡ H² = I, i.e., applying H twice to a state does nothing to it. This is quite amazing since the application of a randomizing operation to a random state produces a deterministic outcome! Of course, only under the laws of quantum mechanics and not of classical mechanics. More generally, the application of the Hadamard transform on n-qubits (represented by $$ H^{ \otimes n} $$) (sometimes known as the Walsh transform), each initially in the $$ \left |0 \right\rangle $$ state, results in

where the sum is over all possible 2n mutually orthogonal states of n qubits or values of x. Thus, the $$ H^{ \otimes n} $$ gate produces an equal superposition of all $$ 2^{n} $$ possible computational basis states which can be viewed as the binary representation of the numbers from 0 to 2n − 1, and that this is done extremely efficiently (as compared to a classical computer) by using only n gates, i.e., n qubits can simultaneously carry an exponentially large number, 2n, integers by being in that many superposed states.

7.4.2 2-Qubit Controlled-not Gate

The controlled-not gate , Cnot, is a must have gate in quantum computing. It is a generalization of the classical, irreversible XOR gate. The Cnot gate is a particular case of a general class of controlled-U transformations described in Sect. 7.4.7. The term controlled-U means that the form of U depends on the logical value of the control qubit . A qubit so designated controls what happens to the other qubit(s) associated with it.

The effect of Cnot acting on a 2-qubit system in state $$ \left |a \right\rangle $$ ⊗ $$ \left |b \right\rangle $$ ≡ $$ \left |a, b \right\rangle $$ can be written as Cnot $$ \left |a, b \right\rangle $$ = $$ \left |a, a \oplus b \right\rangle $$, where ⊕ signifies the exclusive-or (XOR) operation (i.e., modulo 2 addition).¹¹ It means that the output is “true” if and only if exactly one of the operands has a value of “true.” This gate cannot be decomposed into a tensor product of two 1-qubit transformations, that is, one cannot find two unitary operators O₁ and O₂ such that Cnot = O₁ ⊗ O₂. The list of state changes shown in Table 7.4 is the analog of the truth table for a classical binary logic gate. Note that $$ \left |00 \right\rangle $$, $$ \left |01 \right\rangle $$, $$ \left |10 \right\rangle $$, $$ \left |11 \right\rangle $$ form an orthonormal basis for the state space of a 2-qubit system. By convention, they are associated with the standard 4-tuple basis as just noted.

Table 7.4

2-qubit controlled-not gate

Name

Symbol

Operation

Gate (bra-ket form)

Gate (matrix form)

Controlled-not

Cnot

$$ \left |00 \right\rangle $$ → $$ \left |00 \right\rangle $$

$$ \left |01 \right\rangle $$ → $$ \left |01 \right\rangle $$

$$ \left |10 \right\rangle $$ → $$ \left |11 \right\rangle $$

$$ \left |11 \right\rangle $$ → $$ \left |10 \right\rangle $$

In the matrix form, the columns correspond to $$ \left |00 \right\rangle $$, $$ \left |01 \right\rangle $$, $$ \left |10 \right\rangle $$, and $$ \left |11 \right\rangle $$, respectively, from left to right

Examples

The singular importance of the Cnot gate is that any multiple qubit logic gate may be composed from Cnot and single qubit gates.¹² It is the quantum parallel of the universality of the classical NAND gate.

7.4.3 Creating Entangled Bell States

The Cnot gate along with the H gate provides a convenient method for entangling two qubits , a crucial operation in most quantum algorithms. Entanglement is a uniquely quantum mechanical resource. It plays a key role in the design of efficient quantum algorithms. In fact, entanglement is a fundamental resource of Nature as are energy, information, and entropy. To entangle, apply H ⊗ I followed by Cnot gates to a 2-qubit system. For each of possible four states of a 2-qubit system, we have

The resulting states are known as the Bell states (after John Bell) or sometimes as the EPR states or EPR pairs (after Albert Einstein, Boris Podolsky, and Nathan Rosen who had come up with the EPR-paradox; see Chap. 4, Sect. 4.​5.​2). We shall encounter these states in quantum computation frequently. They are interesting because they are all entangled states, i.e., in an unfactorizable joint state. If the states of the entangled particles are used to encode qubits , then the entangled joint state represents what is called an ebit.

Unlike a classical bit or a qubit, an ebit is intrinsically a shared resource. The information an ebit carries is always distributed between two or more qubits and their states are always correlated, but unknown until measured. An ebit therefore provides the basis for a restricted kind of quantum communication channel in the sense that once a member qubit comprising an ebit is measured, the states of other qubits in the ebit correspondingly change. However, the channel is restricted because one cannot send an intentional message between parties simply by having each party measure the qubit it holds of an ebit because individually the measurements would appear random (see Chap. 1, Sect. 1.​5 on teleportation).

Of all possible unitary operators acting on a pair of qubits , Cnot is an important example of a subset called a “controlled U” which can be written as

where U is some 1-qubit gate (see Sect. 7.4.7 for the more general form). Other logical operations require further qubits , an example of which is the Toffoli gate described in Sect. 7.4.5.

7.4.4 Bit Copying—An Application of the Controlled-not Gate

The no-cloning theorem (see Chap. 4, Sect. 4.​2) shows that it is impossible to make a copy of an unknown quantum state reliably. However, it is possible to use quantum gates to copy classical information encoded as a $$ \left |0 \right\rangle $$ or a $$ \left |1 \right\rangle $$ (note that these are a priori known states). We show this by means of the controlled-not gate. Consider a qubit in an unknown state $$ \left| \psi \right\rangle = a\left| 0 \right\rangle + b\left| 1 \right\rangle $$ and another qubit in the state $$ \left| 0 \right\rangle $$. The input state of the two qubits may be written as

The action of the controlled-not gate to this input is to negate the second qubit when the first qubit is 1, and thus the output is simply $$ a\left| {00} \right\rangle + b\left| {11} \right\rangle $$. So, in this special case where $$ \left| \psi \right\rangle = 0 $$ or $$ \left| \psi \right\rangle = 1 $$ (classical bit states) the gate indeed copies, i.e., we have produced $$ \left| \psi \right\rangle \otimes \left| \psi \right\rangle $$. But in the general case where a and b are unknown, measuring either qubit of the state a $$ \left |00 \right\rangle $$ + b $$ \left |11 \right\rangle $$ produces either 0 or 1 with probabilities |a|² and |b|², respectively. However, once one qubit is measured, the state of the other one is completely determined, and no additional information can be gained about a and b. Thus, any extra hidden information carried in the original qubit $$ \left| \psi \right\rangle $$ is lost forever in the first measurement. If, however, the qubit had actually been copied, the state of the other qubit would have still contained some of that hidden information. Therefore, the control-not operation does not violate the no-cloning theorem as it does not make a copy of a qubit of unknown state.

7.4.5 3-Qubit Toffoli Gate

The quantum Toffoli gate¹³ is a 3-qubit gate that takes three input bits and gives three output bits. The Toffoli gate is a controlled-controlled-not gate, which negates the last of three bits if and only if the first two are 1. That is, $$ T\left| {a, b, c} \right\rangle = \left| {a, b, c \oplus ab} \right\rangle $$ (see Table 7.5). It can be used to construct a complete set of Boolean connectives. The Toffoli gate is its own inverse, i.e.,

Table 7.5

3-qubit Toffoli gate

Name

Symbol

Operation

Gate (bra-ket form)

Gate (matrix form)

Toffoli

T

$$ \left|000 \right\rangle $$ → $$ \left|000 \right\rangle $$

$$ \left|001 \right\rangle $$ → $$ \left|001 \right\rangle $$

$$ \left|010 \right\rangle $$ → $$ \left|010 \right\rangle $$

$$ \left|011 \right\rangle $$ → $$ \left|011 \right\rangle $$

$$ \left|100 \right\rangle $$ → $$ \left|100 \right\rangle $$

$$ \left|101 \right\rangle $$ → $$ \left|101 \right\rangle $$

$$ \left|110 \right\rangle $$ → $$ \left|111 \right\rangle $$

$$ \left|111 \right\rangle $$ → $$ \left|110 \right\rangle $$

Three qubits are necessary to permit the whole operation to be unitary. Here is an interesting application of the Toffoli gate . Consider

The gate takes x and y as inputs and calculates the function $$ f\left( {x, y} \right) = x \wedge y $$. To evaluate $$ f\left( {x, y} \right) $$ for all possible values of $$ \left( {x, y} \right) $$ we do the following: Take 3 qubits , in the initial state $$ \left| {000} \right\rangle $$, where the first qubit is for taking input x, the second qubit for taking input y, and the third qubit for saving $$ f\left( {x, y} \right) $$.

To create the four possible 2-qubit inputs (x, y), we apply the Hadamard gate to the first two qubits :

An application of the Toffoli gate to this results in

The resulting four superposed state of the 3-qubit system encodes a truth table in which the values of x, y, and $$ f\left( {x, y} \right) $$ are entangled in such a way that measuring the system will give only one line of the truth table. Note that the qubits can be measured in any order. Measuring the third qubit will project the state of the system to a subspace in which all input values produce the measured result while measuring the inputs (x, y) will project the third qubit to the corresponding value of $$ f\left( {x, y} \right) $$. Of course, measuring at this point provides no advantage over classical parallelism because only one result is obtained, and moreover, one cannot choose the result one wants. To obtain all the results, the computations and measurements must be repeated many times. This will reveal each of the possible results as frequently as dictated by the probabilities associated with the amplitude of each result. In this case, the probabilities are all equal.

Toffoli gates can be constructed using six Cnot gates and several 1-qubit gates.¹⁴ The n-qubit Toffoli gate is a generalization of the 3-qubit Toffoli gate . It takes n qubits $$ x_{1} , x_{2} , \ldots , x_{n} $$ as inputs and outputs n qubits . The first $$ n - 1 $$ output qubits are just $$ x_{1} , \ldots , x_{n - 1} $$; the last output qubit is $$ \left( {x_{1} \,{\text{AND}}\,x_{2} \,{\text{AND}}\, \ldots \,{\text{AND}}\,x_{n - 1} } \right)\,{\text{XOR}}\,x_{n} $$.

Toffoli Gate in Classical Computing

Any classical logical circuit can be replaced by an equivalent reversible classical circuit by using the classical version of the Toffoli gate (i.e., it takes only classical bits as inputs and produces classical bits as outputs) because it can be used to simulate the NAND gate (see Sect. 7.5.1) as well as to carry out the FANOUT operation. With these two operations, it is possible to simulate all the other elements in a classical logic circuit. The classical Toffoli gate is, therefore, a classical universal gate (see also Chap. 9, Sect. 9.​5.​4). The quantum version of the Toffoli gate , which simply permutes computational basis states in the same way as the classical Toffoli gate , nevertheless cannot carry out the FANOUT operation because of the restriction posed by the no-cloning theorem applicable to qubits of unknown states. The quantum Toffoli gate is not a universal gate in quantum computing.

7.4.6 3-Bit Fredkin Gate

The 3-bit Fredkin gate ¹⁵ functions as a “controlled-swap” gate and is reversible. It is defined as

where Us is the swap operator:

The Fredkin gate has three input bits (a, b, c) and three output bits (a′, b′, c′), where c is the control bit, whose value is not changed by the action of the Fredkin gate , i.e., c′ = c. If c is set to 0 then a and b are left alone, i.e., a′ = a, b′ = b. If c is set to 1, a and b are swapped, i.e., a′ = b, b′ = a (see Table 7.6). The Fredkin gate is its own inverse. It has the further interesting property that it conserves in the output the number of 0’s and l’s that are present at the input. For this reason, the gate is sometimes referred to as a conservative-reversible logic gate. The classical 3-bit Fredkin gate is a universal gate in classical computing.

Table 7.6

3-qubit Fredkin gate

Name

Symbol

Operation

Gate (bra-ket form)

Gate (matrix form)

Fredkin

F

$$ \left |000 \right\rangle $$ → $$ \left |000 \right\rangle $$

$$ \left |001 \right\rangle $$ → $$ \left |001 \right\rangle $$

$$ \left |010 \right\rangle $$ → $$ \left |010 \right\rangle $$

$$ \left |011 \right\rangle $$ → $$ \left |101 \right\rangle $$

$$ \left |100 \right\rangle $$ → $$ \left |100 \right\rangle $$

$$ \left |101 \right\rangle $$ → $$ \left |011 \right\rangle $$

$$ \left |110 \right\rangle $$ → $$ \left |110 \right\rangle $$

$$ \left |111 \right\rangle $$ → $$ \left |111 \right\rangle $$

7.4.7 Controlled-U Gate

For arbitrary unitary gates U₁ and U₂, the “conditional” gate

is also unitary. The controlled-not gate, the Toffoli gate , and Fredkin gate mentioned above are three such examples. We provide a fourth example in the form of a natural extension of the controlled-not gate, called the controlled-U gate . In this gate, U is any unitary matrix acting on n qubits , so U can be regarded as a quantum gate on those qubits . Such a gate has a single control qubit and n target qubits . When U is applied, and the control qubit is 0, the target qubits are untouched. When the control qubit is 1, the gate U is applied to all the target qubits . When $$ U = X $$, we have the controlled-not gate. Before proceeding further, recall that any arbitrary 2 × 2 unitary matrix may be decomposed as a product of rotations¹⁶ (see Sect. 7.3.3)

where α, β, γ, and δ are real-valued. Its usefulness lies in the following corollary (presented here without proof), which is the key to the construction of controlled multi-qubit unitary operations. The factor eiα is the global phase shift.

Corollary

Suppose U is a unitary gate on a single qubit . Then there exist unitary operators A, B, and C on a single qubit such that ABC = I and U = eiα AXBXC, where α is some overall phase factor.¹⁷

To implement the 2-qubit controlled-U operation for arbitrary single qubit U, using only single qubit operations and the controlled-not gate, we proceed as follows. Keeping in mind the corollary, we first apply the phase shift exp(iα) on the target qubit , controlled by the control qubit . That is, if the control (first) qubit is $$ \left |1 \right\rangle $$, a phase shift exp(iα) is applied to the target (second) qubit, otherwise not:

Next, suppose the control qubit is set. Then the operation eiα AXBXC = U is applied to the second qubit . If the control qubit is not set, then the operation ABC = I is applied to the second qubit, that is, no change is made. This completes the 2-qubit controlled-U implementation.

In the more general case, one may use $$ n + k $$ qubits and select U as a k qubit unitary operator to define the controlled operation $$ C^{n} \left( U \right) $$ by the equation

where $$ x_{1} x_{2} \ldots x_{n} $$ in the exponent of U means the product of the bits $$ x_{1} x_{2} \ldots x_{n} $$. That is, the operator U is applied to the last k qubits if the first n qubits are all equal to 1, otherwise nothing is done. However, we shall not discuss this general case further except to note that we have seen a simple example of this in the form of the Toffoli gate .

7.5 Universal Set of Gates

7.5.1 Universal Set of Classical Gates

Classical computers compute using Boolean expressions. Any Boolean expression may be constructed from a non-unique set of universal logic gates. For example, the AND, OR, and NOT gates form a universal set. Alternatively, AND and NOT (also known as NAND), or OR and NOT, or AND and XOR form three other possible universal sets. The truth table for logic gates is provided in Table 7.7. In addition to gates, extra working bits (called ancilla) are needed to create working space (scratch pad) during computations. A functioning classical computer (a Turing machine) can be built using a set of universal gates (the NAND, which can simulate the AND, XOR, and NOT gates, is rather popular), connectors joining gates (e.g., wires), ancilla bits, and FANOUT devises. The resulting physical circuitry can have appropriately designed feedback loops.

Table 7.7

Truth table for logic gates

A

B

A AND B

A OR B

A XOR B

NOT B

0

0

0

0

0

1

0

1

0

1

1

0

1

0

0

1

1

1

1

1

1

1

0

0

The choice of a universal set has little effect on the complexity of computing. This follows from Muller’s theorem. It states that the complexity of the simplest circuits needed to compute any reasonable Boolean function is affected by at most a constant multiplicative factor.¹⁸ Note that the operations of AND, OR, and XOR gates are many-to-one. Therefore, they are logically irreversible. Only the NOT gate is reversible since it is a one-to-one operation. Clearly then, the information content on the right-hand side of $$ \left( {a, b} \right) \to {\text{NOT}}\,\left( {a\,{\text{AND}}\,b} \right) $$ is less than on the left-hand side.

7.5.2 Universal Set of Quantum Gates

All quantum gates are required to be reversible and unitary. Therefore, certain features allowed in classical computation are not allowed in quantum computation. These are (1) feedback loops from one part of the quantum circuit to another (i.e., quantum circuits are acyclic); (2) FANIN, i.e., bitwise OR of the inputs (as it is not reversible and hence not unitary); and (3) FANOUT, i.e., the making of copies of a bit of unknown state (the inverse of FANIN). For those who delight in dealing with the impossible the following quote from Leonid Levin would bring a smile:

Sometimes it is good that some things are impossible. I am happy there are many things that nobody can do to me. (As quoted in Nielsen and Chuang [15, p. 138].)

Any unitary operation on n qubits can be implemented exactly by stringing together operations composed of single qubit and controlled-not gates.¹⁹ For example, the Hadamard, phase,²⁰ controlled-not, and $$ \pi /8 $$ gates provide a set of universal gates for which fault-tolerant constructions are known. This set is the standard set of universal gates. There is a second set of universal gates comprising the Hadamard, phase, controlled-not, and Toffoli gates , which may also be constructed in a fault-tolerant manner. We omit the proofs. In fact, there are many more choices for the universal quantum gate than in classical reversible computing due to the greater power of quantum computing. For example, DiVincenzo²¹ showed that 2-qubit universal quantum gates are indeed possible and Barenco²² extended this knowledge to show that almost any 2-qubit quantum gate (within a certain restricted class) is universal. Subsequently, Lloyd²³ and Deutsch et al.²⁴ have shown that almost any 2-qubit or n-qubit (n ≥ 2) quantum gate is also universal.

Although the controlled-not and single qubit unitary operations together form a universal set for quantum computation, it is not always easy to implement an arbitrary unitary operation using them in an error-resistant manner. Fortunately, there exists a discrete set of gates, which can approximate any unitary operation, and make it error-resistant using quantum error-correcting codes (see Chap. 11). What we mean by an approximation is the following.

Suppose U and V are two unitary operators on the same state space where U is the target operator we wish to implement, and V is the approximation to it. Then we define the approximation error as

where the maximum is over all normalized quantum states $$ \left| \psi \right\rangle $$ in the state space. It can be shown that if $$ E\left( {U, V} \right) $$ is small, then any measurement performed on the state $$ V\left| \psi \right\rangle $$ will give approximately the same measurement statistics as a measurement of $$ U\left| \psi \right\rangle $$, for any initial state $$ \left| \psi \right\rangle $$.

Thus, if $$ E\left( {U, V} \right) $$ is small, then measurement outcomes occur with similar probabilities, regardless of whether U or V is performed. Furthermore, if we perform a sequence of gates $$ V_{1} , \ldots , V_{m} $$ to approximate the gate sequence $$ U_{1} , \ldots , U_{m} $$, then the errors add at most linearly,

Note that we can only approximate the gate $$ U_{j} $$ by the gate $$ V_{j} $$. To ensure that a desired approximation is met within a tolerance Δ > 0 of the correct probabilities, it suffices that

7.6 Some Basic Quantum Operations

The essence of the design of quantum algorithms lies in the clever choice of computational and measurement bases, initial states, sequence of unitary gates and measurement points. As we shall see, one can do certain things, such as finding global information about a function, more efficiently than is possible on classical computers.

7.6.1 Random Number Generation

Quantum particles are perfect random number generators. All you must do is prepare a qubit in the state $$ \left |0 \right\rangle $$, send it through a Hadamard gate to change its state to ($$ \left |0 \right\rangle $$ + $$ \left |1 \right\rangle $$)/√2, and then measure the state. The result will be $$ \left |0 \right\rangle $$ or $$ \left |1 \right\rangle $$ with equal probability. Alternatively, one may prepare a qubit in the state |1〉 and send it through the Hadamard gate to produce ($$ \left |0 \right\rangle $$ − $$ \left |1 \right\rangle $$)/√2, and then measure the state. The result will once again be $$ \left |0 \right\rangle $$ or $$ \left |1 \right\rangle $$ with equal probability.

7.6.2 n-Qubit Hadamard Gate

Recall from Sect. 7.4.1, when the Hadamard gate is applied to n qubits individually, we get

or

where the sum is over all possible $$ 2^{n} $$ mutually orthogonal states of n qubits or values of x. Thus, the Hadamard transform produces an equal superposition of all $$ 2^{n} $$ possible computational basis states, i.e., x can be viewed as the binary representation of the numbers from 0 to $$ 2^{n} - 1 $$, and that this is done extremely efficiently (as compared to a classical computer) by using only n gates. The n-qubit Hadamard gate is sometimes known as the Walsh–Hadamard gate.

Consider a single qubit in state $$ \left| x \right\rangle $$ where $$ x \in \left\{ {0, 1} \right\} $$. Then

Notice that both are captured in the form

Likewise, for an n qubit system $$ \left| x \right\rangle = \left| {x_{1} } \right\rangle \ldots \left| {x_{n} } \right\rangle = \left| {x_{1} \ldots x_{n} } \right\rangle $$, we have

$$ x \cdot y $$ is the bitwise inner product of x and y modulo 2, and $$ \left| y \right\rangle = \left| {y_{1} } \right\rangle \ldots \left| {y_{n} } \right\rangle = \left| {y_{1} \ldots y_{n} } \right\rangle $$. The reader should become visually familiar with the above formula (last line); we will encounter it frequently.

7.6.3 A 3-Qubit Gate for AND and NOT Operations

The 3-qubit Toffoli gate can be used to construct a complete set of Boolean connectives. The AND and NOT operators for this purpose are shown below:

7.7 Taking Stock of Gates

In the real world of computing, two important issues arise—efficient storage and processing of information with available resources. In classical computers, the memory units are bits that hold two-valued variables; in quantum computers, they are two-valued quantum variables called qubits , which may be simulated by a two-level quantum system, the levels represented by eigenstates and labeled $$ \left| 0 \right\rangle $$ and $$ \left| 1 \right\rangle $$, in correspondence with the discrete states of a classical bit, 0 and 1. However, the qubit as a quantum entity can also be in a state of superposition of states. This is the principal difference between a bit and a qubit . Even though the qubit has discrete eigenstates, it can also be in a continuous, infinite range of superposition of states. Perhaps the most significant paper that brought the nature of this difference in the open was by David Deutsch²⁵ in 1985. The second and even more intriguing feature of a qubit (and a quantum system, in general) is that any attempt to measure the state of a qubit makes it lose its quantum character and effectively reduces it to a bit by bringing it to an eigenstate, i.e., lose its state of superposition. The third difference is that unlike classical bits, two or more qubits can interfere with one another and create a macroscopically coherent superposition; that is, n qubits will have $$ 2^{n} $$ product eigenstates (or dimensions) in their Hilbert space. These eigenstates then form the computational basis of an n-qubit quantum computer. Of these eigenstates, entangled states are rather intriguing where the state of one qubit is determined by a measurement on the other by setting up quantum correlations between two observables. These differences play a fundamental role in the design and choice of building and using quantum gates . All quantum gates are necessarily reversible, conventional classical gates usually are not. The fact that classical gates can come in reversible versions with negligible penalty in performance means that a quantum computer would be able to do anything that a classical computer can do. This also means that classical reversible logic is contained in quantum logic. The main advantage in pursuing quantum computing is that quantum superposition allows a qubit to hold far greater amount of information than a classical bit.

Irreversible logic: Two-bit NAND gates simulate all Boolean functions. Reversible logic: Three-bit Toffoli gates simulate all reversible Boolean functions. The XOR gate plays a crucial role in universality, entanglement, and error correction in quantum computing.

The mathematical description of gates provided in this book assumes that they can be simulated or implemented physically. Further, in line with classical computing, we use the base-2 arithmetic to describe the entire computational process. Gates are the building blocks on which we build algorithms. Hence the need to identify a universal set of gates (simple functions) that can be used repeatedly and in any desired sequence. The gates are naturally restricted to operating on a small number of inputs, say two or three at a time. But these gates can be arranged in sequences to create composite functions that mimic more complicated functions and take arbitrarily large inputs and produce large outputs.

In classical computation, the set of two-bit gates AND and OR, and the one-bit gate NOT are universal; they are often represented using truth tables. The AND and OR gates have two inputs and one output, while the NOT gate has one input and one output.²⁶ The truth tables for these three gates are:

The AND gate outputs 1 if and only if both inputs are 1, the OR gate outputs 1 if either or both inputs are 1, and the NOT gate inverts the input (0 to 1 and vice versa). These three gates can be reduced to a single gate called the NAND gate²⁷:

Thus, we have the beautiful result that the two-bit NAND gate is sufficient for classical Boolean logic. The discussion so far indicates classical computation with such a universal set is irreversible, i.e., from the output we cannot recover the input if we wanted to. The only reversible gate here is the NOT gate, the others are not. Irreversible gates lose (erase) information when producing an output. Thermodynamically, erasure of information means dissipation of heat in the physical computer that does the computation and hence an increase in entropy. Landauer showed that the minimum energy loss per bit is (kT ln 2), where k is the Boltzmann constant and T is the absolute temperature. In actual physical devices in use today, the heat dissipated is much more than that. In an important paper in 1973, Bennett²⁸ showed that a reversible model of a Turing machine²⁹ can be constructed. What it meant was that logical reversibility implies that implementation of such a machine would also be physically reversible. In a classic review paper, Bennett³⁰ elaborated on this.

It turns out that for classical reversible computing, the three-bit Toffoli gate can provide all Boolean functions in a reversible way. Since reversible logic gates are symmetric with respect to the number of inputs and outputs, we can represent them in ways other than truth tables that explicitly bring out this symmetry, as is now customary. There is another significant difference between classical irreversible and reversible logic; two-bit gates are not sufficient for universal reversible computing—a three-bit gate is sufficient, e.g., the Toffoli gate , because within it resides the NAND gate. When the third bit is fixed to be 1, the Toffoli gate writes the NAND of the first two bits on the third. Indeed, all reversible Boolean functions are special cases of unitary transformation. This is important since all quantum gates are unitary, meaning that quantum computation is a reversible process, logically and physically.

The typical quantum unitary gate represents a quantum time evolution, where the Hamiltonian $$ H(t) $$ describing the quantum system is exponentiated to produce a unitary operator ,

The integral immediately shows that it is possible to achieve the same unitary gate with different Hamiltonians. This allows one to tailor the computer hardware according to available technologies.

The Cnot gate is also called the XOR gate since it performs an exclusive OR operation on the two input bits and writes the output in the second bit. The XOR gate can clone eigenstates but not superpositions. This makes quantum error correction difficult to carry out, but researchers soon found that entangled states created by XOR provides an alternative means for making error corrections (see Chap. 11).

Note that while unitary matrices are reversible operations on a system of qubits , the read-out process is a classical measurement process that collapses the quantum system to an eigenstate and the process is irreversible. Also note that a product of two unitary matrices is another unitary matrix.

7.8 Concluding Remarks

Once it was realized that reversible computing was possible within the paradigm of classical physics, i.e., both logically and thermodynamically by a physical apparatus dissipating arbitrarily little energy,³¹ the idea of building quantum computers appeared feasible. We now generally assume that quantum computers, in principle, are more powerful than classical computers as the algorithms described in Chaps. 8 and 10 seem to indicate. The unique quantum features of quantum computers are their ability to put qubits in superposed and entangled states and the ability of unitary operators to act in parallel on all the superposed states of a qubit or a group of qubits . As we shall see, e.g., in Chap. 8, Sect. 8.​2.​6, it is possible to calculate the value of a function $$ f(x) $$ for multiple values of x in parallel. Classical computers achieve parallelism by running parallel circuits for each value of x; in quantum computers a single circuit suffices. Indeed, quantum parallelism comes for free. Further, the Hadamard gate allows genuine random numbers to be generated, and hence it allows quantum computers to efficiently simulate a non-deterministic classical computer, such as the Probabilistic Turing Machine (PTM) (see Chap. 10, Sect. 10.​5). However, there is an important difference between a PTM and a quantum computer. In a PTM, the various computing alternatives (say, chosen by the throw of a “dice”) exclude one another in a given computer run (execution of the algorithm); in a quantum computer one can try all alternatives in parallel. Moreover, in a quantum computer, the various alternatives can interfere with one another because of the wave property of qubits , a phenomenon that can be used in novel ways, e.g., to yield some global property of the function $$ f(x) $$. As we shall see in Chaps. 8 and 10, the essence of designing quantum algorithms lies in the clever use of quantum superposition of states, entanglement of quantum states, wave interference, and collapse of superposed quantum states.

References

1.

A. Barenco et al., Elementary gates for quantum computation. arXiv:​quant-ph/​9503016v1, 23 Mar 1995. https://​arxiv.​org/​pdf/​quant-ph/​9503016v1.​pdf. Also as: A. Barenco, C.H. Bennett, R. Cleve, D.P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J.A. Smolin, H. Weinfurter, Elementary gates for quantum computation. Phys. Rev. A 52, 3457–3467 (1995)

2.

A. Barenco, A universal two-bit gate for quantum computation (1995). arXiv:​quant-ph/​9503016v1, http://​arxiv.​org/​PS_​cache/​quant-ph/​pdf/​9505/​9505016v1.​pdf

3.

C.H. Bennett, Logical reversibility of computation. IBM J. Res. Dev. 17, 525–532 (1973). https://​www.​math.​ucsd.​edu/​~sbuss/​CourseWeb/​Math268_​2013W/​Bennett_​Reversibiity.​pdf

4.

C.H. Bennet, The thermodynamics of computation—a review. Int. J. Theor. Phys. 21, 905–940 (1982). https://​www.​cc.​gatech.​edu/​computing/​nano/​documents/​Bennett%20​-%20​The%20​Thermodynamics%20​Of%20​Computation.​pdf. Reprinted in H.S. Leff, A.F. Rex (eds.), Maxwell’s Demon: Entropy, Information, Computing, Chapter 4, Section 4.2 (CRC Press, Boca Raton, 1990)

5.

A. Church, An unsolvable problem of elementary number theory. Am. J. Math. 58, 345–363 (1936)

6.

D. Deutsch, Quantum theory, the Church-Turing principle and the universal quantum computer. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 400(1818), 97–117 (1985). http://​www.​qubit.​org/​oldsite/​resource/​deutsch85.​pdf

7.

D. Deutsch, A. Barenco, A. Ekert, Universality in quantum computation. Proc. R. Soc. Lond. A 449, 669–677 (1995)

8.

P.A.M. Dirac, The Principles of Quantum Mechanics, 4th edn. (Oxford University Press, Oxford, 1958)

9.

D.P. DiVincenzo, Two-bit gates are universal for quantum computation. Phys. Rev. A 50, 1015–1022 (1995). Also at https://​arxiv.​org/​abs/​cond-mat/​9407022

10.

E. Fredkin, T. Toffoli, Conservative logic. Int. J. Theor. Phys. 21, 219–253 (1982). http://​calculemus.​org/​logsoc03/​materialy/​ConservativeLogi​c.​pdf

11.

R. Landauer, Irreversibility and heat generation in the computing process. IBM J. Res. Dev. 5(3), 183–191 (1961). Reprinted in IBM J. Res. Dev. 44(1–2), 261–269 (2000). http://​www.​pitt.​edu/​~jdnorton/​lectures/​Rotman_​Summer_​School_​2013/​thermo_​computing_​docs/​Landauer_​1961.​pdf

12.

S. Lloyd, Almost any quantum logic gate is universal. Phys. Rev. Lett. 75(2), 346–349 (1995). http://​capem.​buffalo.​edu/​rashba/​p346_​1.​pdf

13.

D.E. Muller, Complexity in electronic switching circuits. IRE Trans. Electr. Comput. 5, 15–19 (1956).

14.

A. Muthukrishnan, Classical and quantum logic gates: an introduction to quantum computing, in Quantum Information Seminar, Rochester Center for Quantum Information, 03 Sept 1999. http://​www2.​optics.​rochester.​edu/​~stroud/​presentations/​muthukrishnan991​/​LogicGates.​pdf

15.

M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000). [Errata at http://​www.​squint.​org/​qci/​]

16.

V.V. Shinde, I.L. Markov, On the CNOT-cost of TOFFOLI gates, 15 Mar 2008, arXiv. Available at http://​arxiv.​org/​abs/​0803.​2316; also as Quant. Inf. Comp. 9(5–6), 461–486 (2009)

17.

T. Toffoli, Reversible computing, in MIT Technical Report MIT/LCS/TM-151 (1980). http://​pm1.​bu.​edu/​~tt/​publ/​revcomp-rep.​pdf

18.

A.M. Turing, On computable numbers, with an application to the Entscheidungsproblem. Proc. Lond. Math. Soc. S2-42(1), 230–265 (1936–1937). https://​www.​cs.​virginia.​edu/​~robins/​Turing_​Paper_​1936.​pdf. Correction at: A.M. Turing, On computable numbers, with an application to the Entscheidungsproblem. A correction. S2-43(1), 544–546 (1938). http://​www.​turingarchive.​org/​viewer/​?​id=​466&​title=​02

Footnotes

1

Dirac [8, p. 47].

2

Dirac [8, p. 52].

3

Landauer [11].

4

Bennett [4].

5

Toffoli [17].

6

Fredkin and Toffoli [10].

7

Deutsch [6, p. 107].

8

Although we use the electron in an atom as an example here, one may, instead, use two different polarizations of a photon or the alignment of a nuclear spin in a uniform magnetic field or the two orthogonal spin states (up and down) of an electron, etc.

9

Permutation: each of several possible ways in which a set or number of things can be ordered or arranged.

10

A half-silvered mirror effects this transformation on a photon when it encounters the mirror.

11

In modulo 2 arithmetic , the following addition rules for adding two binary numbers are quite obvious:

Analogously, the XOR (or Cnot) operation is:

12

See Barenco et al. [1].

13

It is named after Tommaso Toffoli, who in 1980 showed that the classical version is universal for classical reversible computation. See Toffoli [17]. To know more about Toffoli, visit http://​pm1.​bu.​edu/​~tt/​vita.​pdf.

14

Shinde and Markov [16].

15

Named after Edward Fredkin. See Fredkin and Toffoli [10].

16

A proof is available in Nielsen and Chuang [15, pp. 174–7]. See also: Barenco et al. [1].

17

For a proof and implementation example, see Nielsen and Chuang [15, p. 176 and pp. 180–182].

18

Muller [13].

19

See, e.g., Nielsen and Chuang [15, pp. 189 and 191].

20

Even though S = T², i.e., the phase gate S is included because of its natural role in the fault-tolerant constructions. See Nielsen and Chuang [15, p. 189].

21

DiVincenzo [9].

22

Barenko [2].

23

Lloyd [12].

24

Deutsch et al. [7].

25

Deutsch [6].

26

Muthukrishnan [14].

27

Muthukrishnan [14].

28

Bennett [3].

29

Turing [18]. See also: Church [5].

30

Bennett [4].

31

See, e.g., Landauer [11]; Bennett [3]. Bennett, in particular, showed that a Turing machine “may be made logically reversible at every step, while retaining their simplicity and their ability to do general computations.” He also discussed the biosynthesis of messenger RNA as a physical example of reversible computation.

© Springer Nature Singapore Pte Ltd. 2020

R. K. BeraThe Amazing World of Quantum ComputingUndergraduate Lecture Notes in Physicshttps://doi.org/10.1007/978-981-15-2471-4_8

8. Unusual Solutions of Usual Problems

Rajendra K. Bera1  

(1)

Acadinnet Education Services India, Bangalore, Karnataka, India

Rajendra K. Bera

Email: rajendrabera@yahoo.com

Like mathematics, computer science will be somewhat different from the other sciences, in that it deals with artificial laws that can be proved, instead of natural laws that are never known with certainty.

—Donald Knuth

Abstract

This chapter describes how certain problems that may be computationally solved on a Universal Turing Machine have a superior algorithmic solution in quantum computing. As examples, inter alia, we describe the Deutsch, the Deutsch–Jozsa, the Elitzur–Vaidman, and a few other algorithms. These problems and their related solution algorithms are easy to understand. They are also a convenient way of introducing certain frequently used computational steps in developing quantum algorithms, e.g., computing a function in parallel for multiple values of its argument.

8.1 Introduction

Quantum mechanics provides some unusual solutions to some usual problems as we shall see in the examples below. The reader is advised to go through them carefully and slowly as it will help them to understand the more sophisticated algorithms presented in Chap. 10. The examples presented here show that quantum computers can do certain things that classical computers based on classical mechanics cannot do. For example, quantum computers can produce genuine random numbers (as opposed to pseudo-random numbers by classical computers), distribute cryptographic keys with complete security, and teleport unknown information (see Chap. 1, Sect. 1.5) without violating the no-cloning theorem. Further, it is from Bell’s inequality that we learn that entanglement is a powerful resource not available in classical physics. Therefore, by using this resource in a computational problem, we open a new world of possible algorithms unimaginable in classical computations.

The reader should bear in mind Donald Knuth’s’ observation: “algorithms are concepts that have existence apart from any programming language.” This is also clear from the nature of the universal Turing machine described by Alan Turing and the Church–Turing thesis that asserts an equivalence between a rigorous mathematical concept—function computable by a Turing machine—and the intuitive concept of what it means for a function to be computable by an algorithm.¹

8.1.1 Mach–Zehnder Interferometer

Before we look at any algorithms, let us consider the eerie behavior of a photon in a Mach–Zehnder interferometer .² The setup has a photon source, a pair of half-silvered mirrors, a pair of fully silvered mirrors, and a pair of photon detectors arranged as shown in Fig. 8.1.

Fig. 8.1

Mach–Zehnder interferometer

Photons from the source impinge on a half-silvered mirror (bottom-left corner); half of the photons take the horizontal path and the other half the vertical path. Along each path between the two half-silvered mirrors is a fully silvered mirror which deflects the photons it receives in a perpendicular direction as shown in the figure. Finally, the two beams meet at the half-silvered mirror at the top right corner of the figure. Our commonsense tells us that after passing through the half-silvered mirror at the top-right corner, the beam will split, and detectors A and B would each detect the photon with equal probability. Will it? Amazing as it sounds, only detector B will catch the photons! We emphasize that what we have seen is a property of single photons. Each individual photon, even though it remains one photon (it does not split at the half-silvered mirrors) somehow feels out both routes that are open to it so that at the final half-silvered mirror both beams are “present” simultaneously and the two beams interfere to produce just the single possibility of hitting detector B. The Mach–Zehnder interferometer starkly presents the conflict between the wave and particle theories of light in an acute form. Here is the resolving explanation.

A photon in state $$ \left| h \right\rangle $$ impinges on a half-silvered mirror³ and its state evolves into a superposition $$ \left| {h_{1} } \right\rangle + i\left| {v_{1} } \right\rangle $$. We now try to bring the two split beams (or the two parts of the photon state) together again by first reflecting each beam with a fully silvered mirror. After reflection, the photon state $$ \left| {h_{1} } \right\rangle $$ evolves into another state $$ i\left| {v_{2} } \right\rangle $$ while $$ \left| {v_{1} } \right\rangle $$ evolves into $$ i\left| {h_{2} } \right\rangle $$. Finally, the two beams meet at the half-silvered mirror at the top right corner of the figure. Each individual photon, even though it does not split, somehow feels out both routes that are open to it so that at the final half-silvered mirror both beams are present simultaneously and the two beams interfere to produce just the single possibility $$ \left| b \right\rangle $$. The photon dynamics is described below in steps numbered 1–6:

1.

2.

3.

4.

5.

6.

In the last step, the two beams, one in state $$ \left| {h_{2} } \right\rangle $$ and the other in state $$ \left| {v_{2} } \right\rangle $$, interfere; the result of which is state $$ \left| b \right\rangle $$. This explains why all the photons end up at detector B.

Note that the half-silvered mirror is a physical implementation of the single qubit Hadamard gate, H:

The Mach–Zehnder interferometer was invented in 1891, well before quantum mechanics was born. With the benefit of hindsight and the fact that the mirrors act as unitary quantum gates , and hence can act reversibly (i.e., from the output one can retrace back to the input) and the symmetrical set up of the apparatus, it is clear that from symmetry considerations the photon must end up at detector B even if the photons were released very slowly, one at a time, so that at any given time only one photon could possibly be travelling in the apparatus. This is because a photon can interfere with itself!

The above Mach–Zehnder interferometer may be generalized by placing phase shifters along one or more path segments of the photon. For example, by placing a phase shifter set at $$ \phi_{0} $$ in the path of $$ \left| {h_{1} } \right\rangle $$ and a phase shifter set at $$ \phi_{1} $$ in the path of $$ \left| {h_{2} } \right\rangle $$, the phase shifters in the two paths can be tuned to effect any prescribed relative phase shift $$ \phi = \phi_{1} - \phi_{0} $$ and to direct the particle with probabilities $$ (1/2) (1 + \cos \phi ) $$ and $$ (1/2) (1 - \cos \phi ) $$, respectively, to detectors A and B. For example, when $$ \phi = 0 $$ all photons will end up at detector A, and when $$ \phi = \pi $$ they will end up at detector B. The second half-silvered mirror (top right corner) effectively erases all information about the path taken by the particle (path $$ \left| {h_{1} } \right\rangle $$ or path $$ \left| {h_{2} } \right\rangle $$) which is essential for observing quantum interference in the experiment.⁴

The secret of designing quantum algorithms lies in the intelligent use of quantum interference and quantum entanglement. It pays to view quantum computation as multi-particle interferometry, i.e., as multi-particle interferometers with phase shifts that result from operations of some quantum logic gates. Recall that in quantum computing, unitary operators are used (under Postulate 1 of quantum mechanics) to rotate state vectors that represent wave functions.

As an aside, note that quantum algorithm designers may jump among one of three basic notations of state vector representation: integer kets, binary kets, and vectors. For example,

represent the same state in 3-qubit space.

8.2 Some Simple Quantum Algorithms

The key to understanding the following algorithms is to refer to the postulates of quantum mechanics (Chap. 2, Sect. 2.​7) whenever in doubt. It takes a while to get a hang of dealing with Hilbert spaces and the fact that measurement in quantum mechanics has a very different meaning from that in classical mechanics.

8.2.1 Computing x ∧ y

Take 3 qubits , each prepared in state $$ \left| 0 \right\rangle $$. The first qubit is the placeholder for x, the second for y, and the third for the result of x ∧ y. To create the 4 possible inputs of x and y, $$ \left\{ {\left| {00} \right\rangle + \left| {01} \right\rangle + \left| {10} \right\rangle + \left| {11} \right\rangle } \right\} $$, apply the Hadamard gate to the first two qubits :

An application of the Toffoli gate , T, now produces

Notice that the 3-qubit system is put into equal superposition of the four possible results of $$ x \wedge y $$. The result of ANDing the first two qubits appears in the third qubit.

8.2.2 Computing x + y

Start with $$ T\left| {\psi_{1} } \right\rangle $$ from above:

To the first two qubits of $$ \left| \phi \right\rangle $$, apply the Cnot gate, to get

where the second qubit is the sum and the third qubit is the carry bit. Note that the carry bit in the adder is the result of an AND operation. The carry and AND are really the same thing. The sum bit comes from an XOR gate (i.e., the Cnot operation).⁵

8.2.3 Swapping States

The following sequence of operations on a computational basis state $$ |a,b\rangle $$ shows how the state of two qubits $$ |a,b\rangle $$ can be changed to $$ \left| {b,a} \right\rangle $$ by three applications of the Cnot gate

where ⊕ denotes modulo 2 addition.⁶

8.2.4 The Deutsch Algorithm

Consider the Boolean functions f that map $$ \left\{ {0, 1} \right\} $$ to $$ \left\{ {0, 1} \right\} $$. There are exactly four such functions, which may be categorized into two sets:

(1)

two “constant” functions: $$ f(0) = f(1) = 0\;{\text{and}}\;f(0) = f(1) = 1 $$;

(2)

two “balanced” functions: $$ f(0) = 0,\;f(1) = 1\;{\text{and}}\;f(0) = 1,f(1) = 0 $$.

Our interest is in identifying the set to which the binary numbers $$ f(0) $$ and $$ f(1) $$ belong to in only one measurement. Note that we do not seek the values of $$ f(0) $$ and $$ f(1) $$ but only a global property of f.

The solution provided here is reported in Cleve et al. [4],⁷ which is an improved version of the original Deutsch [5] solution.⁸ Deutsch’s solution had three possible outcomes: “balanced,” “constant,” and “inconclusive” with 50% probability that it would produce an “inconclusive” result. Nevertheless, it is a task that no classical computation can accomplish. The solution described here always correctly identifies “constant” and “balanced” functions.

Take a 2-qubit computer in the initial state $$ \left| \psi \right\rangle = \left| {01} \right\rangle $$ and apply the Hadamard gate to each qubit:

Now consider the gate $$ U_{f} $$ defined by the map $$ \left| {x, y} \right\rangle \to \left| {x, y \oplus f\left( x \right)} \right\rangle $$. Then

Bear in mind that $$ f(x) $$ can take only one of two values: 0 or 1.

The reader should make note of this clever jugglery that led to the same formula for both values of $$ f(x) $$ since it frequently appears in many quantum algorithms. Therefore,

Applying the above formula to our state $$ \left| {\psi_{1} } \right\rangle $$ yields

We now apply the Hadamard gate to the first qubit :

Here, we note that if $$ f(x) $$ is constant, then

And if $$ f\left( x \right) $$ is balanced, then

The state $$ \left| {\psi_{3} } \right\rangle $$ can now be written in the compact form

We therefore have

which can be concisely written as

What has been cleverly accomplished is that by measuring the first qubit , we may determine $$ \left| {f(0) \oplus f(1)} \right\rangle $$ which will tell us if $$ f(0) = f(1) $$ or not. In essence, just one evaluation of $$ f(x) $$ has allowed us to find a global property of $$ f(x) $$, namely $$ f(0) \oplus f(1) $$. A classical computer would have required at least two. The Deutsch algorithm provided the first example that a quantum computer could do something that a classical computer cannot. This was a remarkable demonstration.

8.2.5 The Deutsch–Jozsa Algorithm

The Deutsch–Jozsa algorithm⁹ solves a generalized Deutsch problem. Given a black box $$ U_{f} $$ whose action is to perform $$ \left| {x,y } \right\rangle \to \left| {x,y \oplus f(x)} \right\rangle $$ for $$ x \in \{ 0, 1\}^{n} \equiv \{ 0, \ldots , 2^{n} - 1\} $$ and $$ f(x) \in \{ 0,1\} $$, and that it is known beforehand that $$ f(x) $$ is one of two kinds: Either $$ f(x) $$ is constant for all values of x, or else $$ f(x) $$ is balanced, that is, equal to 1 for exactly half of all possible x, and 0 for the other half. The significance of this problem is that a classical solution, in the worst-case, would require $$ 2^{n - 1} + 1 $$ evaluations of $$ f(x) $$ before determining the answer with certainty since one may receive $$ 2^{n} /2 $$ 0s before finally getting a 1. Remarkably, the Deutsch–Jozsa algorithm requires only one evaluation of $$ f(x) $$.

We begin with n qubits , initially in state $$ \left| 0 \right\rangle^{ \otimes n} $$ and an auxiliary qubit in state $$ \left| 1 \right\rangle $$. Thus

We then apply the Hadamard gate to each qubit in $$ \left| {\psi_{0} } \right\rangle $$:

Next, the function $$ f(x) $$ is evaluated using $$ U_{f} :\left| {x,y } \right\rangle \to \left| {x,y \oplus f(x)} \right\rangle $$, giving

We then apply the Hadamard gate to the first n qubits of $$ \left| {\psi_{2} } \right\rangle $$ and obtain

Note that the last qubit has remained unchanged from states $$ \left| {\psi_{1} } \right\rangle $$ to $$ \left| {\psi_{3} } \right\rangle $$. The solution to the Deutsch–Jozsa problem can be found in the first n qubits . For the case where $$ f\left( x \right) $$ is constant, the amplitude of $$ \left| 0 \right\rangle^{ \otimes n} $$ is +1 or −1, depending on the constant value $$ f(x) $$ takes. Because $$ \left| {\psi_{3} } \right\rangle $$ is of unit length, it follows that all the other amplitudes must be zero, and a measurement will yield 0s for all the first n qubits . If $$ f(x) $$ is balanced, then the positive and negative contributions to the amplitude for $$ \left| 0 \right\rangle^{ \otimes n} $$ cancel, leaving an amplitude of zero, and a measurement must yield a result other than zero on at least one qubit in the first n qubits .

The Deutsch–Jozsa algorithm later inspired two revolutionary quantum algorithms, namely Shor’s factoring algorithm and Grover’s search algorithm . Both algorithms are described in Chap. 10.

8.2.6 Computing f(x) in Parallel

Suppose $$ f(x):\{ 0,1\} \to \{ 0,1\} $$. Now consider a 2-qubit quantum computer which is in the state $$ \left| {x, y} \right\rangle $$. Let $$ U_{f} $$ be the unitary transformation that maps $$ \left| {x,y} \right\rangle \to \left| {x,y \oplus f(x)} \right\rangle $$. Note that the first qubit contains the input value and the second must contain the value of $$ f(x) $$ if $$ y = 0 $$, i.e.,

Now, if the first qubit is in state $$ \left( {\left| 0 \right\rangle + \left| 1 \right\rangle } \right)/\sqrt 2 $$ and $$ U_{f} $$ is applied, we get

This is remarkable. The first term contains information about $$ f(0) $$ along with its input x = 0, and the second term contains information about $$ f(1) $$ along with its input x = 1. That is, the second qubit is in an equal superposition of $$ f(0) $$ and $$ f(1) $$, and the two values were calculated in parallel.

The power of quantum algorithms comes from taking advantage of quantum parallelism and entanglement. So, most quantum algorithms begin by computing a function $$ f(x) $$ on a superposition of all values of x as follows:

Start with an n qubit state $$ \left| {00 \ldots 0} \right\rangle $$ and apply H to all the qubits to get

which is the superposition of all integers $$ 0 \le x &lt; 2^{n} $$. Next, add a k-qubit register in state $$ \left| {00 \ldots 0} \right\rangle $$, large enough to hold the longest value of $$ f(x) $$. Then by linearity

Thus, $$ f(x) $$ is computed in parallel for all values of $$ x \in \left\{ {0, \ldots , 2^{n} - 1} \right\} $$, provided one knows how to construct the appropriate $$ U_{f} $$ that will map a pair of qubit strings $$ \left| {x, 0} \right\rangle $$ into the pair $$ \left| {x,f\left( x \right)} \right\rangle $$. What is unusual here is that quantum parallelism, in a sense, comes for free without the need to have multiple copies of the “processing unit.” The output of this system comes from the constructive interference among the parallel computations.

The trick in constructing Uf is to break down the computation, corresponding to the function $$ f(x) $$, into a set of 1-qubit and 2-qubit unitary operations such that the state $$ \left| {x, 0} \right\rangle $$ is mapped to the state $$ \left| { x, f\left( x \right)} \right\rangle $$ for any input X. Note that the number of qubits required for the second register must be at least sufficient to store the longest result $$ f(x) $$ for any of these computations. In general, in designing quantum algorithms, one assumes that for reasonable functions, a suitable $$ U_{f} $$ can be built.

8.2.7 Hardy’s Reprieve

This interesting example is from von Baeyer.¹⁰ The original version is due to Hardy.¹¹ It has since been refined and simplified by several researchers.

In a prison, two crooked quantum physicists are put in solitary confinement, convicted of unspeakable crimes. The warden is rather anxious to get rid of such convicts and offers them a tantalizing deal. Each of them, in isolation, will be asked to select one of a pair of cards: If they pick the right cards, they go free; if they choose badly, they are executed. The grapevine has it that more prisoners fail at the game than are granted a reprieve. A quick inspection of the game shows why: The warden has rigged the game so that execution is three times more likely than pardon (Fig. 8.2).

Fig. 8.2

Hardy’s reprieve. Baeyer [1, pp. 14–17]

The card game: The warden brings the two physicists together and shows them a diagram, which tells them how their fates depend on each pair of cards they can draw. The first player’s moves are given by the outermost square, and the second player’s moves are given in the square just inside it. Each prisoner gets a chance to choose one card, and together the two cards determine a single colored region on the inside of the diagram. If both prisoners pick the ace, they go free. But in three cases—if both choose the queen, or if one picks the jack—they are both executed.

The warden explains that in separate rooms, each of them will be given one of two possible commands: either “Pick ace or king” or “Pick queen or jack.” Each prisoner then chooses a card according to the command. The commands themselves are chosen at random, by the flip of a coin; so no one has prior knowledge of which command will be given, and there is nothing to prevent both prisoners from receiving the same command.

As they are led back to their separate cells, they struggle to find a way to avoid picking the fatal combinations of queen–queen or ace–jack without sacrificing their chances of freedom with the ace–ace. They quickly conclude that classical physics offers no hope. The game is so deviously rigged that on the one hand, if the physicists play it safe and only agree in advance to pick only kings or jacks, they can avoid the gallows, but they can never go free. The same is true if one physicist promises to choose either aces or jacks while the other picks only kings or queens, or if one promises to pick kings and queens while the other picks kings or jacks. On the other hand, any agreement that offers a chance at the ace–ace combination also poses the risk of death.

Our desperate physicists wonder if quantum mechanics can help. They carefully consider such things as quantum superposition and entanglement, the EPR paradox, Bell’s inequalities, and experiments by Aspect et al. which showed that the Bell inequalities were violated by entangled quantum particles. Perhaps this violation might help them in improving the odds of their pardon. And bingo, they find a solution!

Before they agree to the warden’s deal, they place a pair of elementary particles (electrons, if you will) into an entangled quantum state—a superposition of three states: up-down $$ (\left|01 \right\rangle) $$, down-up $$ (\left|10 \right\rangle) $$, and down-down $$ (\left|11 \right\rangle) $$—all of which have equal probability of being observed.¹² They also agree to the following convention (that is, select their basis) about how to choose cards on the basis of a later observation of the particles:

Thus, the state vector of the entangled electron-pair is given by

where $$ \left|0 \right\rangle $$ denotes spin-up, $$ \left|1 \right\rangle $$ spin-down, the computational basis is {$$ \left|0 \right\rangle$$, $$ \left|1 \right\rangle$$} and measurement bases are {$$\left|0 \right\rangle $$, $$ \left|1 \right\rangle$$} (up-down spin) and {$$ \left|+ \right\rangle $$, $$ \left|- \right\rangle$$} (left-right spin).

Recall that due to the entangled state, when one physicist observes his particle, it immediately imposes limits on the possible states of the other. When the second physicist observes his particle, the prearranged convention enables the physicists to choose cards that preserve their chance of being set free and eliminate the risk of execution. Note that an electron’s spin can point in only left or right directions when measured along the horizontal, and only up or down when measured along the vertical. Each now carries one of the two electrons to his chamber. Each on receiving the warden’s command to pick a card measures the spin in the horizontal or the vertical direction, depending on the command.

How does the whole thing work?

When the first physicist receives, say, “Pick the queen or the jack,” he checks the vertical spin of his particle. Suppose it is spin-up, then he chooses the queen. The measurement also leaves the electron pair in the up-down state only. Now suppose the second physicist is also told to “Pick the queen or the jack,” he will find, on measurement, that his particles spin is down, and so he will choose the jack and avoid the death-bearing double queen.

It turns out that the three-state superposed state has a complicated though predictable effect on spins in the horizontal direction. When the two prisoners’ measurements are made in perpendicular directions—one horizontal, one vertical—the left-down and down-left combinations are ruled out. (Show that it is true to test your skills.) That means that the prisoners can avoid the deadly pairs ace–jack and jack–ace.

The superposed states do allow the left–left, the double ace that leads to a pardon. Thus, with a properly prepared pair of entangled particles, the game loses its risk yet retains a modest chance (about one in twelve) of reward.

8.2.8 The Elitzur–Vaidman Bomb Problem

Consider a bomb with an ultra-sensitive detonator on its nose that even a photon impinging on the detonator’s slightly wobbly mirror can set it off. However, there are problems of quality control during the manufacture of these bombs—in some cases the detonator is jammed and so the bomb fails to explode and is classed a dud. The problem is to determine if a bomb’s plunger is stuck or not without exploding the bomb. The problem and its solution (see Fig. 8.3) were proposed by Elitzur and Vaidman in 1993.¹³ Classical physics cannot determine without exploding the bomb.

The solution is an interesting application of the Mach–Zehnder interferometer . The photon source emits a single photon. Now two possible paths for the photon exist. If the photon takes the $$ \left|h_{1} \right\rangle $$ path (50% probability), the bomb’s mirror will either absorb the photon and wobble and the bomb will explode, or it will simply send the photon along the $$ \left|v_{2} \right\rangle $$ path because the plunger is stuck and the photon will be detected by the detector B. In this case, 50% of good bombs will explode (Fig. 8.3).

Fig. 8.3

Elitzur–Vaidman bomb problem

The more interesting case is when the photon takes the $$ \left|v_{1} \right\rangle $$ path (50% probability). The photon then does not impinge on the bomb’s mirror and so even a good bomb cannot explode. But the photon’s superposed state will still sense if the $$ \left|h_{1} \right\rangle $$ path has a wobbly mirror or not! If it senses a jammed mirror, the photon will end up at detector B. If it senses a wobbly mirror, it can end up at either detector A or at B. Note that if A detects a photon, then the bomb is not a dud. Thus, in half of the cases where an active bomb does not explode will the detector A register a photon. At the end of the tests, we would have found only a quarter of the originally active bombs which are guaranteed to be active. We can repeat the tests on the remaining doubtful bombs till no doubtful bomb is left. Ultimately, we will obtain just one-third (since 1/4 + 1/16 + 1/64 + … = 1/3) of the active bombs that we started with, but now all are guaranteed to be active.

What is remarkable here is that classically there would be no way of saving even a single good bomb, but quantum mechanically one can save one-third of them. With some refinements, the two-thirds wastefulness can be reduced to effectively one-half [8].¹⁴ In 1994, Kwiat, Weinfurter, Herzog, and Zeilinger performed an experiment verifying the Elitzur–Vaidman result, thereby proving that interaction-free measurements are indeed possible.¹⁵

In 1996, Kwiat, Weinfurter, Zeilinger, Herzog, and M. Kasevich devised a method, using a sequence of polarizing devices, which efficiently increases the yield rate to a level arbitrarily close to one.¹⁶ As Roger Penrose notes:

Classically, as the problem is phrased, there is no way of deciding whether the bomb detonator has jammed other than by actually wiggling it—in which case, if the detonator is not jammed, the bomb goes off and is lost. Quantum theory allows for something different: a physical effect that results from the possibility that the detonator might have been wiggled, even if it was not actually wiggled! What is particularly curious about quantum theory is that there can be actual physical effects arising from what philosophers refer to as counterfactuals—that is, things that might have happened, although they did not in fact happen.¹⁷

8.2.9 Securing Banknotes

Around 1970, Stephen Wiesner discovered that quantum mechanical effects could be used to produce banknotes that would be impossible to counterfeit. This revolutionary idea went completely unnoticed, except by his former undergraduate classmate Charles Bennett. Wiesner’s idea led Bennett and Brassard to develop their quantum key distribution algorithm. In fact, the indistinguishability of non-orthogonal quantum states (see Chap. 6, Sect. 6.​2.​3) may be used to protect banknotes from being counterfeited by imprinting the notes with a classical serial number, and a sequence of qubits each in either the state, say, $$ \left|0 \right\rangle $$ or ($$ \left|0 \right\rangle $$ +  $$ \left|1 \right\rangle $$/√2. If the bank maintains a confidential list of matched pairs of each note’s classical serial number and the sequence of the quantum states, then it would be impossible to counterfeit a note exactly. This is because it would be impossible for a would-be counterfeiter to determine with certainty the state of the qubits in the original note without destroying them. The genuineness of a note can be verified by a designated authority to see if the note has a matched pair of serial number and qubit sequence. Wiesner had submitted his paper “Conjugate Coding” to the IEEE Transactions on Information Theory. Unfortunately, it was rejected. The paper was later published in 1983.¹⁸ Wiesner’s quantum money proposal was tested experimentally, assuming a banknote to be an image encoded in the polarization states of single photons by Bartkiewicz et al. in 2017.¹⁹ Wiesner’s proposal was found to be feasible. This means that Wiesner’s method can also be used as a security measure in quantum digital right management.

8.3 Concluding Remarks

There is no doubt that the algorithms described in this chapter, although they solve some simple problems, nevertheless, required considerable ingenuity to construct. The Deutsch algorithm was simply brilliant in conception. In Chap. 10, we shall look at more complex algorithms, mainly Peter Shor’s highly efficient algorithms for finding the prime factors of an integer, and Lov Grover’s search algorithm which efficiently conducts a search through unstructured search space. These algorithms were remarkable breakthroughs, both in terms of superior computational efficiency in comparison with the best-known classical algorithms for the respective problems and in terms of conceptual ideas. So how difficult is it to design quantum algorithms compared to designing classical algorithms? At the moment, the task appears to be very hard! There are two sources of difficulty. The first is our human intuition which is deeply rooted in the classical world and so grossly hinders our ability to think in terms of the postulates of quantum mechanics ; and the second is the compulsive need to design quantum algorithms that are demonstrably superior to corresponding classical algorithms. The demands on the quantum algorithm designers are indeed onerous.

References

1.

H.C. von Baeyer, Tangled tales. The Sciences, Spring 2001, pp. 14–17

2.

K. Bartkiewicz et al., Experimental quantum forgery of quantum optical money. arXiv:​1604.​04453v2 [quant-ph], 2 Mar 2017. https://​arxiv.​org/​pdf/​1604.​04453.​pdf

3.

G. Brassard, Brief history of quantum cryptography: a personal perspective. arXiv:​quant-ph/​0604072v1, 11 Apr 2006. https://​arxiv.​org/​pdf/​quant-ph/​0604072.​pdf

4.

R. Cleve, A. Ekert, C. Macchiavllo, M. Mosca, Quantum algorithms revisited. Proc. R. Soc. Lond. A 454, 339–354 (1998). See preprint at: arXiv:​quant-ph/​9708016, v1 8 Aug 1997, http://​arxiv.​org/​PS_​cache/​quant-ph/​pdf/​9708/​9708016v1.​pdf

5.

D. Deutsch, Quantum theory, the Church-Turing principle and the universal quantum computer. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 400(1818), 97–117 (1985). http://​www.​ceid.​upatras.​gr/​tech_​news/​papers/​quantum_​theory.​pdf

6.

D. Deutsch, R. Jozsa, Rapid solutions of problems by quantum computation. Proc. R. Soc. Lond. A 439, 553–558 (1992). http://​www.​qudev.​ethz.​ch/​phys4/​studentspresenta​tions/​djalgo/​DeutschJozsa.​pdf

7.

A.J. DeWeerd, Interaction-free measurement. Am. J. Phys. 70(3), 272–275 (2002). http://​bulldog2.​redlands.​edu/​facultyfolder/​deweerd/​research/​IFM-AJP.​pdf

8.

A.C. Elitzur, L. Vaidman, Quantum mechanical interaction-free measurements. Found. Phys. 23, 987–97 (1993). Preprint available at http://​arxiv.​org/​PS_​cache/​hep-th/​pdf/​9305/​9305002v2.​pdf

9.

L. Hardy, Nonlocality for two particles without inequalities for almost all entangled states. Phys. Rev. Lett. 71(11), 1665–1668 (1993). https://​pdfs.​semanticscholar.​org/​f940/​11b5ad26ac9b8c8a​7f1e336d3d1f8545​0b31.​pdf

10.

P.G. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, M. Kasevich, Experimental realization of “Interaction-Free” measurements, in Symposium, on the Foundations of Modern Physics, 1994: 70 Years of Matter Waves, ed. by K.V. Laurikainen, C. Montonen, K. Sunnarborg (Frontières, 1994) pp. 129–138. http://​www.​univie.​ac.​at/​qfp/​publications3/​pdffiles/​1994-08.​pdf

11.

P. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, M. Kasevich, Interaction-free measurement. Phys. Rev. Lett. 74, 4763–4766 (1995a). https://​vcq.​quantum.​at/​fileadmin/​Publications/​1995-03.​pdf

12.

P.G. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, M. Kasevich, Experimental realization of interaction-free measurement, in Proceedings of the Conference on Fundamental Problems in Quantum Theory, held in honor of Prof. J.A. Wheeler, UMBC, Baltimore, Maryland June 19–22, 1995. Annals of the New York Academy of Sciences, ed. by D.M. Greenberger, J.A. Wheeler, A. Zeilinger, vol. 755, (1995b). http://​vcq.​quantum.​at/​fileadmin/​Publications/​1994-08.​pdf

13.

L. Mach, Ueber einen Interferenzrefraktor. Zeitschrift für Instrumentenkunde 12, 89–93 (1892)

14.

M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000). Errata at http://​www.​squint.​org/​qci/​

15.

R. Penrose, Shadows of the Mind (Oxford University Press, Oxford, 1994) (Vintage paperback)

16.

L. Vaidman, The Meaning of the Interaction-Free Measurements. Foundations of Physics, 33 (3), 491–510 (2003) http://​www.​tau.​ac.​il/​~vaidman/​lvhp/​m87.​pdf

17.

S. Wiesner, Conjugate coding. Sigact News 15(1), 78–88 (1983). https://​doi.​org/​10.​1145/​1008908.​1008920 (Original manuscript written circa 1970)

18.

L. Zehnder, Ein neuer Interferenzrefraktor. Zeitschrift für Instrumentenkunde 11, 275–285 (1891)

Footnotes

1

Nielsen and Chuang [14, p. 125].

2

It is named after Ludwig Mach (son of Ernst Mach) and Ludwig Zehnder who independently invented it in 1891. See Zehnder [18], and Mach [13].

3

Most efficient half-silvered mirrors are not silvered but made of a thin piece of transparent material of just the right thickness in relation to the wavelength of the light. It achieves its effect by a combination of repeated internal reflections and transmissions, so that the final transmitted and reflected beams are equal in intensity.

4

Cleve et al. [4].

5

Given $$ |x, y\rangle $$, the action of the Cnot gate on $$ |x, y\rangle $$ may be viewed as executing the instruction in a computer programming language: if $$ (|A\rangle $$ = 1) then $$ (|B\rangle $$ = NOT $$ (|B\rangle $$.

6

Modular arithmetic: A form of arithmetic dealing with the remainders after whole numbers is divided by a modulus. In modulo 2, the modulus is 2. Clocks use modular arithmetic with modulus 12 for displaying hours and modulus 60 for displaying minutes. A quick tutorial on modulo arithmetic is provided in Chap. 10, Sect. 10.​3.

7

Cleve et al. [4].

8

Deutsch [5].

9

Deutsch and Jozsa [6].

10

Baeyer [1, pp. 14–17].

11

Hardy [9].

12

The only possible outcomes of measuring an electron’s spin are “up” and “down” with respect to a chosen axis. The chosen axis may be vertical, horizontal, or tilted.

13

Elitzur and Vaidman [8]. See also: Penrose [15, pp. 239–240 and 268–270]; and Bayer [1, p. 16].

14

Elitzur and Vaidman [8]. See also: Vaidman [16], (it clarifies the meaning of the interaction-free measurements).

15

Kwiat et al. [11]. See also: Kwiat et al. [12].

16

Kwiat et al. [10], See also: DeWeerd [7].

17

Penrose [15, p. 240].

18

See Wiesner [17]. Original manuscript written circa 1970. As Brassard notes: “It is fortunate that Wiesner had expounded his ideas to Bennett, for they might otherwise have been lost forever. Instead, Bennett mentioned them occasionally to various people in the subsequent years, invariably meeting with very little sympathy until…” See Brassard [3] for some interesting history.

19

Bartkiewicz et al. [2].

© Springer Nature Singapore Pte Ltd. 2020

R. K. BeraThe Amazing World of Quantum ComputingUndergraduate Lecture Notes in Physicshttps://doi.org/10.1007/978-981-15-2471-4_9

9. Fundamental Limits to Computing

Rajendra K. Bera1  

(1)

Acadinnet Education Services India, Bangalore, Karnataka, India

Rajendra K. Bera

Email: rajendrabera@yahoo.com

Science may set limits to knowledge, but should not set limits to imagination (Russell [32], p. 16).

—Bertrand Russell

Abstract

This chapter introduces certain fundamental limits that mathematics, thermodynamics, information theory, and computational complexity impose on algorithm development. Topics include Hilbert’s second and tenth problem, Turing’s halting problem , resolution of Maxwell’s demon paradox, classification of computational complexity , and a brief discussion on NP-complete problems. The aim is to provide an understanding of the deep issues involved in the development of quantum algorithms and the hurdles that lie ahead.

9.1 Introduction

Before 1800, mathematics was largely a practical matter, concerned with making statements about real world objects. However, in the 1800s mathematicians began to invent, and then reason about, imaginary objects to which they ascribed properties that were not necessarily compatible with “common sense.” The truth or falsity of statements made about such imaginary objects could not be determined by appeal to the real world; hence, David Hilbert advocated a formalist approach to proofs.

To a formalist, symbols cease to have a meaning other than that implied by their relationships to one another. No inference is permitted unless there is an explicit rule that sanctions it, and no information about the meaning of any symbol enters into a proof from the outside. In short, the formalist approach lent itself to automation. The modern emphasis of axiomatization of mathematics began with Hilbert, although Euclid had shown the way in his text book concerned with geometry, algebra, and number theory, called Elements written about 300 B.C. The deep significance of Elements is that geometry was presented as an axiomatic system, most probably for the first time. Many centuries later, René Descartes invented coordinate geometry by assigning number-pairs to the points of plane Euclidean geometry and proved geometrical theorems about points by proving algebraic theorems about numbers. Euclidean geometry was thus reduced to a branch of algebra. The precise relationship between the arithmetic of abstract numbers and the geometry of physical space is amazing.

In this chapter, we explore the limits of axiomatic mathematics and, in particular, raise the question, “Is there a limit to what we can, in principle, compute?” Given that quantum mechanics is built on an axiomatic system, and this book is about quantum computing, the importance of either is self-evident. This chapter looks at certain important limitations imposed on computing. They come from three diverse directions: limitations imposed by the very nature of axiomatic mathematical systems (Sects. 9.2, 9.3 and 9.4); limitations imposed by the laws of thermodynamics on physically realizable computer systems (Sect. 9.5); and limitations imposed by the complexity of the best-known algorithms for solving a given problem (Sect. 9.6).

9.2 Hilbert’s Second Problem

The modern axiomatic view of mathematics begins with David Hilbert, who in 1900 posed a series of 23 problems for mathematicians to investigate.¹ The second problem in the list—The Compatibility of the Arithmetical Axioms—was truly breathtaking in its ambition:

Whether, in any way, certain statements of single axioms depend upon one another, and whether the axioms may not therefore contain certain parts in common, which must be isolated if one wishes to arrive at a system of axioms that shall be altogether independent of one another.

But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms: To prove that they are not contradictory, that is, that a definite number of logical steps based upon them can never lead to contradictory results.

What Hilbert sought was the presentation of mathematics as a completely formal axiomatic system, encased in an artificial language, so that the system, in principle, could be manipulated completely mechanically by machines. Clearly, axiomatic systems were meant to be constructed without any a priori meaning and viewed purely as a game of symbol manipulation according to prescribed rules, where the symbols are no more than marks on paper. Through such mechanized rules-based manipulations or combinatorial play with symbols, machines (or humans working in a mechanistic manner without the benefit of intelligence or insight) would be able to deduce theorems² from axioms. To make the system independent of interpretation, all the rules of the game would have to be described in great detail—the definitions, the elementary concepts, the language and its grammar, and rules of inference—to the extent that all can agree on how mathematics should be done. In a sense, mathematical reasoning was to be “atomized” into such tiny steps that nothing was left to the imagination, and nothing was left out.

The hope was that such a system would help settle, once and for all, the question of whether a piece of mathematical reasoning is correct or not and that it would not produce any paradoxes. His plan seemed reasonable and plausible. If this could be achieved, its consequences would be truly enormous. It would, in principle, allow the possibility of relegating proof and disproof to a formal procedure that could be carried out by a computer! Mathematicians would then be free to make conjectures and computers could do the slave work of proving or disproving those conjectures. Of course, in such a world only very intelligent mathematicians would survive. To physicists, the issue was of fundamental importance because the key to the understanding of Nature lay within an unassailable mathematics. Of course, it is true that vast portions of mathematics today has no obvious connection with physics, although we may be frequently surprised by the discovery of unexpected important applications.

Thirty years later, Kurt Gödel (1906–1978) in 1931, at age 25, electrified the mathematical world by showing that Hilbert’s ideal axiomatic system cannot be constructed.³ He proved that no formal axiomatic system of mathematics sufficiently strong⁴ to allow one to do basic arithmetic (such as Peano arithmetic) can be at once consistent and complete. This result, famously known as Gödel’s (First) Incompleteness Theorem, states that

Every system of arithmetic contains arithmetical propositions, by which is meant propositions concerned solely with relations between whole numbers, which can neither be proved nor disproved within the system.

This theorem is a monumental achievement of twentieth-century mathematics, and perhaps the most important theorem in all of mathematics. To add fuel to fire, Gödel’s (Second) incompleteness theorem stated

If number theory is consistent, then a proof of this fact does not exist using the methods of first-order predicate calculus .

Surprisingly, an inconsistent axiomatic system can prove that it is consistent! So, a consistency proof of T in T is meaningless. However, it may be possible to prove the consistency of T in some system T′ which is in some sense more believable than T itself. It is in this sense that the consistency of Peano arithmetic is proved in Zermelo–Fraenkel set theory. But a proof in T′ is not completely convincing unless the consistency of T′ has already been established without using T. Further, the problem of proving consistency now shifts from T to T′, and for T′ to, say, T″, ad infinitum.

The existence of an incomplete system is not particularly surprising since a system can be incomplete because all the required axioms may not have been discovered. However, the reference to incompleteness here is something deeper; we can never discover the complete list of axioms. Each time we add a statement as an axiom, there will always be another statement out of reach. We can, of course, create an axiom system where we add an infinite number of axioms (for example, all true statements about the natural numbers, in which case we will have a dictionary of facts about numbers rather than an axiomatic system that humans or finite machines can deal with). Unfortunately, such a list will not be a recursive set (see Sect. 9.2.1). Given a random statement, there will be no way to know if it is an axiom of the system or not. Furthermore, if a proof is given that the given statement is true, in general there will be no way for one to check if that proof is valid.

9.2.1 Recursive Set

A set is said to be recursively enumerable if there is an algorithm⁵ that correctly decides whether a number is in the set; the algorithm may give no answer (but never the wrong answer) for numbers not in the set.

A set of natural numbers is called recursive, computable, or decidable if there is an algorithm which terminates after a finite amount of time and correctly decides whether or not a given number belongs to the set. Otherwise, it is called non-computable or undecidable.

If A is a recursive set , then the complement of A is a recursive set . A set is recursive if and only if A and the complement of A are recursively enumerable sets.

Let N be the set of natural numbers and C be the subset of all composite numbers in N and P be the subset of all prime numbers in N. There exist algorithms that will correctly enumerate all composite numbers; therefore, C is recursively enumerable. Likewise, there exist algorithms that will correctly enumerate all prime numbers; therefore, P is also recursively enumerable. Since both C and P are recursively enumerable, and one is the complement of the other, both C and P are recursive.

A recursive process is one in which objects are defined in terms of other objects of the same type. Using some sort of recurrence relation, the entire class of objects can then be built up from a few initial values and a small number of rules. Care, however, must be taken to avoid self-recursion,⁶ in which an object is defined in terms of itself, leading to an infinite nesting.

A function is recursive if there is an algorithm for calculating its value when one is given the value of its arguments. In other words, there is a computer for doing this. If it is possible that this algorithm never halts and the function is thus undefined for some values of its arguments, then the function is called partial recursive. In this sense, a computer is a partial recursive function C(p); its argument p is a binary string and the value of C(p) is the binary string output by the computer C when it is given the program p to execute. If C(p) is undefined, it means that running the program on C produces an unending computation. The set of functions that can be defined recursively in this manner is known to be equivalent to the set of functions computed by Turing machines (see Sect. 9.4) and by the lambda calculus.⁷

Examples of non-halting problem

1.

Find a number that is not the sum of four square numbers.

2.

Find an even number >2 that is not the sum of two primes.

In either case, we can write a computer program that will step through the natural numbers one-by-one in increasing order of magnitude and test whether the sought-after number is found. You can rest assured that for the first problem, no such number will be found because Lagrange provided a complicated proof of this fact in 1770. In the second case (Goldbach conjecture, 1742), so far no one has found such a number by calculation, and no one has found a proof of the existence or otherwise of such a number. The important fact is that we cannot rely on a computer program to find the answer because the program may never halt.

There are many other conjectures and problems in mathematics awaiting an answer. They could be easily resolved if it were possible to write a general computer program that could tell whether another program, if run on a computer, would halt or not. This leads us to Hilbert’s tenth problem in his list of 23 problems.

9.3 Hilbert’s Tenth Problem

Gödel’s theorems showed that as far as Principia Mathematica and related systems are concerned, there is no hope of resurrecting Hilbert’s dream. But, perhaps each statement of arithmetic might be provable in some other formal system. For computers to carry out proofs, a concrete interpretation of “mechanical manner” was required. In 1936, Alan Turing provided that interpretation in a brilliant paper⁸ titled “On computable numbers, with an application to the Entscheidungsproblem” and shattered any remaining hopes of realizing Hilbert’s dream of an unassailable formal system. In this epochal paper, Turing defines the Turing machine, formulates the halting problem, and shows that it as well as the Entscheidungsproblem is unsolvable. The Entscheidungsproblem (German for “decision problem ,” the German word for “decision” is Entscheidung) was the problem of deciding whether or not an arbitrary formula of the predicate calculus is a tautology.⁹ The problem was partially posed by Hilbert as the tenth problem in his famous list of 23 problems mentioned in Sect. 9.2, and later more completely (with Wilhelm Ackermann), at the Bologna International Congress in 1928. Here is the statement of the problem:

Determination of the solvability of a Diophantine equation.

Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers.

In other words, the problem was to find a computational procedure—or algorithm—for deciding, for a given system of Diophantine equations, whether the equations have any common solution. Diophantine equations are polynomial equations, in any number of variables, for which all the coefficients and all the solutions must be integers. Diophantine equations involve only the simple concepts of addition, multiplication, and exponentiation (raising one number to the power of another) of whole numbers. For example, in the following two sets of equations¹⁰:

(i)

(ii)

the first has the solution w = 1, x = 1, y = 2, z = 4 whereas the second has no solution whatever (because, by its first equation, y must be an even number whence, by its second, z must be even also, but this contradicts its third equation, whatever w is, because w²—w is always even, and 3 is an odd number).

The central issue was, “What, in precise terms, is an algorithm?” It was this question that led Turing to propose his own definition of what an algorithm is, in terms of machines now known as Turing machines. Apart from Turing, several others—Church, Kleene, Gödel, Post, and others)—proposed somewhat different procedures at about the same time, but all were soon shown to be equivalent (by Turing and Church). Turing’s approach turned out to be the most influential. Notwithstanding, all this, Hilbert’s actual tenth problem had to wait till 1970, when the Russian mathematician Yuri Matiyasevich finally showed that there can be no computer program (algorithm) which decides yes/no systematically to the question of whether a system of Diophantine equations has a solution.

In fact, James Jones and Yuri Matiyasevich had shown how to translate the operations of Turing’s computer into a Diophantine equation,¹¹ and they found a relationship between the solutions of the equation and the halting problem for the machine’s program. Specifically, if a given program does not ever halt, the related Diophantine equation will have no solution. In effect, the equations provide a bridge linking Turing’s halting problem with simple mathematical operations, such as the addition and multiplication of whole numbers.

The motivation for the entscheidungsproblem stemmed from the trend toward abstraction in mathematics. Turing had heard of the entscheidungsproblem during a course of lectures he attended at Cambridge University in England.

9.4 Turing and the Entscheidungsproblem

To answer the Entscheidungsproblem, Turing asked: “Can a problem be solved (or a question decided) in a finite amount of time on a computer?” This is known as the computability problem. Turing began by defining an abstract mathematical model of a human computer (who were mostly women in his days) acting without the benefit of insight, and succeeded in providing a concrete interpretation of a “mechanical manner” by inventing a simple idealized computer now called the Turing machine (he called it the logical computing machine; others to honor him called it the Turing machine), for which he showed that any classical computation can be simulated by this machine.

The Turing machine comprises a small central processing unit (CPU) with a memory of only a few bits—what is technically known as a “finite-state automaton,” since it only has a finite number of different internal memory states. The CPU can read or write binary digits on a computer tape of infinite length as it moves back and forth according to its own very simple, hard-wired rules. This single tape serves as the Turing machine’s input, program, bulk memory, output, and perhaps “garbage” storage. It essentially describes all classical computers—from Charles Babbage’s analytical engine to modern day supercomputers.

Turing’s approach is as follows. Let f be a function that takes a bit-string and outputs a bit. Then an algorithm for computing f is a set of mechanical rules, such that by following them we can compute f(x) given any input x ∈ {0, 1}. The set of rules being followed is finite and must work for all infinitely many inputs; each rule may be applied arbitrarily many times. Each rule must involve one of the following elementary operations:

Read a bit of the input.

Read a bit from the “scratch pad.”

Write a bit to the scratch pad.

Stop and output either 0 or 1.

Decide which of the above operations to apply based on the values that were just read.

The running time of the algorithm is the number of elementary operations performed.

The Turing “machine,” conceived before there were any real computers, is not a physical object but a piece of abstract mathematics. Turing regarded the human brain to be a “machine” at least when it was engaged in the task of doing mathematics. That task, he felt, was nothing but calculations performed by a human mathematician who has unlimited time and energy, an unlimited supply of paper and pencils, perfect concentration, and worked according to some algorithmic or “rule-of-thumb” method. Thus, in the context of Turing’s paper, the words “computer,” “computable,” and “computation” pertain to human calculators. The usefulness of the Turing machine stems from the fact that it is sufficiently simple to allow mathematicians to prove theorems about its computational capabilities and yet sufficiently complex to accommodate any actual classical digital computer, no matter how it is implemented. Turing abstracted the mathematician’s proof derivation process into four principal ingredients: a set of transformation rules that allowed one mathematical statement to be transformed into another, a method for recording each step in the proof, an ability to go back and forth over the proof to combine earlier inferences with later ones, and a mechanism for deciding which rule to apply at any given moment (Fig. 9.1).

Fig. 9.1

(Left) Turing machine; (Right) von Neumann architecture of a modern digital computer. Source of figure http://​physics.​kenyon.​edu/​coolphys/​thrmcmp/​newcomp.​htm (Turing machine); Source of figure http://​en.​wikipedia.​org/​wiki/​Image:​Von_​Neumann_​architecture.​svg (von Neumann architecture)

Instead of dealing with many possible Turing machines (e.g., a Turing machine for each algorithm or each device such as a mechanical desk calculator), he introduced the idea of a universal Turing machine. This machine is a mimic of all possible Turing machines and hence universal in status. Mimicking is made possible by coding the list of instructions for an arbitrary Turing machine T into a string of symbols, say, 0 and 1s, that can be put on a tape and feeding that tape to the universal Turing machine U. The initial part of the tape essentially provides complete information that U needs to imitate (simulate) exactly a given machine T. This enables the universal machine U to act on the remainder of the input just as T would have done. The modern-day implementation of the universal Turing machine is the von Neumann architecture. It uses a single storage structure to hold both instructions and data. The separation of storage from the processing unit is implicit. The term “stored-program computer” is generally used to mean a computer of this design.

Of his universal machine Turing asked, “What is impossible for such a machine? What can it not do?” And he immediately found a question—the halting problem—the problem of deciding in advance if a Turing machine or a computer program will eventually halt. For example, can it decide the Goldbach conjecture? Of course, one can decide for certain combinations of programs and input data.¹² The real question is, “Can it be done for every possible program and input data?” Note that the question is not whether a program will halt after a certain time limit (which can be answered by running the program up to the time limit) but whether it will ever halt. Turing showed that there is no way to decide in advance if a program will eventually halt. The real problem lies in those programs that will not halt—the problem of not knowing when to give up.

In essence, Turing addressed the computability problem by asking whether a universal Turing machine executing the ith problem acting on the jth input would halt. This is the famous Turing’s halting problem . By adapting Cantor’s diagonal slash argument, he showed that the problem was uncomputable. However, before we deal with the halting problem we note that there are other non-computable problems which are not equivalent to the Turing halting problem. The most famous among them is Gödel’s Incompleteness Theorem (noted above), which establishes that any finitely axiomatic, consistent mathematical system sufficiently complex to embrace arithmetic must be incomplete—that is, there exist some statements whose truth cannot be confirmed or denied from within the system, resulting in undecidability. Interestingly, it is precisely the statement about consistency of the system that is neither provable nor deniable.

9.4.1 Turing’s Halting Problem

Let $$ T_{n} $$ be the nth Turing machine acting on some finite input string of 0 and 1s on a tape. We may regard the input as some number m in base two. We further assume that after a succession of finite number of steps, $$ T_{n} $$ comes to a stop after depositing its output (another binary string representing the number $$ p $$) on the left side of the tape. Symbolically we can write

Alternatively, we can look upon the above relation as

where $$ U $$ is the universal Turing machine which mimics $$ T_{n} $$. Since $$ U $$ is a Turing machine, it too will have a number, say $$ u $$, i.e., it will be the Turing machine $$ T_{u} $$. Clearly, the ability of a Turing machine to act on its own kind provides a self-referential capability that enables one to investigate the capability and limitations of Turing machines using no other instruments other than the machines themselves. Note that the encoding of Turing machines into input forms acceptable by universal Turing machines is achieved by converting the finite description of a Turing machine into a unique non-negative integer. Thus, in a very real sense, Turing machines are just functions from non-negative integers (encoding the input) to non-negative integers (encoding the output).

So how does one decide if a particular Turing machine when fed with some specific input will ever stop? For many $$ \left( {n,m} \right) $$ pairs one may be able to construct procedures to find an answer; but for problems, such as the Goldbach conjecture or an NP-complete problem success has been elusive. What is being sought is a general algorithmic procedure for answering the halting problem completely automatically for any $$ \left( {n,m} \right) $$ pair. Thus, if one could decide whether this Turing machine ever stops, we should have a way of deciding the truth of, say, the Goldbach conjecture. Turing showed that there is no such algorithmic procedure.

His argument is the following. Suppose there is indeed such an algorithm. Then there must be some Turing machine $$ H $$ which “decides” if the nth Turing machine when acting on the number $$ m $$, will eventually stop or not. Let us say that its output is the following:

where $$ T_{n} (m) = { \boxdot } $$ implies those cases for which the Turing machine either does not halt or runs into a problem at some stage because it finds no appropriate instruction to tell it what to do. Now let us imagine an infinite array, which lists all the outputs of all possible Turing machines acting on all possible different inputs as reflected by the function $$ T_{n} (m)H\left( {n;m} \right) $$. Table 9.1 is an example. Note that by assuming $$ H $$ exists, the rows of this table necessarily consist of computable sequences. That is, there is a Turing machine which, when applied to the natural numbers $$ m = 0,1,2,3,4, \ldots $$ in turn, would yield the successive numbers of the sequence. In particular, given $$ \left( {n,m} \right) $$, a universal Turing machine would be able to produce the appropriate entry $$ T_{n} (m)H\left( {n;m} \right) $$ in Table 9.1. Therefore, every computable sequence of natural numbers must appear somewhere (perhaps more than once) amongst its rows.

Table 9.1

m→

n↓

0

1

2

3

4

5

6

7

8

…

0

0

0

2

1

0

0

6

4

1

…

1

1

4

3

0

9

5

2

8

9

…

2

2

4

2

2

1

7

5

0

0

…

3

2

1

3

3

3

5

9

2

2

…

4

9

0

1

3

2

5

3

6

6

…

5

9

4

4

1

0

0

8

0

0

…

6

4

6

7

7

7

7

7

7

4

…

7

6

2

5

4

7

1

1

1

1

…

8

6

8

2

3

3

6

2

1

1

…

…

…

…

…

…

…

…

…

…

…

…

Now take the diagonal elements (shown in bold font) in the table and create the sequence Sdia = (0, 4, 2, 3, 2, 0, 7, 1, 1, …), to each of whose elements add 1 to produce the new sequence Snew = (1, 5, 3, 4, 3, 1, 8, 2, 2, …). This is clearly a computable procedure and, given that our Table was computably generated, it provides us with some new computable sequence, in this case the sequence $$ 1 + T_{n} (n)H\left( {n;n} \right) $$. Since our Table is supposed to contain every computable sequence, $$ S_{\text{new}} $$ must be somewhere in the table. Yet it cannot be, for the new sequence differs from the first row in the first entry, from the second row in the second entry, and so on. From this contradiction, we therefore conclude that the Turing machine H cannot exist! There is no universal algorithm for deciding whether a Turing machine will halt. The halting problem is uncomputable. Note that the undecidability of the halting problem relies on the fact that Turing machines are assumed to have potentially infinite storage: At any one time they can only store finitely many things, but they can always store more and they never run out of memory. If the memory and external storage of a machine is limited, as it is for real computers, then the halting problem for programs running on that machine can be solved with a general algorithm (albeit an extremely inefficient one).

Clearly, the class of functions realized by Turing machines cannot be the same as the whole class of functions from the set of natural numbers to the same set. While the number of Turing machines is only countably infinite because each machine can be mapped into a unique integer, say n, the whole class of functions from natural numbers to natural numbers are known to be uncountably infinite; here the cardinality is the same as that of the set of reals. As noted earlier, the set of functions captured by and identified with Turing machines is the so-called partial recursive functions. In fact, the phrases Turing machines, programs, and partially recursive functions are interchangeable.

The halting problem is the prototypical example of an undecidable problem, i.e., a well posed problem that requires a yes or no answer but is impossible to solve. Note that the class of undecidable problems are impossible to solve not merely because of practical reasons such as time (example: NP-complete problems which can be solved but are computationally very expensive) or memory requirements. Undecidable problems are unsolvable, in principle, regardless of the amount of computational power available. “This statement is unprovable” is an obvious example of an unprovable statement!

The halting problem is clearly a decision problem because the proposition that a certain algorithm will halt given a certain input can be automatically reformulated as a statement about numbers. Since there is no algorithm that can decide if the original statement about algorithms is true or not, it follows that there is no algorithm that can decide whether the corresponding statement about numbers is true or not.

A consequence of the halting problem’s undecidability is that the Entscheidungsproblem is unsolvable; that is, there cannot be a general algorithm that decides whether a given statement about natural numbers is true or not. The Entscheidungsproblem, as noted earlier, is a challenge in symbolic logic to find a general algorithm, which can decide for first-order statements whether or not they are universally valid. A first-order statement is universally valid or logically valid if it follows from the axioms of first-order predicate calculus .

The universal Turing machine is completely equivalent, in its action, to that of a modern general-purpose computer—with the specific idealization that the computer must have access to an unlimited storage capacity. It is due to Turing that the abstract ideas of computers and computation have become an integral part of mathematical thinking. By introducing mathematically well-defined machines, Turing was able to capture the essence of computational processes and algorithms. Not only that, by introducing the notion of computability (i.e., the concept of algorithm)—of distinguishing things that cannot be calculated from those that can—and then deducing incompleteness from uncomputability, he was able to extend Gödel’s incompleteness theorem. With this he showed that undecidability in mathematics was even more widespread than had been anticipated. His theory of computation provided a deep understanding of algorithmic procedures, and in the process, ushered in the modern computer revolution. Indeed, computer science was born before the computer!

9.4.2 The Church–Turing Thesis

The power of the universal Turing machine led to the Church–Turing thesis :

The class of functions computable by a Turing machine corresponds exactly to the class of functions which we would naturally regard as being computable by an algorithm.

This is a deeply rooted gut feeling: The Turing machine model of computation completely captures the notion of computing a function using an algorithm.¹³ The Church–Turing thesis , independently arrived at by Church and Turing, is not a theorem; it is an empirical statement, a belief about the nature of the world. The thesis asserts a perceived equivalence between a rigorous mathematical concept—function computable by a Turing machine—and the intuitive concept of what it means for a function to be computable by an algorithm.

A priori it is not obvious that every function that we would intuitively regard as computable by an algorithm can be computed using a Turing machine. So far, there is no evidence against the Church–Turing thesis , nevertheless, the possibility exists that someday we may discover a natural process which computes a function not computable by a Turing machine (TM). In that event, we would need to redefine computability, and with it, computer science.

There is also a strong form of the Church–Turing thesis which says:

[E]very physically realizable computation model can be simulated by a TM with polynomial overhead, (in other words, $$ t $$ steps on the model can be simulated in $$ t^{c} $$ steps on the TM, where $$ c $$ is a constant that depends upon the model.¹⁴

A consequence of this thesis is that the entire theory of computational complexity , which we discuss later in this chapter, can take on an elegant, model-independent form if the notion of efficiency is identified with polynomial resource algorithms (also to be described later in this chapter). However, not everybody believes in the strong form of the Church–Turing thesis , in particular, the quantum computing community (see Deutsch’s Church–Turing principle below). It is also speculated that computing devices built on the basis of exotic physics, such as string theory, may prove the strong form of the Church–Turing thesis wrong.

We have already seen that quantum computers can compute the same class of functions as is computable by a classical computer, so quantum computers do satisfy the (non-strong) Church–Turing thesis . The difference lies in the efficiency with which the computation is done. We have also seen that there are functions which can be computed much more efficiently on a quantum computer than on a Turing machine with their respective best-known algorithms.

It is now established that Turing machines, recursive functions, λ-definable functions, cellular automata, pointer machines, bouncing billiard balls, Conway’s Game of Life, etc. are equivalent in terms of what they can and cannot compute. Thus, the set of computable problems does not depend on the computational model. It has therefore become standard practice to study questions about computing efficiency using the universal Turing machine.

9.4.3 Deutsch on the Church–Turing Thesis

David Deutsch asked the obvious question: “Whether it was possible for a quantum computer to efficiently solve computational problems which have no efficient solution on a classical computer, even a probabilistic Turing machine¹⁵ (PTM)?” For example, can it efficiently solve a problem in the NP-complete complexity class? Although he did not answer the question, he did suggest, by constructing a simple example, that quantum computers might have computational powers exceeding those of classical computers (see Chap. 8, Sect. 8.​2.​4 The Deutsch algorithm) and proposed reformulating the Church–Turing thesis in physical terms, i.e., into a principle of physics, to:

Every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means.¹⁶

As Deutsch notes, this formulation, which he called the Church–Turing principle, is not only stronger but is both better defined and more physical than the Church–Turing thesis . The qualifications introduced by finitely realizable and “finite means” are important in order to state something useful. The principle does not refer to Turing machines; there are basic differences between the very nature of Turing machines and the principles of quantum mechanics. Turing machines deal with operations on classical bits, while quantum mechanics deals with the evolution of quantum states. Hence, in principle, there is the possibility that the universal Turing machine might not be able to simulate certain behaviors in Nature. The Church–Turing principle is a scientific empirical assertion in the sense of Popper because it is falsifiable [30], that is, there exist potential observations that could contradict it.

9.4.4 Can Quantum Computers Prove Theorems?

It has been possible in some cases, using classical computers, to prove automatically a mathematical theorem (i.e., use algorithmic methods of theorem proving).¹⁷ In fact, several genuine proofs of non-trivial theorems have been found in this way. But in a quantum computer, the details of reasoning cannot be followed. Any attempt to do so converts the quantum computer into a classical computer, that is, any attempt to learn an intermediate step in a proof would require a measurement, which would collapse the quantum computer. Thus, we have a paradoxical situation where a quantum computer would be able to tell you if your theorem is true or false, but it would not be possible to extract the proof from it.

9.5 Thermodynamic Considerations

As physical entities, computing machines are subject to the laws of thermodynamics. We know, e.g., that analog devices can be much faster, while generating less heat, than digital devices. This is because it costs a lot of time and energy to shape, maintain, and then move around a digital signal. But our preference for digital computers is because it provides better error control and ease of programming at the expense of speed. The obvious question is: “Can digital computers be improved so as to minimize or even eliminate the production of heat?” The abstract Turing machine is completely mute on this point. In real life, there is a lot of heat generation due to electrical currents overcoming resistance, etc. The central question we ask is “What is the minimum energy required to carry out a computation?”

Notwithstanding the fact that modern day classical computers are built using solid state devices whose theoretical underpinning is quantum mechanics, such computers are essentially classical devices. The classical Turing machine, in general, is logically irreversible because it frequently performs operations that throw away historical information about the computer’s actions, leaving the computer in a state whose immediate predecessor is logically indeterminable. The logically irreversible operations include erasure or overwriting of data, and branching into program segments accessible by several different transfer instructions. That is, the transition function, which maps each whole-machine state onto its successor, if the state has a successor, lacks a single-valued inverse. Therefore, it is natural to ask if the irreversible actions of a classical Turing machine could be transformed into equivalent reversible actions, and then inventing a Hamiltonian which could cause a quantum system to evolve in a way that would mimic such a reversible Turing machine. Rolf Landauer took the first step in answering the question.

To understand Landauer’s remarkable contribution, we begin with a fundamental process in thermodynamics: the compression of a volume of gas. We begin by considering a dilute gas (no intermolecular forces) contained in some volume V₁ at absolute temperature T. We now shrink this gas isothermally to volume V₂ with the help of a piston of area A. The work done by the piston in compressing the gas with the piston moving a distance δx is given by

where p is the pressure exerted by the gas on the piston. Now, from the kinetic theory of gases we know that

where N is the number of molecules in the gas and k is the Boltzmann’s constant. Using this, and noting that T remains constant during the compression, we have

By convention, we regard work done by the gas as positive and work done on the gas as negative.

Ordinarily, when we compress a gas, it heats up due to the energy imparted to it by the work done on it by the piston. This is a result of the molecules speeding up and gaining kinetic energy according to the law of conservation of energy. Since the temperature of a gas is proportional to the mean kinetic energy of the molecules, it follows that isothermal compression cannot change the mean kinetic energy of the molecules. So where did the energy W added by the piston go? In fact, W was converted into internal gas heat, but was promptly drained off into the thermal bath used to keep the gas at constant temperature. Of course, the compression process has to be done quite slowly, so that at all times both the gas and the thermal bath are in equilibrium. (We are avoiding getting into non-equilibrium thermodynamics.) Thus, the effected change of state where the gas from occupying volume V₁ now occupies volume V₂ occurs such that the total energy of the gas, U, which is the sum of its constituent parts, remains unchanged.

The second law of thermodynamics also kicks in, in the form of the relation

where F is the free energy (i.e., the total amount of energy in a physical system which can be converted to do work), U is the total energy, and S is the entropy. If only small variations at constant temperature are considered, we have

where, as already noted, $$ \delta U = 0 $$ for our compression process. Thus, $$ \delta F = - T\,\delta S $$ is just the “missing” heat energy siphoned off into the thermal bath, that is,

or

9.5.1 The One-Molecule Gas

In statistical thermodynamics , entropy is related to the probability that the gas be in the configuration in which it is found. By configuration, one means a particular arrangement, or cluster of arrangements, of positions and momenta for each of the N constituent molecules (as described by a point in the phase space). Our observations of gases tell us that it is far less likely at the outset that we will find all the gas molecules moving in the same direction or paired up and dancing than their shooting all over the place at random. Entropy quantifies this notion. Roughly speaking, if the probability of a particular gas configuration is w, we have the entropy S given by

Note that like all probabilities, the w’s add, so we can readily calculate the chances of the gas being in some range of configurations. When the gas molecules are all moving in one direction, w is much less than when the molecules are moving randomly.

Since the definition of entropy is essentially statistical, it is perfectly alright to define it for a gas with a single molecule provided we look at time-averaged quantities. While it is not obvious that the laws of thermodynamics apply to a one-molecule gas , rest assured that it does. It is, of course, difficult to get a feeling for concepts like temperature, pressure and volume, never mind free energy and entropy, when you have only one molecule. However, these concepts make sense as long as we consider them to be time-averaged, smoothing out the irregularities of this one particle as it bounces back and forth. Indeed, thermodynamic formulae work if there is this hidden smoothing.¹⁸ Thus, if we compress the volume of the one-molecule gas by half, then we halve the number of spatial positions, and hence the number of configurations w that the molecule can occupy. Before, it could be in either half of the box; now, it can only be in one half. Therefore, this would lead to a decrease in entropy by an amount

We thus note a subtle connection between our knowledge of the possible locations of the molecule and the entropy of the gas. In the initial state, the gas molecule could be hiding anywhere in the volume of the box; after the compression, it must be somewhere within the compressed half of the box. In other words, the molecule can be in fewer places after the compression. When the uncertainty of where the molecule is, is high, the entropy is high, and correspondingly when the uncertainty of where the molecule is, is low, the entropy is low.

9.5.2 Knowledge and Entropy

In information theory, the concept of knowledge is central to the concept of entropy. Since it is not possible to follow the paths and momenta of every molecule in the gas, we are forced to take a statistical approach. Concepts such as temperature, pressure, etc. of a gas are essentially defined as statistical averages of microscopic properties. We assign certain physical properties to each molecule, assume particular distributions for these molecules, and calculate the average by a weighting process: So many molecules will move with this speed, corresponding to one temperature; so many will move with that speed, giving another temperature; and then we just average over everything.

The entropy of a gas is defined statistically, but in a way different from quantities such as temperature and energy. Unlike these quantities, it is not a macroscopic property that arises from a sum of microscopic properties. Rather, it is directly related to the probability that the gas be in the configuration in which it is found. The less we know about the configuration of a gas, the more states it could be in, and the greater the overall w—and the greater the entropy. Thus, when we compress a gas isothermally into a smaller volume, the momenta of the molecules within the container remain the same, but each molecule has access to fewer spatial positions. The gas has therefore adopted a configuration with smaller w, and its entropy has decreased. In general, the less information we have about a state, the higher the entropy.

9.5.3 Information Is Physical

Rolf Landauer viewed computations as engines for transforming free energy into waste heat and mathematical work. He argued that information must be encoded in physical systems, without which it is not possible to store, transmit, process, or receive information. The laws of physics would therefore place natural limits on information processing with the classical and the quantum worlds perhaps providing different information processing capabilities in their respective worlds. These “obvious” conclusions became obvious only in 1961 with the publication of Landauer’s seminal paper on reversible computing.¹⁹ He showed that there is a fundamental asymmetry in the way Nature allows us to process information by proving the surprising result that all but one operation required in computation could be performed in a logically and a thermodynamically reversible manner, thus dissipating no heat.²⁰ For example, copying classical information can be done reversibly and without wasting any energy,²¹ but when information is erased, there is always a minimum energy cost of kT ln 2 per classical bit (about 3 × 10⁻²¹ J at room temperature) to be paid, where k is the Boltzmann constant, and T is the temperature of the environment of the computer. That is, the erasure of information is inevitably accompanied by the generation of heat. Indeed, Landauer’s erasure principle provides a bridge between information theory and physics (Fig. 9.2).

Fig. 9.2

Erasure of information.

Ref. Plenio and Vitelli [29], pp. 25–60

To understand Landauer’s erasure principle, note that classically a degree of freedom is associated with kT amount of thermal energy. Now consider a rectangular box partitioned in the middle into a left and a right compartment. The box is filled with a one-molecule gas that can be on either side of the partition, but we do not know which. We may arbitrarily label the event when the molecule is found in the left partition to be the state 0 and when found in the right partition to be the state 1. Thus, the box serves as a device for storing binary information. To show that erasure of the bit of information encoded in the position of the molecule in this device brings about a change in entropy, we do the following.

Remove the partition and compress the molecule into the left part of the box irrespective of where it was before by placing a piston at the right end of the box and pushing it to the left end. The act of compression erases the information because information about the molecule’s original location is lost forever. After the procedure, the atom is on the left-hand side of the box irrespective of its initial state. The process has decreased the thermodynamic entropy of the gas by k ln 2. The minimum amount of work needed to compress the gas in the box is kT ln 2 (if the compression is isothermal and quasi-static). Furthermore, kT ln 2 amount of heat is dumped into the environment during the compression. Landauer conjectured that this energy-entropy cost cannot be reduced further irrespective of how the information is encoded and erased—it is a fundamental limit (this is known as Landauer’s principle). This principle can be deduced from the second law of thermodynamics and is, in fact, equivalent to it. Thus, Landauer’s principle relates information to physical quantities like thermodynamic entropy and free energy. The essence of the above discussion is that

Classical information is always encoded in a physical system.

The erasure of information causes a generation of $$ kT\ln 2 $$ of heat per bit in the environment.

Thus, if a device is logically reversible and can run backwards (physically reversible), then the second law of thermodynamics guarantees that it will not dissipate any heat. In principle, an irreversible classical computer can always be made reversible by the trivial act of having it save all the information it would otherwise throw away. In practice, in many cases, this would require impractically huge information storage devices. In fact, Landauer showed how any function f(a) could be made one-to-one by keeping a copy of the input a:

Toffoli later developed a gate, known as the Toffoli gate to implement the above operation (see Chap. 7, Sect. 7.​4.​5).

9.5.4 Toffoli Gate

The output of the reversible Toffoli gate can be decomposed into various gates:

where A · C represents an AND gate, A ⊕ B represents an XOR gate, and $$ \bar{A} $$ represents a NOT gate. This is a universal gate since it can perform AND, XOR, NOT, or FANOUT depending on the inputs. Computations exclusively based on Toffoli gates are reversible. Thus, any classical circuit is replaceable by an equivalent circuit containing only the Toffoli gates . The Toffoli gate maintains all input information, so computations made by it can be run backward. The gate behaves as

creating in the process extra junk bits $$ j(a) $$. The price for logical reversibility is the creation and accumulation of junk bits but the advantage is that heat generation is eliminated during computation.

9.5.5 Bennett’s Solution for Junk Bits

In a brilliant paper,²² Charles Bennett solved the problem of junk bits by showing that they could be reversibly erased at intermediate steps in the computation with minimal run-time and memory costs. His solution roughly works as follows²³:

where $$ f^{u} $$ stands for uncomputing $$ f $$. In the first step, $$ f $$ is computed, and the process produces both junk bits and the intended output. In the second step, the FANOUT gate duplicates the output $$ f $$. In the final step, the original f is uncomputed by running its computation backwards, which removes the junk bits and the original output but retains the duplicate. It turns out that primitive erase is not essential in computation.

In other words, Bennett showed that such machines could be made logically reversible at every step, while retaining their simplicity and their ability to do general computations. This result also makes plausible the existence of thermodynamically reversible computers, which could perform useful computations at useful speed while dissipating very little energy per logical step. Clearly, a requirement of logical reversibility, where input and output are uniquely retrievable from each other, does not bar the logical design of computers. However, computations on a reversible computer take about twice as many steps as on an ordinary irreversible one and may require a large amount of temporary storage. On the other hand, reversible computing has applications in low-power computing where managing power density in modern and future computers is a serious problem. Reversible logic has obvious connection with quantum computing and interesting applications in the biosynthesis of messenger RNA.²⁴

9.5.6 Reversible Classical Computation Set the Stage for Quantum Computing

The significance of reversible classical computation is that if we can compute something reversibly, we can also compute it on a quantum computer. The deep understanding provided by Landauer and Bennett about the physical limitations placed on computation due to heat dissipation (Landauer) and that classical, reversible computation was indeed possible and economical (Bennett), laid the foundation for the development of quantum computers. Note that by Postulate 2, inputs provided to a quantum computer can evolve toward its output quantum mechanically only by the application of unitary gates, which must necessarily be invertible, i.e., it must always be possible to “uncompute” a computation on a quantum computer. The work of Landauer and Bennett showed that computations on a classical computer could always be made reversible, if desired, and hence the constraint of reversibility imposed on quantum computation, in principle, did not prevent quantum computers from executing analogous classical algorithms.

9.5.7 Maxwell’s Demon

In 1871, the Scottish physicist James Clerk Maxwell proposed a thought experiment: A wall separates two compartments filled with gas. There is a little demon who sits by a tiny trapdoor in the wall. The demon looks at oncoming gas molecules, and depending on their speeds, it opens or closes the trapdoor. The demon is thus able to eventually collect all the molecules faster than average on one side, and the slower ones on the other side. The result is a hot, high-pressure gas on one side and a cold, low-pressure gas on the other side. Note that in this process, conservation of energy is not violated, but the demon has managed to redistribute the random kinetic energy of the molecules (heat) in such a manner that energy can now be extracted from the system (for example, it can now drive a turbine).

The Maxwell demon experiment is an excellent demonstration of entropy, and how it is related to

the fraction of energy that is not available to do useful work and

the amount of information we lack about the detailed state of the system (Fig. 9.3).

Fig. 9.3

Maxwell’s demon (redraw).

Source https://​commons.​wikimedia.​org/​wiki/​File:​Maxwell%27s_​demon.​svg

In this thought experiment, the demon has apparently managed to decrease the entropy by $$ Nk\ln 2 $$, where $$ N $$ is the total number of molecules in the gas. That is, the demon has increased the amount of energy available to do useful work by increasing its knowledge about the motion of all the molecules. The second law of thermodynamics, of course, says that such a thing is impossible—you can only increase entropy (or rather, you can decrease it at one place provided it is balanced by at least as big an increase somewhere else in a closed system). In fact, the second law forbids any apparatus from doing this (i.e., increase the amount of energy available to do useful work) reliably, even for a gas consisting of a single molecule, without producing a corresponding entropy increase elsewhere in the universe. So where is the flaw?

Several attempts were made to exorcise Maxwell’s demon . In doing so, people had often assumed that the measurement the demon makes to determine whether the molecule is approaching from the left or the right is an unavoidably irreversible act, requiring an entropy generation of at least $$ k { \ln } 2 $$ per bit of information obtained, or that measurement cannot be made without disturbing (and thus heating) the gas. In fact, both assumptions are untrue. Some were even tempted to propose that the second law of thermodynamics could indeed be violated by the actions of an “intelligent being.” It was not until 1929 that Leo Szilard made progress by reducing the problem to its essential components (his demon has since come to be known as Szilard’s engine), in which the demon need merely identify whether a single molecule is to the right or left of a sliding partition. Here’s how it works.

The molecule is originally placed in a box, free to move in the entire volume V as shown in step (a). Step (b) consists of inserting a partition, which divides the box into two equal parts. At this point, the Maxwell’s demon measures in which side of the box the molecule is and records the result (in the figure the molecule is pictured on the right-hand side of the partition as an example). In step (c), the Maxwell demon uses the information to replace the partition with a piston and couple the latter to a load. In step (d), the one-molecule gas is put in contact with a reservoir and expands isothermally to the original volume V. During the expansion, the gas draws heat from the reservoir and does work (= kT ln 2) to lift the load. Apparently, the device is returned to its initial state and it is ready to perform another cycle whose net result is again full conversion of heat into work, a process forbidden by the second law of thermodynamics (Fig. 9.4).

Fig. 9.4

Gas expands converting heat from reservoir to work.

Source Plenio and Vitelli [29]

Szilard’s way out of this predicament was to postulate that the act of measurement, in which the molecule’s position is determined, brings about an increase in entropy enough to compensate for the decrease in entropy brought about during the power stroke.²⁵ Szilard was somewhat vague about the nature and location of the increase in entropy. So, Szilard still had not solved the problem, since his analysis was unclear about whether the act of measurement, whereby the demon learns whether the molecule is to the left or the right, must involve an increase in entropy. In fact, Szilard’s argument had missed an important point: While the gas in the box has returned to its initial state, the mind of the demon has not! In fact, the demon needs to erase the information stored in his mind for the process to be truly cyclic. This is because the information in the brain of the demon is stored in physical objects and cannot be regarded as a purely mathematical concept! However, this realization came some fifty years later!

In the meantime, digital computers were developed, and the physical implications of information gathering, and processing were carefully considered. The thermodynamic costs of elementary information manipulations were analyzed by Landauer and others during the 1960s, and those of general computations by Bennett, Fredkin, Toffoli, and others during the 1970s. It was found that almost anything can, in principle, be done in a reversible manner, that is, with no entropy cost at all. Charles Bennett was then finally able to resolve the paradox.²⁶ Central to this resolution was Bennett’s argument that information processed by the demon must be encoded in a physical system that obeys the laws of physics and therefore will need Landauer’s erasure principle!

Bennett showed that measurements of the sort required by Maxwell’s demon can be made reversibly, provided the measuring apparatus (e.g., the demon’s internal mechanism) is in a standard state before the measurement, so that measurement, like the copying of a bit onto previously blank tape, does not overwrite information previously stored there. Under these conditions, the essential irreversible act, which prevents the demon from violating the second law, is not the measurement itself but rather the subsequent restoration of the measuring apparatus to a standard state in preparation for the next measurement. Thus, the irreversible step is not the acquisition of information, but the loss of information when the demon later clears its memory using kT ln 2 amount of energy. Note that all the work gained by the engine is needed to erase the information in the demon’s mind, so that no net work is produced in the cycle. Furthermore, the erasure transfers into the reservoir the same amount of heat that was drawn from it originally. So, there is no net flow of heat either. There is no net result after the process is completed and the second law is saved!

Maxwell’s demon , therefore, reveals a deep connection between thermodynamics and information theory. Real-life versions of Maxwellian demons (with their entropy lowering effects of course duly balanced by increase in entropy elsewhere) occur in living systems, such as the ion channels and pumps that make our nervous system work, including our minds. Molecular-sized mechanisms are no longer found only in biology; they are ubiquitous in the emerging field of nanotechnology.

9.6 Computational Complexity

A programmer usually has a choice of data structures and algorithms to solve a problem. The choice depends on available resources, e.g., how much time and memory the program will take for a given input. The choice must therefore accommodate and make tradeoffs between Time Complexity and Space Complexity. For example, he may choose a data structure that requires a lot of space to reduce computation time. The art of making choices is facilitated by complexity analysis.

Complexity refers to the rate at which the need for a given resource (such as time, and memory space) grows as a function of the problem size (as measured by the size of the input data). One is not interested in absolute growth since it depends on a variety of factors, such as the computer and the operating system used to run the program, the compiler used to compile the program, etc. What is therefore attempted is a means of describing the inherent complexity of a program independent of machine, compiler, and other considerations. This naturally leads us to a “proportionality” approach whereby we express complexity in terms of its relationship to some known function. That is, we attempt an asymptotic analysis where we would like to know a quantity only approximately to enable comparisons. Let us, therefore, get used to some new symbols and notations for this study.

The symbol O (first introduced by the number theorist Paul Bachmann in [2]²⁷) is used to describe an asymptotic upper bound for the magnitude of a function in terms of another, usually simpler function. Apart from the O symbol, there are other symbols—Ω, Θ, o, and ω—for various other upper, lower, and tight bounds. Informally, the O notation is sometimes incorrectly employed to describe an asymptotic tight bound, but tight bounds are more formally and precisely denoted by the Θ (capital theta) symbol as described later. The distinction between upper and tight bounds is useful, and sometimes critical; hence, it is important that one pays proper attention to the usage of O and Θ.

The O notation

Mathematicians and computer scientists use the O notation (also called the Landau notation, or Bachmann-Landau notation, or asymptotic notation) in slightly different contexts. But their underlying principle is the same. In mathematics, it generally characterizes the residual terms of a truncated infinite series, especially of an asymptotic series. For example,

expresses the fact that the error, the difference $$ {\text{e}}^{x} - \left( {1 + x + x^{2} /2} \right) $$, is smaller in absolute value than some constant times $$ \left| {x^{3} } \right| $$ when $$ x $$ is close enough to zero. In computer science, it is used to analyze the complexity of algorithms. Henceforth, we will discuss its use in computer science.

In computational complexity theory, the big ‘$$ O $$’ notation is often used to describe how the size of input data affects the use of computational resources (such as program execution time and required memory space) by an algorithm. Thus, by writing $$ f(n) = O\left( {g(n)} \right) $$, we denote the collection of functions $$ f(n) $$ of the variable n that exhibit a growth, as $$ n \to \infty $$, that is limited to that of $$ g(n) $$ in some respect. That is, the O notation signifies that $$ f(n) $$ is asymptotically less than or equal to $$ g(n) $$ as $$ n \to \infty $$. More precisely, it means that there are positive constants $$ c $$ and $$ n_{0} $$ such that the number $$ x_{n} $$ represented by $$ O\left( {g(n)} \right) $$ satisfies the condition $$ \left| {x_{n} } \right| \le c\left| {g(n)} \right| $$, for all integers $$ n \ge n_{0} $$. We do not say what the constants $$ c $$ and $$ n_{0} $$ are; in fact, they can be different for each appearance of $$ O $$.²⁸ Does this convey anything useful? For example, we may (correctly) say that $$ n^{2} + 5n + 6 = O\left( {n^{100} } \right) $$ but this is perhaps not very useful since $$ n^{100} $$ is simply too large. In an $$ O $$-analysis, we usually choose a function $$ g(n) $$ to be as small as possible and still satisfy the definition of Big-Oh. Therefore, instead, it is more meaningful to say that $$ n^{2} + 5n + 6 = O\left( {n^{2} } \right) $$. Note also that the constants $$ c $$ and $$ n_{0} $$ that can satisfy the definition of Big-Oh are not unique either in this case or in general.

Clearly, if $$ f(n) = O\left( {g(n)} \right) $$, then it is not necessarily true that $$ g(n) = O\left( {f(n)} \right) $$. The “$$ = $$” sign does not mean equality in the usual algebraic sense. Confusion is less likely if we view “$$ = O $$” as a single symbol.

Example

Given the polynomials $$ f(n) = 6n^{4} - 2n^{3} + 5 $$, and $$ g(n) = n^{4} $$, $$ f(n) $$ has order $$ O\left( {g(n)} \right) $$ or $$ O\left( {n^{4} } \right) $$.

If a function $$ f(n) $$ can be written as a finite sum of other functions, then the fastest growing one determines the order of $$ f(n) $$. For example, if

then $$ f(n) = O\left( {n^{3} } \right) $$ as $$ n \to \infty $$ because one may disregard all other (lower-order) terms since they grow more slowly than $$ n^{3} $$.

Here are a few useful properties of the $$ O $$ notation.²⁹

1.

Constant factors may be ignored: For all $$ k &gt; 0 $$, $$ k * g $$ is $$ O(g) $$.

2.

Higher powers of $$ n $$ grow faster than lower powers: $$ n^{r} $$ is $$ O\left( {n^{s} } \right) $$ if $$ 0 \le r \le s $$.

3.

The growth rate of a sum of terms is the growth rate of its fastest growing term:

If $$ f $$ is $$ O(g) $$, then $$ f + g $$ is $$ O(g) $$. For example, $$ a * n^{3} + b * n^{2} $$ is $$ O\left( {n^{3} } \right) $$.

4.

The growth rate of a polynomial is given by the growth rate of its leading term.

If $$ f $$ is a polynomial of degree $$ d $$, then $$ f $$ is $$ O\left( {n^{d} } \right) $$.

5.

If $$ f $$ grows faster than $$ g $$, which grows faster than $$ h $$, then $$ f $$ grows faster than $$ h $$.

6.

The product of upper bounds of functions gives an upper bound for the product of the functions: If $$ f $$ is $$ O(g) $$ and $$ h $$ is $$ O(r) $$, then $$ f * h $$ is $$ O\left( {g * r} \right) $$, e.g., if f is $$ O\left( {n^{2} } \right) $$ and $$ g $$ is $$ O\left( {\log n} \right) $$, then $$ f * g $$ is $$ O\left( {n^{2} \log n} \right) $$.

7.

An algorithm is said to have polynomial time complexity iff it is $$ O\left( {n^{d} } \right) $$ for some integer $$ d $$. A problem is said to be intractable if no algorithm with polynomial time or of lower time complexity is known for it.

8.

Exponential functions grow faster than powers: $$ n^{k} $$ is $$ O\left( {b^{n} } \right) $$, for all $$ b &gt; 1 $$, $$ k \ge 0 $$.

For example, $$ n^{4} $$ is $$ O\left( {2^{n} } \right) $$ and $$ n^{4} $$ is $$ O\left( {\exp (n)} \right) $$.

9.

Logarithms grow more slowly than powers: $$ \log_{b} n $$ is $$ O\left( {n^{k} } \right) $$ for all $$ b &gt; 1 $$, $$ k &gt; 0 $$.

For example, $$ \log_{2} n $$ is $$ O(n^{0.5} ) $$.

10.

All logarithms grow at the same rate: $$ \log_{b} n $$ is $$ \left( {\log_{d} n} \right) $$ for all $$ b $$, $$ d &gt; 1 $$.

11.

The sum of the first $$ n\,r $$th powers grows as the $$ \left( {r + 1} \right) $$th power:

Notes

A logarithm is an inverse exponential function. Exponential functions grow distressingly rapidly, while logarithms grow placidly. Binary search is an example of an $$ O\left( {\log n} \right) $$ algorithm. If anything is halved on each iteration, then you usually get $$ O\left( {\log n} \right) $$. Asymptotically, the base of the $$ \log $$ does not matter.

$$ O\left( {n^{c} } \right) $$ and $$ O\left( {c^{n} } \right) $$ are very different. The latter grows much, much faster, no matter how big the constant $$ c $$ is (as long as it is greater than 1). A function that grows faster than any power of $$ n $$ is called superpolynomial. One that grows more slowly than any exponential function of the form $$ c^{n} $$ is called subexponential. An algorithm may require time that is both super-polynomial and subexponential; examples of this include the fastest known algorithms for integer factorization.

Note that $$ O\left( {{ \log } \,n} \right) $$ is exactly the same as $$ O\left( {\log \,n^{c} } \right) $$ since $$ O\left( {\log n^{c} } \right) = c\,O\left( {\log \,n} \right) $$. Similarly, logarithms with different constant bases are equivalent. Exponentials of different bases, on the other hand, are not of the same order; for example, $$ 2^{n} $$ and $$ 3^{n} $$ are not of the same order.

Here are some simple operations we can do with the $$ O $$-notation:

The notion of a universal computer has made the job of classifying computational tasks in terms of their difficulty relatively easy. A given algorithm is deemed to address not just one instance of a problem, but a whole class of them. The computational complexity of a problem is determined by the amount of computational resources (time and memory space) required to complete a given computational task. Unfortunately, complexity theory provides strong evidence that many optimization problems are likely to be intractable and have no efficient algorithm.³⁰ That is, each such problem is effectively impossible to solve, not because we cannot find an algorithm to solve the problem, but because all known algorithms consume such vast amounts of space or time as to render them practically useless. It is likely that questions about intractability and quantum computation may help to shed light on the fundamental properties of matter.

To understand the computational efficiency, we study the number of basic operations (such as single digit addition and multiplication) needed to execute an algorithm as the size of the input increases. The input is represented by a string of binary digits,³¹ and the size of the input is the number of binary digits in the binary string. By careful considerations, researchers have been able to study the efficiency of an algorithm independent of the technology and programming language used to execute the algorithm. A related question that naturally arises is whether the best known algorithm is really the best (as is known for, say, Grover’s algorithm, see Chapt. 10, Sect. 10.10) in the sense that there is no algorithm which takes less than a certain number of steps. Unfortunately, such questions remain unanswered for most problems. An important special case of functions mapping strings to strings is the case of Boolean functions, whose output is a single bit. Such functions are related to decision problems .

9.6.1 Classification of Complexity

To deal with efficiency issues, computer scientists have developed a classification system based on the mathematical form of the function (the Oh notations) that describes how the computational cost incurred in solving a problem scales up as larger problems of the same kind are considered. The most common measures of efficiency use the rate of growth of time or memory needed to solve a problem as the size of the problem increases. By size of the problem, we mean the size of the binary string that represents the input to the problem, for example, the number of bits needed to encode the number we want to factorize. Growth rate of running times and memory requirements rather than absolute running times and memory requirements are chosen to factor out variations in performance that may arise due to hardware and architectural differences of computers (such as, size of cache memory, swap space, processor speed, etc.), and software differences. Those interested in studying the topic in greater depth would immensely benefit from a reading of the excellent Introduction to Algorithms by Cormen et al.³²

The O, Ω, Θ, o, ω notations

In our further analysis of complexity and efficiency, we will use the following standard notations to express the rate of growth of functions rather than their precise behavior. If f and g are two functions from N to N, then we say that

(1)

$$ f = O\left( g \right) $$ if there exists a constant $$ c $$ such that $$ f(n) \le c \cdot g(n) $$ for every sufficiently large $$ n $$;

(2)

$$ f =\Omega (g) $$ if $$ g = O(f) $$;

(3)

$$ f = {{\varTheta}}(g){\text{ if }}f = O(g){\text{ and }}g = O(f) $$;

(4)

(5)

$$ f = \omega (g){\text{ if }}g = o(f) $$.

The above considerations lead to a classification system that is independent of factors that do not change the functional relationship between input size and the number of elementary computer operations (and hence the time) needed by an algorithm to solve a problem. Sometimes, beneficial tradeoffs between time and space are possible. Although complexity measures are independent of the physical hardware, they are related to a particular mathematical model of the computer such as a deterministic Turing machine (DTU) or a probabilistic Turing machine (PTM) or a quantum (Turing) machine (QTM). In brief, a complexity class is then the set of problems solvable in a particular model under particular resource constraints. Thus, there are special complexity classes for a DTM, a PTM, and a QTM.³³ These models are simple enough to allow mathematical analysis yet general enough to be useful in a wide context.

Two of the most important complexity classes are called P (for polynomial time) and NP (for non-deterministic polynomial time). A problem is placed in a particular class based on the most efficient algorithm known to solve the given problem. Thus, if the best-known algorithm for a problem solves the problem in a certain number of execution steps (as a function of input size) bounded by a polynomial, then the problem is said to belong to class P, and hence is deemed tractable. A simple example of this class is the multiplication of two numbers. Of course, it is possible for a problem in class P to have a large polynomial order, such as 14, making a supposedly tractable problem rather difficult in practice. Fortunately, such large polynomial growth rates are seldom encountered, and the P-NP distinction is a pretty good indicator of difficulty. Note that exponential growth will always exceed polynomial growth eventually, regardless of the order of the polynomial.

It should not come as a surprise that our choice of polynomial algorithms as the mathematical concept that is supposed to capture the informal notion of ‘practically efficient computation’ is open to criticism from all sides. […] Ultimately, our argument for our choice must be this: Adopting polynomial worst-case performance as our criterion of efficiency results in an elegant and useful theory that says something meaningful about practical computation, and would be impossible without this simplification.—Christos Papadimitriou. [As quoted in Nielsen and Chuang [27], p. 138]

Problems not in class P are hard problems, by which we do not mean that the problem is impossible to solve or non-computable—just that the number of steps needed to handle such problems increase faster than any polynomial in input size. The classes of problems that belong to NP are problems which require super-polynomial number of steps (such as exponential growth, which will always exceed polynomial growth eventually, regardless of the order of the polynomial) in input size to solve but whose solutions can be verified in polynomial number of steps in input size and hence are deemed intractable. A well-known example of this kind is the factorization of a number into its prime factors. The best-known algorithm for solving the problem of determining the prime factors places the problem in the NP class, but the algorithm for the problem of verifying the factors is well known to be trivial and belongs to the P class.

Clearly, it is possible to change the complexity class of a problem to a more efficient class by inventing new and better algorithms should such algorithms exist.³⁴ It is also clear that P is a subset of NP, but whether there exist problems in NP that are not in P is not known. In fact, “is P ≠ NP?” is an outstanding unsolved problem in computer science.³⁵ It is widely believed that NP contains problems that are not in P, that is, P is a proper subset of NP, but there is no proof.

There is a subclass of NP problems called NP-complete. Any given NP-complete problem is in some sense “at least as hard” as all other problems in NP. Around 1971, Stephen Cook³⁶ and Leonid Levin³⁷ independently discovered the notion of NP-completeness and gave examples of combinatorial NP-complete problems whose definition seems to have nothing to do with Turing machines. In 1972, Richard Karp³⁸ showed that 21 intractable combinatorial computational problems are all NP-complete. Since then, many thousands of problems have been discovered to belong to the NP-complete class.

Examples of NP-complete problems

Travelling salesman problem. Given a set of $$ n $$ cities, find the shortest route connecting them all, with no city visited more than once.

Hamilton cycle problem. Given an undirected graph, is there any way to connect all nodes with a simple cycle? That is, starting at some node, can we “visit” all other nodes and return to the original node, visiting every node in the graph exactly once?

Partition problem. Given a set of integers, can they be divided into two sets whose sum is equal?

Integer linear programming problem. Given a linear program, is there a solution in integers?

Multiprocessor scheduling problem. Given a deadline and a set of tasks of varying length to be performed on two identical processors, can the tasks be arranged so that the deadline is met?

Vertex cover problem. Given a graph and an integer $$ n $$, is there a set of fewer than $$ n $$ vertices which touch all the edges?

Independent set. Given a graph $$ G $$ and a number $$ k $$, is there a $$ k $$-size independent subset of $$ G $$’s vertices?

Decision problems

Some of the simplest examples of NP-complete problems come from propositional logic. These are the so-called decision problems —problems with a simple yes or no answer.³⁹ A Boolean formula over the variables $$ u_{1} , \ldots ,u_{n} $$, consists of the variables and the logical operators $$ {\text{AND}} $$ (∧), $$ {\text{OR}} $$ (∨), and $$ {\text{NOT}} $$ (¬). For example, (a∧b) ∨ (a∧c) ∨ (b∧c) is a Boolean formula that is TRUE if and only if the majority of the variables a, b, c are TRUE. If φ is a Boolean formula over variables $$ u_{1} , \ldots ,u_{n} $$, and $$ z \in \left\{ {0,1} \right\}^{n} $$, then φ(z) denotes the value of φ when the variables of φ are assigned the values z (where TRUE = 1, and FALSE = 0). The formula is satisfiable if there is an assignment to the variables that evaluates φ to TRUE else unsatisfiable.

Every statement in logic consisting of a combination of multiple $$ {\text{AND}} $$, $$ {\text{OR}} $$, and $$ {\text{NOTs}} $$ can be written in conjunctive normal form (CNF). A Boolean formula is in this form if it is an $$ {\text{AND}} $$ of $$ {\text{ORs}} $$ of variables or their negations. Examples are: a; (a∨b) ∧ (¬a∨c); a∨b; a∧(b∨c). This fact is known as universality of the operations $$ {\text{AND}} $$, $$ {\text{OR}} $$, and $$ {\text{NOT}} $$. If all clauses (terms in brackets) contain at most k literals (a literal is a variable or its negation), the formula is a kCNF. Cook, in 1971, showed that deciding whether a given Boolean formula in conjunctive normal form has an assignment that makes the formula TRUE or not (the satisfiability or SAT problem) is an NP-complete problem. This result enables us to easily prove that many other problems are NP-complete. Instead of directly proving that a problem is NP-complete, we instead prove that it is in NP and that SAT reduces to it.

It turns out that an important restricted case of SAT is also NP-complete—the 3-satisfiability problem (3SAT), which is concerned with formulas in 3-conjunctive normal form. The 3-SAT problem is to determine whether a formula in 3-conjunctive normal form is satisfiable or not. What is rather interesting is that 2SAT, which is concerned with formulas in 2-conjunctive normal form, can be solved in polynomial time! So, in a sense, 3SAT is the NP-complete problem since it has minimal combinatorial structure and is thus easy to use in reductions. In fact, it is the basis for countless proofs that other problems are NP-complete. If you suspect a problem might be NP-complete, look up Garey and Johnson, Computers and Intractability.⁴⁰ It contains a list of several hundred problems known to be NP-complete. Either what you are looking for will be there or you might find a closely related problem to use in a reduction. Stephen Cook, Jack Edmonds,⁴¹ Richard Karp, and Leonid Levin redefined computing around the notion of tractability and produced the most influential milestone in post-Turing computer science.

9.6.2 NP-Complete Problems Stand or Fall Together

The class of NP-complete problems has an interesting property: A NP-complete problem can be converted into another NP-complete problem by the action of some polynomial time algorithm. The name NP-complete thus comes from the facts that the problems in the class are in NP and consist of the “complete” set of problems that can be mapped into one another in polynomial time. Thus, the fate of one NP-complete problem is intimately bound to its kin. Either all NP-complete problems are tractable or none of them are! They stand or fall together . More specifically, an algorithm to solve a specific NP-complete problem can be adapted to solve any other problem in NP, with a small overhead. In particular, if P ≠ NP, then it will follow that no NP-complete problem can be efficiently solved on a classical computer. So far, no efficient algorithm, either classical or quantum, has been found to solve any NP-complete problem.

9.7 Concluding Remarks

We have discussed fundamental limits to computing with respect to algorithms in the idealized setting of the UTM that can simulate all digital computers in everyday use. The mathematical theory that underpins the UTM (Turing excluded quantum physics when considering universality), the Church–Turing thesis , and the von Neumann hardware architecture (stored program) used by all modern digital computers were important in the sense that a single computer design could be effectively used in many diverse applications with the added benefit of providing economies of scale, standardization of various aspects of computer design, exact reproducibility of computations, communication protocols, etc. in computer hardware design and software implementation. Their great virtue is that they are technology-agnostic, conceptually simple, and therefore have significant impact in being put into practice. Even the P-NP distinction is cautiously worded in terms of worst-case rather than average case performance. These aspects have kept prices of computers, despite their phenomenal increase in performance, within reach of billions of people and proliferated the use and embedding of computers in myriad electronic gadgets. The UTM serves as a versatile general purpose deterministic, computing machine as is evidenced in the diverse functionalities it provides in portable inexpensive smartphones. Indeed, general purpose computing has transformed entire industries, e.g., newspapers, banking, photography, commerce, etc. and launched new applications, e.g., video conferencing, GPS navigation, online shopping, networked entertainment, etc. They have brought about a sea change in the socio-economic structure of society. Advances in computing technologies today is invariably tied to costs and capital when creating, sustaining, and competing in new markets in a fundamental way. It is so because the world is under tremendous pressure due to population growth and an intensely competitive global economy. This is the world the millennials now inhabit and in which humans communicate with computers via user-friendly interfaces and domain-specific programming languages.

While we have not discussed how the special quantum mechanical aspects of quantum computing may affect the limits to computing, we now briefly remark that for the moment it holds promise in niche applications, especially in scientific research where it can be of immense help in simulating quantum-chemical phenomena and reveal new fundamental limits.⁴² In building quantum computers, one must bear in mind Heisenberg’s uncertainty principle , especially as related to energy–time since faster computation requires greater energy. In addition, there are substantial overheads in providing fault tolerance to combat decoherence in quantum computers. In the future, technology-specific limits will likely face tradeoffs arising from conflicting performance parameters and properties of material.

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K. Gödel, Über formal unentseheid-bare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik 38, 173–198. (Leipzig: 1931). http://​www.​ddc.​net/​ygg/​etext/​godel/​, (English translation: On formally undecidable propositions of Principia Mathematica and related systems, I. http://​www.​cs.​colorado.​edu/​~hirzel/​papers/​canon00-goedel.​pdf The theorem appears as Proposition VI of the paper. Part II of the paper was never published.)

17.

D. Hilbert, Mathematical problems, in Lecture Delivered Before the International Congress of Mathematicians at Paris in 1900. http://​aleph0.​clarku.​edu/​~djoyce/​hilbert/​problems.​html Dr. Maby Winton Newson translated this address into English with the author’s permission for Bull. Am. Math. Soc. 8, 437–479 (1902). A reprint of appears in Mathematical Developments Arising from Hilbert Problems, ed. by Felix Brouder, American Mathematical Society, 1976. The original address “Mathematische Probleme” appeared in Göttinger Nachrichten, 1900, pp. 253–297, and in Archiv der Mathematik und Physik, (3) 1 (1901), 44–63 and 213–237. [A fuller title of the journal Göttinger Nachrichten is Nachrichten von der Königl. Gesellschaft der Wiss. zu Göttingen.]

18.

R.M. Karp, Reducibility among combinatorial problems, in Complexity of Computer Computations (Proceedings of a Symposium IBM Thomas J. Watson Research Center, Yorktown Heights, NY, 1972) (Plenum, New York, 1972), pp. 85–103

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R.W. Keyes R. Landauer, Minimal energy dissipation in logic. IBM J. Res. Dev. 14, 152–157 (1970)

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22.

R. Landauer, Irreversibility and heat generation in the computing process. IBM J. Res. Dev. 5(3), 183 (1961). Reprinted in IBM Journal of Research and Development, 44(1/2) January/March 2000. https://​www.​pitt.​edu/​~jdnorton/​lectures/​Rotman_​Summer_​School_​2013/​thermo_​computing_​docs/​Landauer_​1961.​pdf

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L. Levin, Universal sorting problems, Probl. Peredaci Inf. 9:115–116 (1973). Original in Russian. English translation in Probl. Inf. Transm. USSR 9:265–266 (1973)

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I.L. Markov, Limits on fundamental limits to computation. arXiv:​1408.​3821v2 [cs.ET] 8 Jan 2015. https://​arxiv.​org/​pdf/​1408.​3821.​pdf Also as: Nature 512, 147–154 (14 August 2014). https://​doi.​org/​10.​1038/​nature13570

25.

Y. Matiyasevich, Enumerable sets are diophantine. Dokl. Akad. Nauk SSSR 191, 279–282 (1970); English translation with addendum, Soviet Math. Doklady 11, 354–357 (1970)

26.

Y. Matiyasevich, Hilbert’s Tenth Problem (MIT Press, 1993)

27.

M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2000). [For errata: http://​www.​squint.​org/​qci/​]

28.

R. Penrose, Shadows of the Mind (Oxford University Press, 1994) (Vintage paperback)

29.

M.B. Plenio V. Vitelli, The physics of forgetting: Landauer’s erasure principle and information theory. Contemp. Phys. 42, 25–60 (2001). https://​arxiv.​org/​pdf/​quant-ph/​0103108.​pdf

30.

K.R. Popper, Conjectures and Refutations: The Growth of Scientific Knowledge (Routledge, London, 1963)

31.

W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, “Is matrix inversion an N³ process?” §2.11, in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd edn. (Cambridge University Press, Cambridge, England, 1989)

32.

B. Russell, History of Western Philosophy (Simon and Schuster, 2008)

33.

V. Strassen, Gaussian elimination is not optimal. Numerische Mathematik 13, 354–356 (1969)

34.

L. Szilard, D. Über, Entropieverminderung in einem thermodynamischen system bei eingriffen intelligenter wesen. Zeitschrift für Physik. 1929, 53, 840–856. (In German) Szilard, L. Z. Physik (1929) 53: 840. https://​doi.​org/​10.​1007/​BF01341281 Translation by A. Rapoport and M. Knoller “On the decrease of entropy in a thermodynamic system by the intervention of intelligent beings” reprinted in Harvey S. Leff and Andrew F. Rex, Maxwell’s Demon: Entropy, Information, Computing (Princeton: Princeton University Press, 1990); Second edition: Maxwell’s Demon 2: Entropy, Classical and Quantum Information, Computing (Institute of Physics Publishing, 2003; pp. 124–133 (1st edition) and pp. 110–119 (2nd edition)

35.

A. Turing, On computable numbers, with an application to the Entscheidungsproblem, in Proceedings of the London Mathematical Society, Series 2, vol. 42, pp. 230–265. http://​www.​turingarchive.​org/​viewer/​?​id=​466&​title=​01bb, and at https://​www.​cs.​virginia.​edu/​~robins/​Turing_​Paper_​1936.​pdf. (Errata (1937): Vol. 43, pp. 544–546. http://​www.​abelard.​org/​turpap2/​tp2-ie.​asp

Footnotes

1

Hilbert [17].

2

Theorems are mathematical assertions derived from axioms and, when available, previously derived theorems, according to a prescribed set of rules.

3

Gödel [16].

4

The phrase “sufficiently strong” means that the axiomatic system contains enough arithmetic to carry out the coding constructions needed for the proof of the first incompleteness theorem. In that sense, for example, Gödel’s theorems do not apply to Presburger arithmetic, which proves every true first-order statement involving only addition. However, it applies to Peano arithmetic.

5

An algorithm is a step-by-step problem-solving procedure, often an established, recursive computational procedure for solving a problem in a finite number of steps.

6

E.g., I am the square root of −1. Who am i? Source: http://​mathworld.​wolfram.​com/​Self-Recursion.​html.

7

Church [7].

8

Turing [35].

9

Tautology: needless repetition of the same sense in different words.

10

Example from Penrose [28], pp. 28–30.

11

Matiyasevich [25]. See also: Matiyasevich [26]. To “show that Hilbert’s original problem about ordinary polynomial Diophantine equations is unsolvable required proving that exponentiation can be represented by a Diophantine equation, and this was finally done by Yuri Matiyasevich in 1969.” https://​www.​wolframscience.​com/​reference/​notes/​1161a.

12

Given a specific algorithm, one may often show that it must halt for any input, and in fact computer scientists often do just that as part of a correctness proof. However, each such proof requires new arguments: there is no mechanical, general way to determine whether algorithms halt.

13

There are certain computational tasks that cannot be performed by evaluating any function, e.g., there is no function that generates a true random number. Consequently, a Turing machine can only feign the generation of random numbers.

14

Arora and Barak [1], p. 26.

15

An alternative model of classical computation is a probabilistic Turing machine (PTM). It is a deterministic Turing machine in which some transitions are based on random choices from among finitely many alternatives.

16

Deutsch [11]. The reader is encouraged to read this classic paper in quantum computing.

17

Gallier [14].

18

Feynman [13], p. 141.

19

Landauer [22]. Landauer’s principle, and indeed, the second law of thermodynamics, can be understood as a logical consequence of the underlying reversible laws of physics as seen in the general Hamiltonian formulation of mechanics and in the unitary time evolution of quantum mechanics. See also: Keyes and Landauer [19].

20

For any deterministic device to be logically reversible, its input and output must be uniquely retrievable from each other. For the device to be thermodynamically reversible, it must be physically reversible, i.e., it must be capable of running backwards. For such a device, the second law of thermodynamics guarantees that it will not dissipate any heat.

21

For a proof see: Feynman [13], pp. 155–160.

22

Bennett [3]. See also: Bennett [5].

23

See, e.g., Braunstein [6].

24

Bennett [3, 4].

25

Szilard [34].

26

Bennett [4].

27

It first appeared in Paul Bachmann’s book, Analytische Zahlentheorie (Analytic Number Theory), pt. 2 Leipzig: B. G. Teubner, 1894. The notation was popularized in the work of another number theorist, Edmund Landau. See also: Landau [21].

28

Knuth [20], p. 107.

29

See, e.g., Analysis of Algorithms—Review. Adapted from notes of S. Sarkar of UPenn, Skiena of Stony Brook, etc. COMP 171, Fall 2005. http://​www.​cs.​ust.​hk/​~huamin/​COMP171/​algo_​2.​ppt#1.

30

Quantum mechanics shattered the platonic view of a reality amenable to noninvasive observation; tractability has clobbered classical notions of identity, randomness, and knowledge.

31

Simple encodings can be used to represent general mathematical objects—integers, pairs of integers, graphs (by their adjacency matrices), vectors, matrices, etc.—as strings of bits.

32

Cormen et al. [10], Chap. 3.

33

UTM, DTM, PTM, QTM are discussed in Chap. 10, Sect. 10.​5.

34

For example, in 1971 it was discovered that a faster algorithm for multiplication exists that uses Fast Fourier Transform (see A. Schönhage and V. Strassen, “Schnelle Multiplikation großer Zahlen,” Computing 7 (1971), pp. 281–292). It multiplies two n-digit numbers using cn log n operations where c is some absolute constant independent of n. Similarly, the classic Gaussian elimination technique for solving linear algebraic equations which use O(n³) basic arithmetic operations to solve n equations over n variables was improved upon in the 1960s by Volker Strassen who did it in O(n2.81) operations (see Strassen [33]); the best current method for the problem uses O(n2.376) (see Coppersmith and Winograd [9]). See also: Press et al. [31], pp. 95–98.

35

It is literally a million-dollar question; that is how much prize money the Clay Mathematics Institute will award to someone who resolves this question. See http://​www.​claymath.​org/​millennium/​.

36

Cook [8].

37

Levine [23].

38

Karp [18].

39

For example, is the given binary bit on or off.

40

Garey and Johnson [15].

41

Edmonds [12].

42

Markov [24].

© Springer Nature Singapore Pte Ltd. 2020

R. K. BeraThe Amazing World of Quantum ComputingUndergraduate Lecture Notes in Physicshttps://doi.org/10.1007/978-981-15-2471-4_10

10. The Crown Jewels of Quantum Algorithms

Rajendra K. Bera1  

(1)

Acadinnet Education Services India, Bangalore, Karnataka, India

Rajendra K. Bera

Email: rajendrabera@yahoo.com

Programming is the art of algorithm design and the craft of debugging errant code.

—Ellen Ullman

Abstract

This chapter provides detailed descriptions of the most intellectually valued algorithms in quantum computing, including Peter Shor’s factoring algorithm and Lov Grover search algorithm, among others. An attempt is made to explain the subtle aspects of the algorithms and why such algorithms are valued.

10.1 Introduction

In Chap. 8, we saw some unusual quantum solutions to problems whose classical solutions are well known. What was unusual was the use of superposition and entanglement in the rather weird Hilbert space, and the clever use of measurement. As Cleve et al. succinctly state:

Quantum computation is based on two quantum phenomena: quantum interference and quantum entanglement. Entanglement allows one to encode data into non-trivial multi-particle superpositions of some preselected basis states, and quantum interference, which is a dynamical process, allows one to evolve initial quantum states (inputs) into final states (outputs) modifying intermediate multi-particle superpositions in some prescribed way. Multi-particle quantum interference, unlike single particle interference, does not have any classical analogue and can be viewed as an inherently quantum process.¹

Multi-particle quantum systems are described in terms of tensor products on Hilbert space (from Postulate 4), which introduces non-local interactions (entanglement) between components of a quantum system. The Bell states , and quantum entanglement, in general, provide the essential novel, non-classical ingredient for quantum computing. They allow information to be encoded in non-local correlations between different parts of a physical system (teleported). The idea is to exploit both entanglement and quantum interference of probability waves . The trick lies in suppressing non-answers through destructive interference and amplifying answers through constructive interference. Quantum computers thus function as multi-particle interferometers with phase shifters (quantum logic gates). The wave function transports information via teleportation rather than by moving charges or a pulse as in a classical computer. Once the computations are done, a final measurement provides the answer, or in the case of multiple answers, a series of measurements gives their probability distribution from which the answer can be calculated.

In quantum algorithms, an observable-operator O, e.g., momentum, energy, position, etc., acts on a wave function $$ \left| \psi \right\rangle $$, collapses it to an eigenvector $$ \left| {s_{i} } \right\rangle $$ of O, and produces a classical output which is the eigenvalue $$ \lambda_{i} $$ associated with the eigenvector $$ \left| {s_{i} } \right\rangle $$. Thus, what we measure are the eigenvalues of an operator. Manipulating relative phase factors is the key to quantum algorithm development.

10.2 General Remarks on Quantum Algorithms

Presently, there are three broad classes of quantum algorithms which provide an advantage over known classical algorithms:

Those based on Fourier transform, which are also widely used in classical algorithms. The Shor’s algorithm (Sect. 10.8) for factoring, finding the discrete logarithm (not discussed²), and finding a hidden subgroup (not discussed³) belongs to this class. The quantum version of the Fourier transform (QFT) provides exponential speed up compared to the classical version. Therefore, QFT can easily break many of the most widely used cryptosystems now in use, including the RSA system.

Quantum search algorithms (Sect. 10.10), e.g., those used to extract statistics, such as the minimal element, from an unordered data set. For some problems, especially problems for which the best-known algorithm is a brute force search, they provide substantial speed up.

Quantum simulation algorithms, where quantum computers are used to simulate a quantum system (not discussed).

In the future, we may find other classes of quantum algorithms that surpass classical algorithms in the same class in some sense.

The key to designing quantum algorithms lies in clearly knowing the postulated differences between classical and quantum mechanics and the ability to master bitwise operations. Such operations operate on one- or two-bit patterns or binary numerals⁴ at the level of their individual bits. For that, we need to understand modulo arithmetic.

10.3 Modulo Arithmetic

Modulo arithmetic is a kind of “arithmetic of remainders.” The idea was formalized by Carl Friedrich Gauss in his influential book Disquisitiones Arithmeticae⁵ in 1801. Congruence arithmetic is familiar to us from our use of the clock. Clocks use modular arithmetic with modulus 12 for displaying hours (as there are 12 h on the clock face) and modulus 60 for displaying minutes (as there are 60 min in an hour). Modulo arithmetic is a way of doing arithmetic with a fixed set of consecutive numbers such that when you count beyond the top you start from zero again. In the 12-h clock example, if to 11 o’clock you add 5 h, normal arithmetic will give you 16 o’clock which is beyond the top 12 o’clock on the clock face, but because the clock uses modular arithmetic you will get (16 − 12 = 4) o’clock on the clock face.

Let m ≠ 0 be an integer. We say two integers a and b are congruent modulo m if there is an integer k such that a − b = km, and in this case, we write

Note that the symbol “≡” is not equality but congruence⁶, and the “mod m” simply signifies that the modulus is m. It should be read as “a is congruent to b, modulo m.”

10.3.1 Some Important Properties of Congruence

Let a ≡ a′ mod m and b ≡ b′ mod m. Then⁷:

1.

Equivalence: a ≡ b mod 0 ⇒ a = b. (The symbol ⇒ means “implies.”)

2.

Determination: either a ≡ b mod m or $$ a \not\equiv b $$ mod m.

3.

Reflexivity: a ≡ a mod m.

4.

Symmetry: a ≡ b mod m ⇒ b ≡ a mod m.

5.

Transitivity: a ≡ b mod m and b ≡ c mod m ⇒ a ≡ c mod m.

6.

$$ a + b \equiv a^{\prime} + b^{\prime} $$ mod m.

7.

$$ a - b \equiv a^{\prime} - b^{\prime} $$ mod m.

8.

$$ ab \equiv a^{\prime}\,b^{\prime} $$ mod m.

9.

a ≡ b mod m ⇒ ka ≡ kb mod m.

10.

a ≡ b mod m ⇒ an ≡ bn mod m.

11.

a ≡ b mod m₁ and a ≡ b mod m₂ ⇒ a ≡ b mod [m₁, m₂], where [m₁, m₂] is the least common multiple.

12.

ka ≡ kb mod m ⇒ a ≡ b mod m/(k, m), where (k, m) is the greatest common divisor.

13.

If a ≡ b mod m, then P(a) ≡ P(b) mod m, for P(x) a polynomial.

Notice that in the above properties although x ≡ y mod n, it is certainly not an equation, yet in many ways congruences can be handled in the same way as equations. For example, the same integer can be added to both members, and that both members can be multiplied by the same integer.

10.3.2 Congruence Classes

The relation of congruence modulo n is an equivalence relation (reflexivity, symmetry, and transitivity) on the set of integers Z = {…, −2, −1, 0, 1, 2, …}.⁸ As with any equivalence relation, the equivalence classes for congruence modulo n form a partition of Z; that is, the equivalence classes separate Z into mutually disjoint subsets, which are called congruence classes or residue classes. We see that there are n distinct congruence classes modulo n, given by

Here is an example. When n = 4, the classes are

The set of n distinct congruent classes noted above is usually denoted as

10.3.3 Modulo 2 Arithmetic

Modulo arithmetic is done, digit by digit, on binary numbers. Each digit is considered independent of its neighbors. Numbers are not carried or borrowed.

The following addition rules for adding two binary numbers are obvious:

In modulo 2 arithmetic , addition is the XOR operation. It is carried out on the corresponding binary digits of each operand according to the following rule:

The above rules may also be interpreted as “the sum of even numbers is even,” “the sum of an even number and an odd number is odd,” “the sum of an odd number and an even number is odd,” and “the sum of two odd numbers is even,” respectively.

The following multiplication rules for multiplying two binary numbers are also obvious:

In modulo 2 arithmetic , multiplication is the Boolean AND operation represented by ∧:

10.4 Bits and Qubits

A bit is a scalar that belongs to {0, 1}, whereas a qubit is a vector that belongs to a 2-dimensional complex Hilbert space. A classical n-bit register, at a time, can contain one binary string from

which allows for 2n entities to be indexed by, say, numbers $$ \{ 0,1,2,3, \ldots ,2^{n} - 1\} $$ where each number uniquely pairs with a binary string, say, according to some agreed upon convention. We are interested in bits because, at a fundamental level, a computer can be manipulated efficiently using binary symbols. Use of just unary symbols is inefficient because the number of memory locations needed grows exponentially with the size of the problem. Information theory tells us that bits (and hence qubits ) are enough to compute efficiently. Thus, it is not necessary to work with notations with an “alphabet” of more than two symbols. This has greatly simplified the design of computers and the analysis of computations. Strings using an alphabet with more than two symbols can always be mapped to an equivalent string using an alphabet with two symbols without an exponential increase in memory requirement with the size of the problem. To manipulate a string comprising only binary symbols (bits), it is sufficient to be able to manipulate the bits one at a time or in pairs. A binary logic gate takes two bits x, y as inputs and calculates a function f(x, y). Since f can be 0 or 1 (two possibilities), and there are four possible inputs, there are 2⁴ = 16 possible functions f. This set of 16 different logic gates is called a universal set, since by combining such gates in series any transformation of n bits can be carried out. Indeed, only one, the NAND (abbreviation for NOT AND) gate, can perform the role of the universal set.

It can be shown that there are no universal–reversible 2-bit gates in classical computing, but there exist universal–reversible 3-bit gates, e.g., the Toffoli gate T. The third bit (target bit) flips only if the first two bits (control bits) are each set to 1, otherwise not. That is, $$ T\left( {a,b,c} \right) \to T(a,b,c \oplus ab) $$. Therefore, depending on the input, the Toffoli gate sets the third bit to

The Toffoli is a universal gate because it can perform AND, XOR, NOT, or FANOUT depending on the inputs. A crucial difference between classical and quantum computing is that in quantum computing, the combination of 2-qubit controlled-not gate and 1-qubit unitary gates (all unitary gates are reversible) is adequate for universal quantum computation. In quantum computing, FANIN and FANOUT are prohibited.

10.4.1 Bitwise Operators

The important bitwise operators are listed in Table 10.1.

Table 10.1

Bitwise operators

Operator

Action

Remarks

NOT

A unary operation that performs logical negation on each bit, forming the ones’ complement of the given binary value

Example: NOT 0111 = 1000

In all the following operations, the operator takes two bit patterns of equal length and produces another one of the same length by matching up corresponding bits and performs the respective logical operation on each pair of corresponding bits. Specifically note that the respective counterpart Boolean operators treat their operands as Boolean values and not the individual bits that represent the value

OR

In each pair, the result is 1 if the first bit is 1 or the second bit is 1 or both bits are 1; otherwise the result is 0

Example: 0101 OR 0011 = 0111

XOR (exclusive or)

In each pair, the result is 1 if the two bits are different, and 0 if they are the same

This operation is also known as parity addition because it is similar to binary addition, except that there is never a carry

The bitwise XOR can be used to toggle flags in a set of bits. Example: Given the bit pattern 0010, the first and third bits may be toggled simultaneously by a bitwise XOR with another bit pattern containing 1 in the first and third positions (1010):

0010 XOR 1010 = 1000

AND

In each pair, the result is 1 if both bits are 1, otherwise the result is 0

The AND operator can be used to perform a bit mask operation, e.g., it may be used to isolate part of a string of bits, or to determine whether a particular bit is 1 or 0

Example: Given a bit pattern 0011, to find if the third bit is 1, it is bitwise ANDed with another bit pattern of equal length with 1 in the third bit and 0 in the other bits (0010):

0011 AND 0010 = 0010

The result (0010) shows the third bit is 1 in the original pattern (0011). Note that all other bits except the third have been “masked” to 0

NAND

It takes two bit patterns of equal length and performs the logical NOT followed by the AND operation on each pair of corresponding bits

10.4.2 String Manipulation Leads to Algorithms

Prior to the 1930s, the notion of an algorithm was rather vague, generally seen as a step-by-step problem-solving procedure, often an established, recursive computational procedure for solving a problem in a finite number of steps. One of the oldest is the two-thousand-year old algorithm devised by Euclid for finding the greatest common divisor of two positive integers. During the period 1930s–1950s, Alan Turing, Alonzo Church, and others provided our modern insight into the abstract nature of algorithms and the power of computation.

The fundamental model for algorithms is now the universal Turing machine (UTM). Over time, mathematicians have developed a theory to decide which algorithmic problems are computable and which are not, and they have discovered that many interesting algorithmic tasks are uncomputable or undecidable, i.e., no classical computer could solve them on certain inputs. Developments in classical computer science—the abstract universal Turing machine, the Church–Turing thesis , Turing’s halting theorem (see Chap. 9, Sect. 9.4)—have furthered our understanding of both classical and quantum computation. For example, some of the fast algorithms for implementation on quantum computers are based upon the Fourier transform used in many classical algorithms. Once it was realized that quantum computers could perform a type of Fourier transform much faster than classical computers, it became possible to develop many important quantum algorithms.

10.5 UTM, DTM, PTM, and QTM

The Universal Turing machine (UTM) described in Chap. 9 (Sect. 9.4) is called a deterministic Turing machine (DTM). A probabilistic Turing machine (PTM) is a non-DTM in which some transitions are made by randomly choosing from a finite set of allowed alternatives according to some probability distribution. This means that if the same problem is run many times, a PTM will produce stochastic results; each run for a given input may have a different run time, or it may not halt at all; further, it may accept an input in one execution and reject the same input in another execution. Interestingly, certain problems can often be solved quickly on a PTM compared to a DTM. But in the PTM model, there are often trade-offs between the time it takes to return an answer and the probability that the returned answer is correct. Alternatively, if one requires the answer to be correct, then there is uncertainty regarding the time the PTM will take to provide it. Despite these differences, it can be shown that anything computable by a PTM can be computed by a DTM.

A quantum Turing machine (QTM) may be thought of as a quantum mechanical generalization of a PTM. The key difference between a PTM and a QTM is that in a PTM only one particular computational trajectory is followed in a run while in a QTM all computational trajectories are followed in parallel in the same run. In classical computers (DTM and PTM), the states of the tape and of the head are always readable and writable, data can always be copied, and everything is uniquely defined. A mathematical theory of computation based on quantum mechanics involving superposition and entanglement is bound to be different. In his classic 1985 paper titled Quantum Theory, the Church–Turing Principle and the Universal Quantum Computer,⁹ David Deutsch described how a computer might run using the strange rules of quantum mechanics and why such a computer would differ fundamentally from ordinary computers. In fact, Deutsch showed that it is possible to construct reversible quantum gates for any classically computable function and suggested that quantum superposition might allow quantum evolution to perform many classical computations in parallel. Deutsch’s paper described the first true quantum Turing machine (QTM). In a QTM, read, write, and shift operations are all accomplished by quantum mechanical interactions and its “tape” can exist in states that are quantum mechanical. For example, it can simultaneously encode many inputs to a problem on the tape and perform a calculation on all the inputs at the same time (quantum parallelism). In 1997, Samuel Bernstein and Umesh Vazirani¹⁰ showed that it is possible to conceive a universal QTM. In such a construction, we must provide enough qubits that correspond to the tape of a Turing machine. The QTM is sufficiently simple and at the same time universal to prove various theorems about quantum computation.

10.5.1 Are Quantum Computers More Powerful?

It is generally believed but not yet proven that quantum computers are more powerful than classical computers because we do not know how the classical and the quantum worlds are bridged. It may turn out that quantum computers are only as powerful as classical computers: That any problem which can be efficiently solved on a quantum computer can also be efficiently solved on a classical computer. It all depends on how clever we are in designing algorithms and the nature of algorithms permitted by the axiomatic systems we use in designing them. Thus, there are issues of computability (which computational problems computers can or cannot solve), issues of algorithmic complexity (how efficiently computers can solve the problems they can), affordability and availability of resources needed to compute, etc.

D-Wave began to sell quantum computers commercially in May 2011 with its model D-Wave One (DW1). It used a 128-qubit chip-set, that was orders of magnitude faster than existing supercomputer technology. The D-Wave One was purchased by some research labs and defense contractors such as Lockheed. D-Wave Two (DW2) was even faster and uses a 512-qubit array. It is about 300,000 times faster than its predecessor. And, of course, it was much smaller than a classical computer but operates in an extreme environment of low temperature and low pressure.¹¹ The D-Wave 2000Q™ System has up to 2048 qubits , operates in an environment of 0.015° above absolute zero, in a vacuum where the pressure is 10 billion times lower than atmospheric pressure, shielded to 50,000 times less than Earth’s magnetic field, and consumes less than 25 kW of power.¹² D-Wave’s customers include Lockheed Martin, Google, and Los Alamos National Laboratory. The D-Wave computers are Quantum annealers, intended for use in specific technical applications. There are several quantum chips being developed by various companies, including Google’s 72-qubit Bristlecone processor (announced on 05 March 2018) along with Cirq, an open-source software toolkit to program it¹³; Intel’s 49-qubit super-conducting quantum test chip “Tangle Lake” (announced on January 8, 2018)¹⁴; IBM’s 50-qubit Q 50 prototype (announced on November 10, 2017)¹⁵; and Rigetti’s 19-qubit general purpose quantum processor (announced on December 18, 2017). More powerful chips are under development.¹⁶

10.6 The Quantum Fourier Transform

A word of caution. When dealing with algorithms, such as the Fourier transform, algorithm developers tend to freely move between decimal, binary, and other symbols (e.g., spin states ↑ and ↓) depending on what appears to be most intuitively appealing in a given step of the algorithm. For the first-time reader, it is advisable to study algorithms slowly to get used to this free movement between symbols. Note further that there is an implicit assumption that given a function $$ f(x) $$, it is possible for a clever person to break down its computation to the application of a set of 1-qubit and 2-qubit unitary operations. Typically, the sequence of operations is designed to map the state $$ \left| {x,0} \right\rangle $$ to the state $$ |\left. {x,f(x)} \right\rangle $$ for any input x. The number of qubits required to represent x and $$ f(x) $$ are chosen to be large enough to accommodate them before starting computations.

10.6.1 Background

Fourier transforms map from the time domain to the frequency domain. In other words, they map functions of period r to functions that have non-zero values only at multiples of the frequency $$ 2\pi /r $$. The discrete Fourier transform (DFT) operates on N equally spaced data samples in the interval $$ [0, 2\pi] $$ for some N and outputs a function whose domain is the integers between 0 and N − 1. The DFT of a sampled function of period r is a function concentrated near multiples of $$ N/r $$. If the period r divides N evenly, the result is a function that has non-zero values only at multiples of $$ N/r $$. Otherwise, the result will approximate this behavior, and there will be non-zero terms at integers close to multiples of $$ N/r $$. In the special case when N is a power of 2, the DFT results in a version known as the Fast Fourier Transform (FFT).

10.6.2 Quantum Fourier Transform

The discrete Fourier transform takes as input a vector of complex numbers, $$ x_{0} ,x_{1} , \ldots ,x_{N - 1} $$ where N is a fixed positive integer and outputs another vector of complex numbers $$ y_{0} ,y_{1} , \ldots ,y_{N - 1} $$ defined by

The quantum Fourier transform (QFT) is exactly the same transform. When written in the Dirac notation on an orthonormal basis $$ \left| 0 \right\rangle , \ldots ,|\left. {N - 1} \right\rangle $$, it is defined to be a linear operator with the following action on the basis states,¹⁷

Equivalently, the action on an arbitrary state may be written as

where the amplitudes yk are the discrete Fourier transform of the amplitudes $$ x_{j} $$, and k and j both range over the binary representations for the integers between 0 and N − 1. We see that it corresponds to a vector notation for the Fourier transform (first equation of this subsection) for the case $$ N = 2^{n} $$. If the state were to be measured after the Fourier transform was performed, the probability that the result was $$ \left| k \right\rangle $$ would be $$ \left| {y_{k} } \right|^{2} $$. Note that the QFT does not output a function the way the $$ U_{f} $$ transformation does, i.e., no output appears in an extra register. It can be shown that QFT is unitary. The Hadamard transform is a special case of the Fourier transform.

The best classical algorithms for computing the discrete Fourier transform of $$ 2^{n} $$ elements are algorithms such as the Fast Fourier Transform (FFT), which takes roughly $$ N\log N = n\,2^{n} $$ steps to Fourier transform $$ N = 2^{n} $$ numbers. On a quantum computer, it takes about $$ \log^{2} N = n^{2} $$ steps, an exponential reduction! Of course, we need to deal with the fact that the results of such parallel calculations are not available to us if we go about it in a straightforward manner.

Before getting into Shor’s algorithm, it is important to get the notations used here fixed in one’s mind. A careful reading of the notational convention is therefore suggested. The case when N is a power of 2 simplifies the mathematics without loss of generality. In this case, the operator can be represented as a product using an orthonormal basis.

Given the number $$ N = 2^{n} $$ and the number x represented in the form

we can create the orthonormal basis comprising the ket vectors indexed as $$ \left| 0 \right\rangle ,\left| 1 \right\rangle , \ldots ,\left| {2^{n} - 1} \right\rangle $$ where each basis state index can be represented in binary form as

The vectors $$ \left| {x_{1} } \right\rangle ,\left| {x_{2} } \right\rangle , \ldots ,\left| {x_{n} } \right\rangle $$ thus form the computational basis for the n-qubit system and span its state space.

We now associate with an integer x in $$ \left\{ {0,1, \ldots ,2^{n} - 1} \right\} $$ the rational number in the interval [0,1] whose binary representation is

where the first dot inside the square brackets means

The central result we now derive is that the discrete Fourier transform in the computational basis is such that it maps

into the tensor product

This form makes explicit the unitary operators that can implement the Fourier transform. This form may even be used as an alternative definition of the quantum Fourier transform .

If we take $$ N = 2^{n} $$, where n is some integer,¹⁸ and the basis $$ \left| 0 \right\rangle , \ldots , \left| {2^{n} - 1} \right\rangle $$ as the computational basis for an n qubit quantum computer, it is convenient to write the state $$ \left| j \right\rangle $$ using the shorthand notation $$ j = j_{1} j_{2} \ldots j_{n} $$ for $$ j = j_{1} 2^{n - 1} + j_{2} 2^{n - 2} + \cdots + j_{n} 2^{0} $$. It is also convenient to adopt the notation $$ 0.j_{l} j_{l + 1} \ldots j_{m} $$ to represent the binary fraction $$ \frac{{j_{l} }}{2} + \frac{{j_{l + 1} }}{4} + \cdots + \frac{{j_{m} }}{{2^{m - l + 1} }} $$. Then

Thus, we have

From this, it is rather obvious that unitary operators of the form

are expected to play a significant role.

Let us now see what sequences of unitary operators will allow the QFT to be calculated. Consider the state $$ |\left. {j_{1} \ldots j_{n} } \right\rangle $$ as input. An application of the Hadamard gate to the first qubit produces the state

since $$ e^{{2\pi \,i\,\left( {0.j_{1} } \right)}} = - 1 $$ when $$ j_{1} = 1 $$ and it is +1 otherwise. Now an application of the controlled-R₂ gate (this is a conditional rotation) produces the state

If we continue applying the controlled-R₃, R₄ through Rn gates, we find that each application adds an extra bit to the phase of the coefficient of the first $$\left| 1\right\rangle = 1 $$. Thus, we end with the state

Next, we perform a similar procedure on the second qubit , namely the application of a Hadamard gate followed by controlled-R₂ through Rn−1 gates, to obtain

If we continue in this fashion for each successive qubit , we arrive at the final state

Finally, the swap operation (see Chap. 8, Sect. 8.2.3) is used to reverse the order of the qubits to provide the final result

As already noted above, the best classical algorithms for computing the discrete Fourier transform of $$ 2^{n} $$ elements are algorithms such as the Fast Fourier Transform (FFT), which compute the transform using $$ O\left( {n\,2^{n} } \right) $$ gates. However, on a quantum computer this can be done using $$ O\left( {n^{2} } \right) $$ gates.¹⁹ This spectacular reduction in computational steps is not feasible on a classical computer even for rather small values of n.²⁰ The importance of QFT lies in the fact that many quantum algorithms seem to follow the generic sequence: a Fourier transform, followed by a f-controlled-$$ U $$, followed by another Fourier transform, or has this sequence as an important component.²¹

10.7 Computing the Period of a Sequence

Given the sequence $$ f(0),f(1), \ldots ,f\left( {2^{n} - 1} \right), $$ find its period.

To start with, we take two registers, the first of n qubits to store the $$ 2^{n} $$ argument values, x, of the function f, and the second of k qubits , long enough to store the longest value of $$ f(x) $$. With an appropriately chosen $$ U_{f} $$, we get

A measurement on the second register now will collapse it to some value of f that corresponds to some $$ x = x_{0} $$. Given that $$ f\left( {x + kr} \right) = f(x) $$, where r is the period of the function f and k is an integer, the first register will correspondingly attain a superposition of state vectors $$ \left\{ { \ldots ,x_{0} - r,\;x_{0} ,\;x_{0} + r,\;x_{0} + 2r, \ldots } \right\} $$. The state of the combined registers is thus

For notational convenience, we now choose $$ x_{0} $$ to be the lowest $$ x \in \left\{ {0, \ldots ,2^{n} - 1} \right\} $$ such that $$ f(x) = f\left( {x_{0} } \right). $$ Thus, the index j in the summation will take the values $$ 0,1, \ldots ,A - 1 $$. We now apply the QFT to the first register

A measurement of the first register will yield a $$ |\left. y \right\rangle $$ with probability

Since

When r divides $$ 2^{n} $$ exactly, $$ A = 2^{n} /r $$ and $$ A/2^{n} = 1/r, $$ the probability becomes

If $$ y = A,2A, $$ etc., then

since $$ jy/A $$ is an integer and

But if y is incommensurate (irrational) with A, then we will hit a range of points around the circle, eventually filling the whole circle so the resulting interference will be totally destructive, and we will have $$ p(y) = 0 $$. Therefore, in effect, the measurement will return a $$ y \in \left\{ {A,2A, \ldots ,rA} \right\} $$. From this, we can easily find A, and knowing the range $$ 2^{n} $$ we can find $$ r = 2^{n} /A $$ (see Fig. 10.1).

Fig. 10.1

The plot is idealized. Constructive interference produces narrow peaks at multiples of 1/r. The discretized approximation means that the actual peaks will have non-zero width. $$ q = 2^{n} $$. Source Braunstein [10]

In general, r will not divide 2n exactly, so measurements will return y scattered around integer multiples of 1/r, such that the constructive interferences produce narrow peaks at multiples of 1/r. Thus, we will obtain only a random multiple of the inverse period. To extract the period itself, we will need to repeat this quantum computation roughly $$ \log \log r/n $$ times in order to have a high probability for at least one of the multiples to be relatively prime²² to the period r, thereby uniquely determining the period. This algorithm yields only a probabilistic answer but the probability can be made as high as we like. The actual determination of r can be made by using the continued fractions algorithm. This is a method for determining for an arbitrary real number, x, an expansion of the form²³

Consider, for example, 31/13. This is successively split into its integer and fractional parts,

A truncated continued fraction is called a convergent. The kth convergent is written $$ \left[ {a_{0} , \ldots ,a_{k} } \right] $$. The decomposition terminates when the fraction has a numerator or a denominator of 1. The method terminates after a finite number of “split and invert” steps for any rational number, since successive numerators are strictly decreasing.

One can show that if $$ \varphi = s/r $$ is a rational number and s and r are L bit integers, then the continued fraction expansion for φ can be computed using $$ O\left( {L^{3} } \right) $$ operations—$$ O\left( L \right) $$ “split and invert steps” and $$ O\left( {L^{2} } \right) $$ gates for elementary arithmetic. There is also a useful theorem about continued fractions.

Theorem 1

If s/r is any rational number satisfying $$ \left| {s/r - \varphi } \right| \le 1/\left( {2r^{2} } \right) $$then s/r is a convergent of the continued fraction for φ Moreover, the convergent is such that $$ { \gcd }\left( {s,r} \right) = 1. $$

We can use this theorem to find the closest rationals to the terms s/r and hence find the period r. Furthermore, given φ the continued fractions algorithm efficiently produces numbers $$ s^{\prime} $$ and $$ r^{\prime} $$ with no common factor, such that $$ s^{\prime}/r^{\prime} = s/r^{{\prime}{\prime}} $$. Euclid’s algorithm for finding the greatest common divisor of two numbers is intimately related to continued fractions.²⁴

To find the period, r, measure the state of the first register. This effectively samples from the discrete Fourier transform and returns some number $$ y^{\prime} $$ that is some multiple s of q/r; that is $$ y^{\prime}/q \approx s/r $$ for some positive integer s. To determine the period r we need to estimate s. This is accomplished by computing the continued fraction expansion for $$ y^{\prime}/q $$. By repeating the quantum computation several times (roughly $$ \log \log r/n $$ times), we create a set of samples of the discrete Fourier transform in the first register. This gives samples of multiples of 1/r as $$ s_{1} /r,s_{2} /r,s_{3} /r, $$… for various integers $$ s_{i} $$. Once sufficient samples have been obtained from the first register, we can use the continued fraction technique to compute as to what the $$ s_{i} $$ could be and hence to guess r. Computing the period of a sequence lies at the heart of efficiently factoring numbers, which we discuss below.

10.8 Shor’s Factoring Algorithm

In 1994, Peter Shor surprised the world by describing a polynomial time quantum algorithm for factoring integers.²⁵ Shor’s algorithm became a killer application. The difficulty of factorization underpins the security of many common methods of encryption, e.g., RSA,²⁶ the most popular public key crypto-system which is often used to protect electronic bank accounts. The security of such systems comes from the difficulty of factoring large numbers. Potential use of quantum computation for code breaking purposes then provided the much needed impetus for building future quantum computers. With this algorithm, quantum computing came of age. It prompted a flurry of activity in building quantum computers and theoreticians trying to find other quantum algorithms. Shor’s factoring algorithm has two parts: the quantum procedure within the algorithm and the classical algorithm that calls the quantum procedure.

The algorithm relies upon a result from number theory that relates the period of a particular periodic function to the factors of an integer. In fact, the crux of the algorithm is the quantum method for computing the period r of the function $$ f(a) = x^{a} \bmod N $$ for $$ a = 0,1, \ldots $$. That is exponentially more efficient than any known classical method. Thus, given an integer N (the number to be factored), construct $$ f(a) = x^{a} \bmod N $$ where $$ x &lt; N $$ is a randomly chosen number which is coprime²⁷ to N. It can be shown that $$ f(a) $$ thus constructed is periodic, that is,

Here, r is the first non-trivial power where xr ≡ 1 mod N, and that $$ f(a) $$ is periodic with period r, i.e., $$ f(a) = f\left( {a + r} \right) = f\left( {a + 2r} \right) = \ldots . $$ Obviously, different values of x may produce different periodicities.

The periodicity of $$ f(a) $$ can be determined by using the quantum algorithm described earlier. For a given N and a chosen x if the period r is an even number, we can write

in the alternative form

This implies that the product of the two factors on the left is a multiple of the number N. So, unless $$ x^{r/2} \equiv 1 \pm \bmod \,N $$, at least one of these factors must be in common with N. Thus, we have a good chance of finding a factor of N by computing, $$ { \gcd }\left( {x^{r/2} + 1,N} \right) $$ and $$ { \gcd }\left( {x^{r/2} - 1,N} \right) $$.

If r is odd, we choose another x and redo the previous steps till an x is found for which r is even. (With randomly chosen x, an even r will happen 50% of the time.) Once a factor u is found, Shor’s algorithm is repeated on $$ N_{1} = N/u $$, and so on till all the factors of N are found.

The quantum part of Shor’s algorithm is determining the period r, and the classical part is computing the function $$ { \gcd }() $$, which is efficiently done using Euclid’s algorithm.

Example

Find the factors of N = 91.

Choose x = 3, and find the sequences

The quantum algorithm would find the period r = 6, which is an even number. Now,

Thus, either gcd(28, 91) = 7 and/or gcd(26, 91) = 13 will be factor(s). In this case, both are: 91 = 7 × 13.

10.8.1 Shor’s Algorithm Implemented

On December 19, 2001, IBM announced that it had built under Isaac L. Chuang’s leadership a 7-qubit quantum computer based on seven atoms which, because of the physical properties of those atoms, were able to work together as both the computer’s processor and memory, and that they were able to use the computer to show that Shor’s algorithm works by correctly identifying 3 and 5 as the factors of 15.²⁸ In 2012, Lucero et al. reported in Nature Physics, their successful factorization of the number 15 using Shor’s algorithm in a solid-state system: a circuit made up of four superconducting qubits .²⁹ In March 2016, it was reported that Shor’s algorithm was successfully implemented using five trapped ions for factoring the number 15 in which prior knowledge of the factors was not used to simplify the computational procedure.³⁰ All previous implementations had used such knowledge to simplify the circuit.³¹ Various other implementations of Shor’s algorithm are being tried to gain further insight as to the nature of the resources that may be successfully used.³² Other numbers, e.g., 51 and 85, have also been factorized.³³ Very efficient implementations are expected to be discovered in the future.

10.8.2 Computational Complexity of Shor’s Algorithm

On classical computers (Turing machines), the best known factoring algorithm (the number field sieve) runs in $$ O\left( {\exp \left[ {\left( {64/9} \right)^{1/3} \left( {\ln N} \right)^{1/3} \left( {\ln \ln N} \right)^{2/3} } \right]} \right) $$ steps. This algorithm, therefore, scales exponentially with the input size $$ \log N $$. To give an example, in 1994 a 129-digit number was successfully factored using this algorithm on approximately 1600 workstations scattered around the world; the entire factorization took eight months. It was estimated that factoring a 250-digit number would take roughly 800,000 years and a 1000-digit number would require 10²⁵ years (longer than the age of the universe!). Shor’s algorithm runs in $$ O\left( {\left( {\log N} \right)^{3} } \right) $$ steps. This is roughly quadratic in the input size, so factoring a 250-digit number would require only a few billion steps.

10.9 Phase Estimation Problem

The problem we now look at is the following: Given a unitary operator U which has an eigenvector $$ \left| \psi \right\rangle $$ associated with the eigenvalue $$ {\text{e}}^{2\pi i\theta } $$, find the value of θ. We assume that we know how to construct black boxes that can prepare the state $$ \left| \psi \right\rangle $$ and perform the controlled-$$ U^{{2^{j} }} $$ operations, for suitable non-negative integers j. The phase estimation problem uses two registers. The first contains n qubits initially in the state $$ |\left. 0 \right\rangle^{ \otimes n} $$. The choice of n depends on the desired computing accuracy for estimating θ and the probability with which we wish the phase estimation procedure to succeed. The second register begins in the state $$ \left| \psi \right\rangle $$ and contains as many qubits as necessary to store $$ \left| \psi \right\rangle $$.

The initial state of the two registers is thus $$ \left| 0 \right\rangle^{ \otimes n} \left| \psi \right\rangle $$. To the first register we apply the Hadamard transform, $$ H^{ \otimes n} $$. This puts the register into the state

This is followed by the application of n controlled-$$ U^{{2^{j} }} $$ operations on the second register. Note that U is a unitary operator with eigenvector $$ |\left. \psi \right\rangle $$ such that $$ U\left| \psi \right\rangle = {\text{e}}^{2\pi i\theta } \left| \psi \right\rangle $$. Thus

The job of a controlled-$$ U^{{2^{j} }} $$ with $$ 0 \le j \le n - 1 $$ is to act on the second register only if its corresponding control bit in the first register is $$ \left| 1 \right\rangle $$.

After the n controlled-$$ U^{{2^{j} }} $$ operations have concluded, and using

the state of the first register can be expressed in the form

as we show below. Here, $$ \left| k \right\rangle $$ denotes the binary representation of k. Note that the left-hand side of the above equation is the product form for the Fourier transform. Therefore, if the expression for the phase θ had the exact n-bit representation $$ \theta = 0.\theta_{1} \theta_{2 \cdots } \theta_{n} $$, then

interchanged, giving the final

The state of the control bits can now be rewritten as

which is the Fourier transform of the basis state

When we rewrite the state of the n control bits, we get it in the form

which is the form of a Fourier transformed state. To this more familiar form, we now apply the inverse quantum Fourier transform to get

Jointly, the two registers are in the state

Note that in the entire process, the contents of the second register have not changed. A measurement in the computational basis therefore gives θ exactly!

When θ is not represented exactly with a n bit binary expansion, the above procedure still produces a pretty good approximation for θ with high probability.³⁴

10.10 Grover’s Search Algorithm

In 1996, Lov Grover found an efficient quantum algorithm for the problem: Given an unstructured list of N elements, find a specific element $$ x_{0} $$ in the list. Grover’s algorithm is much faster than any known classical algorithm.³⁵

For convenience, each element is mapped to a unique index, which is just a number in the range 0 to N − 1. Furthermore, assume that $$ N = 2^{n} $$ (the index can be stored in n bits) and that the search problem has exactly M solutions, with $$ 1 \le M \le N $$. An instance of the search problem can be conveniently represented by a function f, which takes an integer x as input in the range 0 to $$ N - 1 $$. By definition $$ f(x) = 1 $$ if x is a solution to the search problem, otherwise $$ f(x) = 0 $$.

We now introduce the notion of a quantum oracle—a black box unitary operator O. The oracle’s unique ability is to recognize solutions to the search problem. It does this by setting an oracle qubit $$ |\left. q \right\rangle $$ in the following manner

where $$ \left| x \right\rangle $$ is the index register. To check if x is a solution to our search problem, we prepare $$ |\left. x \right\rangle |\left. 0 \right\rangle $$, apply the oracle, and check if the oracle qubit has been flipped to $$ \left| 1 \right\rangle $$. (Recall that a similar trick was used in the Deutsch problem in Chap. 8, Sect. 8.​2.​4.)

Here, instead, we cleverly apply the oracle with the oracle qubit initially set to $$ \left( {\left| 0 \right\rangle - \left| 1 \right\rangle } \right)/\sqrt 2 $$. If x is not a solution (i.e., $$ f(x) = 0 $$), applying the oracle to the state $$ \left| x \right\rangle \left( {\left| 0 \right\rangle - \left| 1 \right\rangle } \right)/\sqrt 2 $$ does not change the state. But if x is a solution, $$ \left| 0 \right\rangle $$ and $$ |\left. 1 \right\rangle $$ are interchanged, giving the final state $$ - \left| x \right\rangle $$ $$ \left( {\left| 0 \right\rangle -\left| 1 \right\rangle } \right)/\sqrt 2 $$, i.e.,

When written in this form, the oracle qubit appears unchanged and indeed remains unchanged throughout! Hence, it is customary to depict the action of the oracle by omitting it as follows:

and to note that the oracle marks the solution to the search problem by shifting the phase of the solution. There is, however, a subtle point here; the oracle does not know the answer, and it can only recognize the answer! It is, of course, possible to do the latter without being able to do the former. For example, finding the prime factors of a very large number is a known difficult problem. But if the prime factors are given, we can easily verify if their product is indeed the given number. The point is that even without knowing the prime factors, one can explicitly construct an oracle, which recognizes the factors when it sees one.

Grover’s search algorithm begins with an n qubit register in the state $$ |\left. 1 \right\rangle^{ \otimes n} $$, to which the Hadamard transform is applied and an oracle qubit in the state $$ |\left. 0 \right\rangle $$ to which HX is applied:

To this, we iteratively apply the Grover operator G, comprising the following 4 steps:

1.

Application of the oracle O.

2.

Application of the Hadamard operator $$ H^{ \otimes n} $$.

3.

Performing a conditional phase shift on the computer, with every computation basis state except $$ \left| 0 \right\rangle $$ receiving a phase shift of −1.

4.

Application of the Hadamard operator $$ H^{ \otimes n} $$.

The oracle O marks solutions to the search problem by shifting the phase of the solution. Thus,

where $$ f(x) = 1 $$ if x is a solution else $$ f(x) = 0 $$.

The third step requires a conditional phase shift to be applied to every computational basis state, except $$ |\left. 0 \right\rangle $$, so that $$ |\left. 0 \right\rangle \to |\left. 0 \right\rangle $$ and $$ |\left. x \right\rangle \to - |\left. x \right\rangle $$ for $$ x &gt; 0 $$. This is accomplished by the operator $$ 2|\left. 0 \right\rangle \left\langle 0 \right.| - I $$. Thus, G is

where the operator $$ 2|\left. 0 \right\rangle \left\langle 0 \right.| - I $$ flips states about the $$ |\left. 0 \right\rangle $$ axis, i.e., it reverses every basis state except for $$ |\left. 0 \right\rangle $$. Further let $$ |\left. \psi \right\rangle = H^{n} |\left. 0 \right\rangle $$. Then since $$ H^{2} = I $$, we can rewrite G as

One may show that the operation $$ 2|\left. \psi \right\rangle \left\langle \psi \right.| - I $$ applied to a general state $$ \mathop \sum \limits_{k} a_{k} |\left. k \right\rangle $$ produces

Here, $$ \left\langle \alpha \right\rangle $$ is the mean value of the $$ \alpha_{k} $$. For this reason, $$ 2|\left. \psi \right\rangle \left\langle \psi \right. - |I $$ is sometimes referred to as the inversion about the mean operation.

So, what does the Grover iteration do? For one thing, it can be regarded as a rotation in the two-dimensional space spanned by the starting vector $$ |\left. \psi \right\rangle $$ and the state consisting of a uniform superposition of solutions to the search problem. To see this clearly, let $$ \sum\nolimits_{x}^{\prime } {()} $$ indicate a sum over the M values of x which are solutions to the search problem, and $$ \sum\nolimits_{x}^{{\prime \prime }} {()} $$ a sum over the remaining N − M values of x. Define normalized states as follows:

Hence, the initial state $$ |\left. \psi \right\rangle $$ may now be re-expressed as

So, the initial state of the computer is in the space spanned by $$ |\left. \alpha \right\rangle $$ and $$ |\left. \beta \right\rangle $$. On this space, the oracle O essentially performs a reflection about the vector $$ |\left. \alpha \right\rangle $$ in the plane defined by $$ |\left. \alpha \right\rangle $$ and $$ |\left. \beta \right\rangle $$. That is,

Similarly, $$ 2|\left. \psi \right\rangle \left\langle \psi \right.| - I $$ also performs a reflection in the plane defined by $$ |\left. \alpha \right\rangle $$ and $$ |\left. \beta \right\rangle $$ about the vector $$ |\left. \psi \right\rangle $$. And the product of two reflections is a rotation! We notice two things here. First, after the kth iteration, the state $$ G^{k} |\left. \psi \right\rangle $$ remains in the space spanned by $$ |\left. \alpha \right\rangle $$ and $$ |\left. \beta \right\rangle $$ for all k. Second, it gives us the rotation angle. To get the rotation angle, let $$ \cos (\theta /2) = \sqrt {\left( {N - M} \right)/N} $$, so that

The two reflections that comprise G take $$ |\left. \psi \right\rangle $$ to

so the rotation angle is in fact θ. Hence, k applications of G take the state to

Thus, G produces a rotation in the two-dimensional space spanned by $$ |\left. \alpha \right\rangle $$ and $$ |\left. \beta \right\rangle $$, rotating the space by θ radians per application of G (see Fig. 10.2). Repeated application of G rotates the state vector close to $$ |\left. \beta \right\rangle $$. After R iterations, an observation in the computational basis produces, with high probability, one of the outcomes superposed in $$ |\left. \beta \right\rangle $$, i.e., a solution to the search problem.

Fig. 10.2

G produces a rotation

Thus, in the $$ |\left. \alpha \right\rangle $$, $$ |\left. \beta \right\rangle $$ basis, Grover’s iteration can be written as

where θ is a real number in the range 0 to π/2.

How is R chosen? Note that the initial state of the system was

Hence, $$ |\left. \psi \right\rangle $$ needs to be rotated by $$ \frac{\pi }{2} - \frac{\theta }{2} $$ from its initial position to reach the solution state $$ |\left. \beta \right\rangle $$. Note also that

Therefore, rotating through

radians takes the system to $$ |\left. \beta \right\rangle $$. Let CI(x) denote the integer nearest to the real number x, where by convention we round halves down—CI(3.5) = 3, for example—so that $$ {\text{CI(}}x )\le x $$. Then, applying the Grover iteration

times rotates $$ |\left. \psi \right\rangle $$ to within an angle $$ \theta /2 \le \pi /4 $$ of $$ |\left. \beta \right\rangle $$. Observation of the state in the computational basis then yields a solution to the search problem with probability at least one-half. In fact, for specific values of M and N, it is possible to achieve a much higher probability of success. For example, when $$ M \ll N $$ we have $$ \theta \approx \sin \theta \approx 2\surd \left( {M/N} \right), $$ and thus the angular error in the final state is at most $$ \theta /2 \approx \surd \left( {M/N} \right) $$, giving a probability of error of at most $$ M/N $$. Note that R depends on the number of solutions M, but not on the identity of those solutions. Thus, applying Grover’s iteration, G, R times, for the case M = 1, we have

The result $$ x_{0} $$ is found by measuring the first n qubits .

Boyer et al.³⁶ have shown that if there is a single solution $$ x_{0} $$, then after $$ \left( {\pi /8} \right)\sqrt {\left( {2^{n} } \right)} $$ number of Grover iterations the failure rate is 0.5. After $$ \left( {\pi /4} \right)\sqrt {\left( {2^{n} } \right)} $$ iterations the failure rate drops to 2−n. However, additional iterations will increase the failure rate! For example, after $$ \left( {\pi /2} \right)\sqrt {\left( {2^{n} } \right)} $$ iterations the failure rate is close to 1. The reason is that unitary transformations are rotations of complex space, and thus while a repeated application of a quantum transformation may rotate the state closer and closer to the desired state for a while, eventually it will rotate past that state and get farther and farther away. Thus, to obtain useful results from a repeated application of a quantum transformation, one must know when to stop.

10.10.1 Grover’s Algorithm Verified

In 1997, Isaac L. Chuang (IBM, Almaden), Neil A. Gershenfeld (MIT, Cambridge), and Mark G. Kubinec (Univ. of California, Berkeley) actually built a simple 2-qubit NMR quantum computer using liquid chloroform (CHCl₃) and successfully ran Grover’s algorithm.³⁷ In 2000, Chuang and his team reported the experimental implementation of Grover’s algorithm on a 3-qubit NMR quantum computer comprising molecules of ¹³C-labeled CHFBr₂, in which the three weakly coupled spin-1/2 nuclei behave as qubits and are initialized, manipulated, and read out using magnetic resonance techniques.³⁸ A more recent implementation of the algorithm using three qubits was provided by Caroline Figgatt et al. in 2017.³⁹ They reported “results for a complete 3-qubit Grover search algorithm using the scalable quantum computing technology of trapped atomic ions, with better-than-classical performance.”

10.10.2 Computational Complexity of Grover’s Algorithm

It is well known that classical methods for this search problem require $$ n/2 $$ searches on average to find a solution. Grover’s algorithm requires $$ O\left( {\sqrt n } \right) $$ steps. Not only that, Grover’s algorithm is also the fastest even among all possible quantum algorithms for this problem.⁴⁰ Note that the task remains computationally hard, i.e., it is not transferred to a new complexity class, but it is remarkable that such a seemingly hopeless task can be speeded up at all. Any problem for which finding solutions is hard, but testing a candidate solution is easy, can as a last resort be solved by an exhaustive search. In such cases, Grover’s algorithm may prove very useful.

10.10.3 Remarks on Grover’s Algorithm

There are variations of Grover’s algorithm, which can find the largest or smallest value in a list, or the modal value,⁴¹ and so on. So, it is quite a versatile searching tool. However, in practice, searching a physical database is unlikely to become a major application of Grover’s algorithm for non-algorithmic reasons, at least so long as classical memory remains cheaper than quantum memory. For since the operation of transferring a database from classical to quantum memory (bits to qubits ) would itself require $$ O(n) $$ steps, Grover’s algorithm would improve search times by at best a constant factor, which could also be achieved by classical parallel processing. Grover’s algorithm becomes really useful when searching through lists that are not stored in memory but are themselves generated on the fly by a computer program.

10.11 Dense Coding and Teleportation

Dense coding uses an entangled pair to encode and transmit two classical bits worth of information. Since entangled pairs can be distributed ahead of time, only one qubit needs to be physically transmitted from sender to receiver to communicate two bits of information. The idea is due to Bennett and Wiesner⁴². This is an example of transmitting ordinary classical information using a quantum channel.

Teleportation uses two classical bits to transmit a single qubit (see Chap. 1, Sect. 1.5). Teleportation is surprising in light of the no-cloning principle, in that it enables the transmission of an unknown quantum state. The idea is due to Bennett et al.⁴³

In both, dense coding and teleportation , the 2-qubit entangled Bell state

plays a crucial role. Recall that measurement on any qubit in a group of entangled qubits immediately readjusts the state of all other qubits in the group. Thus, in the above Bell state , if either qubit is measured, that measurement will force both qubits to be in either state $$ \left| {00} \right\rangle $$ or $$ |\left. {11} \right\rangle $$, i.e., the measurement outcomes are correlated.

10.11.1 Dense Coding

If Alice and Bob wish to communicate, each is sent a qubit from an entangled pair in the state

Let Alice get the first qubit and Bob the second, thereby establishing a quantum communication channel between them. Note that there are four mutually orthogonal states $$ |\left. 00 \right\rangle $$ + $$ |\left. 11 \right\rangle $$ $$ |\left. 00 \right\rangle $$ − $$ |\left. 11 \right\rangle $$, $$ |\left. 01 \right\rangle $$ + $$ |\left. 10 \right\rangle $$ $$ |\left. 01 \right\rangle $$ − $$ |\left. 10 \right\rangle $$, which form the Bell basis. When Alice receives two classical bits of information for transmission, say encoding a number from 0 through 3, she performs one of the following transformations on her qubit depending on the value of the encoded number and puts it in the new state:

Value

Transformation

New State

Remark

0

$$ \left| \psi_{0} \right\rangle $$ = (I ⊗ I) $$ \left| \psi_{0} \right\rangle $$〉

(1/√2) ($$ |\left. 00 \right\rangle $$ + $$ \left| 1\,1 \right\rangle $$)

I: $$ \left| 0 \right\rangle $$ → $$ \left| 0 \right\rangle $$; $$ \left| 1 \right\rangle $$ → $$ \left| 1 \right\rangle $$

1

$$ \left| \psi_{1} \right\rangle $$ = (X ⊗ I) $$ \left| \psi_{0} \right\rangle $$

(1/√2) ($$ \left| 10 \right\rangle $$ + $$ \left| 01 \right\rangle $$)

X: $$ \left| 0 \right\rangle $$ → $$ \left| 1 \right\rangle $$; $$ \left| 1 \right\rangle $$ → $$ \left| 0 \right\rangle $$

2

$$ \left| \psi_{2} \right\rangle $$ = (Y ⊗ I) $$ \left| \psi_{0} \right\rangle $$

(1/√2) (−$$ \left| 10 \right\rangle $$ + $$ \left| 01 \right\rangle $$)

Y: $$ \left| 0 \right\rangle $$ → − $$ \left| 1 \right\rangle $$; $$ \left| 1 \right\rangle $$ → $$ \left| 0 \right\rangle $$

3

$$ \left| \psi_{3} \right\rangle $$ = (Z ⊗ I) $$ \left| \psi_{0} \right\rangle $$

(1/√2) ($$ \left| 00 \right\rangle $$ − $$ \left| 11 \right\rangle $$)

Z: $$ \left| 0 \right\rangle $$ → $$ \left| 0 \right\rangle $$; $$ \left| 1 \right\rangle $$ → − $$ \left| 1 \right\rangle $$

Since there are four possibilities, her choice of operation represents two bits of classical information. Note that transforming just one qubit of an entangled pair means performing the identity transformation (I) on the other qubit.

Alice then sends her qubit to Bob who must deduce which Bell basis state the qubits are in (i.e., the new state). Bob first applies a controlled-not to the two qubits using the first (Alice’s) qubit as control. The result is shown in the second column in the table below. (For clarity, the third and fourth column explicitly show the state of the first and second qubit , respectively, after application of the controlled-not gate.)

New state

State after Cnot

First qubit

Second qubit

$$ \left| \psi_{0} \right\rangle $$ = (1/√2) ($$ |\left. 00 \right\rangle $$ + $$ \left| 11 \right\rangle $$)

(1/√2) ($$ |\left. 00 \right\rangle $$ + $$ \left| 10 \right\rangle $$)

(1/√2) ($$ \left| 10 \right\rangle $$ + $$ \left| 1 \right\rangle $$)

$$ \left| \psi_{1} \right\rangle $$ = (1/√2) ($$ \left| 10 \right\rangle $$ + $$ \left| 01 \right\rangle $$)

(1/√2) ($$ \left| 11 \right\rangle $$ + $$ \left| 01 \right\rangle $$)

(1/√2) ($$ \left| 1 \right\rangle $$ + $$ \left| 0 \right\rangle $$)

$$ \left| \psi_{2} \right\rangle $$ = (1/√2) (−$$ \left| 10 \right\rangle $$ + $$ \left| 01 \right\rangle $$)

(1/√2) (−$$ \left| 11 \right\rangle $$ + $$ \left| 01 \right\rangle $$

(1/√2) (−$$ \left| 1 \right\rangle $$ + $$ \left| 0 \right\rangle $$)

$$ \left| \psi_{3} \right\rangle $$ = (1/√2) ($$ |\left. 00 \right\rangle $$ − $$ \left| 11 \right\rangle $$)

(1/√2) ($$ \left| 00 \right\rangle $$ − $$ \left| 10 \right\rangle $$)

(1/√2) ($$ \left| 0 \right\rangle $$ − $$ \left| 1 \right\rangle $$)

Bob then measures the second qubit . If the measurement returns $$ \left| 0 \right\rangle $$, the encoded value was either 0 or 3; otherwise, the value was either 1 or 2. Bob now applies H to the first qubit :

Value

First qubit

After H applied to first qubit )

Second qubit

0

(1/√2) ($$ \left| 0 \right\rangle $$ + |$$ \left| 1 \right\rangle $$)

(1/√2) ((1/√2) ($$ \left| 0 \right\rangle $$ + $$ \left| 1 \right\rangle $$) + (1/√2) ($$ \left| 0 \right\rangle $$ − $$ \left| 1 \right\rangle $$)) = $$ \left| 1 \right\rangle $$

1

(1/√2) ($$ \left| 1 \right\rangle $$ + $$ \left| 0 \right\rangle $$)

(1/√2) ((1/√2) ($$ \left| 0 \right\rangle $$ − $$ \left| 1 \right\rangle $$) + (1/√2) ($$ \left| 0 \right\rangle $$ + $$ \left| 1 \right\rangle $$)) = $$ \left| 0 \right\rangle $$

2

(1/√2) (−$$ \left| 1 \right\rangle $$ + $$ \left| 0 \right\rangle $$)

(1/√2) (−(1/√2) ($$ \left| 0 \right\rangle $$ − $$ \left| 1 \right\rangle $$) + (1/√2) ($$ \left| 0 \right\rangle $$ + $$ \left| 1 \right\rangle $$)) = $$ \left| 1 \right\rangle $$

3

(1/√2) ($$ \left| 1 \right\rangle $$ − $$ \left| 1 \right\rangle $$)

(1/√2) ((1/√2) ($$ \left| 0 \right\rangle $$ + $$ \left| 1 \right\rangle $$) − (1/√2) ($$ \left| 0 \right\rangle $$ − $$ \left| 1 \right\rangle $$)) = $$ \left| 1 \right\rangle $$

and measures that bit. This allows him to distinguish between 0 and 3, and 1 and 2, as shown in the table. Based on his measurements of the two qubits , Bob can read the corresponding value in column 1 of the table.

In principle, dense coding can permit secure communication: The qubit sent by Alice will only yield the two classical information bits to someone in possession of the entangled partner qubit. But more importantly, it shows why quantum entanglement is an information resource. It reveals a relationship between classical information, qubits , and the information content of quantum entanglement.

10.11.2 Teleportation

Teleportation was described in Chap. 1, Sect. 1.5 to acquaint you with the unusual aspects of quantum mechanics. Recall that teleportation is the ability to transmit the quantum state of a given particle using classical bits and reconstruct the exact quantum state at the receiver. The no-cloning principle is not violated since the process requires the quantum state of the given particle to be necessarily destroyed. We had also noted that information transfer at superluminal speed was also not possible because communication between Alice and Bob required classical means. (Of course, superluminal group velocities have been observed in barrier tunneling in condensed matter and this is not forbidden by special theory of relativity.⁴⁴ It is the front velocity of a wave packet that cannot exceed the speed of light. Although neither the speed of light nor theory of relativity is in the picture in non-relativistic quantum mechanics, the matter comes up in exploring its limits. It turns out that “it is impossible, by means of local quantum operations, to transmit any information whatsoever from one observer to the other, without transmitting real material objects between them.”⁴⁵) Note also that if we impose a specific quantum state on one member of an entangled pair of particles, then we would be instantly imposing a predetermined quantum state on the other member of the entangled pair. It also shows that one shared EPR pair together with two classical bits of communication is a resource at least the equal of one qubit of communication. In Chap. 11 we will see that teleportation is intimately connected with the properties of quantum error-correcting codes.

An interesting application of teleporting qubits is to shunt quantum information around inside a quantum computer or among quantum computers. Quantum information can be transferred with perfect fidelity, but in the process the original must be destroyed. This is especially useful if some qubit needs to be kept secret. Teleportation lets a qubit move around without ever being transmitted over an insecure channel and so securely that only one copy of it is available at any time. Furthermore, any eavesdropper would have to steal both the entangled particle and the classical particle in order to have any chance of capturing the information. However, some technological hurdles exist before teleportation can be used in secure communication. The most important is the need to maintain entangled states long enough to allow classical messages to go through. A large-scale deployment of quantum teleportation would require stockpiles of entangled particles to be kept indefinitely, so that they are available on demand. Recent research indicates that this may be possible soon.⁴⁶

10.12 Concluding Remarks

Alan Turing⁴⁷ showed that given an algorithm its execution is mechanizable and Rolf Landauer⁴⁸ showed that computing requires a physical system (information is physical ). Developing algorithms requires both intelligence and knowledge of the laws of Nature. Our present knowledge shows there is a world of difference in the way we understand Nature in terms of classical physics and quantum physics. These differences lead to fundamental differences in the way we can design and efficiently execute algorithms on physical computers. Interest in quantum computers emerged when it was shown that reversible Turing machines were possible and therefore Turing machines could be mimicked using unitary operators in the quantum world. Today, physical quantum computers exist, and the technologies needed to improve and scale them are advancing rapidly.

The algorithms described in this book will, of course, come efficiently coded into reusable libraries for quantum computers, but there are many other novel and non-obvious algorithms waiting in the wings to emerge. It is the excitement of developing and discovering them that induces people to delve into quantum computing in the hope that phenomenally powerful algorithms beyond the reach of classical computing machines will emerge if quantum superposition, entanglement, teleportation , and state vector collapse are intelligently used in the Hilbert space. We now know that quantum computers are not just faster versions of ordinary computers, but something much stranger.⁴⁹

The central theme in designing quantum algorithms is to manipulate the computer in such a way that the probability of obtaining the right answer is continually reinforced while the chances of getting a wrong answer are suppressed.

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H. Johnston, Shor’s algorithm is implemented using five trapped ions. PhysicsWorld.com, 04 March 2016 (2016). http://​physicsworld.​com/​cws/​article/​news/​2016/​mar/​04/​shors-algorithm-is-implemented-using-five-trapped-ions#comments

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R. Landauer, Irreversibility and heat generation in the computing process. IBM J. Res. Dev. 5(3), 183–191 (1961). (Reprinted in IBM Journal of Research and Development, Vol. 44, No. 1/2 January/March 2000, pp. 261–269.). http://​www.​research.​ibm.​com/​journal/​rd/​441/​landauerii.​pdf

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E. Lucero et al., Computing prime factors with a Josephson phase qubit quantum processor. Nat. Phys. 8, 719–723 (2012)

33.

T. Monz et al., Realization of a scalable Shor algorithm. Science 351(6277), pp. 1068–1070 (2016). 04 March 2016

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A. Peres, How the no-cloning theorem got its name (2002). arXiv:​quant-ph/​0205076v1, 14 May 2002. http://​arxiv.​org/​PS_​cache/​quant-ph/​pdf/​0205/​0205076v1.​pdf. (As Asher reports, the title of the paper was contributed by John Wheeler.)

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E. Rieffel, W. Polak, An introduction to quantum computing for non-physicists. ACM Comput. Surv. 32(3), 300–335 (2000). http://​math.​vassar.​edu/​Classes/​280/​papers/​rieffelpolak.​pdf; http://​xxx.​lanl.​gov/​abs/​quant-ph/​9809016

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P.W. Shor, Algorithms for quantum computation: discrete log and factoring, in Proceedings of the 35th Annual Symposium on Foundations of Computer Science, pp. 124–134, Nov 1994. ftp://​netlib.​att.​com/​netlib/​att/​math/​shorquantum.​algorithms.​ps.​Z

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P.W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Sci. Stat. Comput. 26(5), 1484–1509 (1997). Preprint at http://​www.​arxiv.​org/​PS_​cache/​quant-ph/​pdf/​9508/​9508027.​pdf

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J.A. Smolin, G. Smith, A. Vargo, Pretending to factor large numbers on a quantum computer, arXiv:​1301.​7007v1 [quant-ph] (2013). http://​arxiv.​org/​abs/​1301.​7007

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A. Turing, On computable numbers, with an application to the Entscheidungsproblem. Proc. Lond. Math. Soc. Ser. 2 42, 230–265 (1936). http://​www.​turingarchive.​org/​viewer/​?​id=​466&​title=​01bb (Errata (1937): 43, pp. 544–546. http://​www.​abelard.​org/​turpap2/​tp2-ie.​asp

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L.M.K. Vandersypen, M. Steffen, M.H. Sherwood, C.S. Yannoni, G. Breyta, I.L. Chuang, Implementation of a three-quantum-bit search algorithm. Appl. Phys. Lett. 76(5), 646–648 (2000). 31 Jan, 2000. Available at https://​arxiv.​org/​pdf/​quant-ph/​9910075.​pdf

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L.M.K. Vandersypen, M. Steffen, G. Breyta, C.S. Yannoni, M.H. Sherwood, I.L. Chuang, Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance. Nature 414, 883–887 (2001). 20–27 Dec 2001. http://​www.​fisica.​uniud.​it/​~giannozz/​Corsi/​FisMod/​Testi/​nature.​pdf

46.

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Footnotes

1

Cleve et al. [12].

2

The reader may refer to Nielsen and Chuang [34], pp. 238–240 to learn more.

3

The reader may refer to Nielsen and Chuang [34], pp. 240–242 to learn more.

4

A number is an abstract idea. The symbolic representation of a number (such as 1, 2, …, or I, II, …, or i, ii, …, etc.) is called a numeral. However, in common usage, the word number is used for both the idea and the symbol. Our notion of a number has enlarged over many centuries to include such numbers as negative numbers, rational and irrational numbers, and complex numbers. The notion of a negative number took several centuries before it found acceptance by mathematicians.

The notion of complex numbers, which require taking the square root of negative numbers, took place in the sixteenth century. It was originally used as a mathematical artifice to enable square roots to be taken with impunity. Its remarkable properties were later discovered by a bunch of talented mathematicians. It is difficult to imagine the creation of quantum mechanics without complex numbers. The secrets of nature lie hidden in the numbers.

5

Gauss [17].

6

Coinciding exactly when the remainders are compared.

7

See, e.g., Weisstein, Eric W., “Congruence.” From MathWorld—A Wolfram Web Resource. http://​mathworld.​wolfram.​com/​Congruence.​html.

8

From the Latin integer, which means with untouched integrity, whole, entire. The symbol Z comes from the German word Zahlen, which means numbers.

9

Deutsch [14].

10

Bernstein and Vazirani [7].

11

Tarantola [41]. See also: http://​www.​dwavesys.​com/​, Website of D-Wave.

12

The D-Wave 2000Q™ System: Technology Overview. https://​www.​dwavesys.​com/​sites/​default/​files/​D-Wave%20​2000Q%20​Tech%20​Collateral_​0117F.​pdf.

13

Kelly [29].

14

Intel [25].

15

IBM [24].

16

See, e.g., Quantum Computing Market Forecast 2020–2025. Market Research Media, 16 May 2019. https://​www.​marketresearchme​dia.​com/​?​p=​850; and ScienceDaily.com at https://​www.​sciencedaily.​com/​ for new developments.

17

The quantum version was worked out by Coppersmith [13] and Deutsch (1994) [unpublished] independently. See also: Ekert and Jozsa [15] and Barenko [3].

18

To keep the discussions simple, we shall not consider the case when N ≠ 2n. However, we do remark that the larger the power of 2 used as a base for the transform, the better is the approximation. See Rieffel and Polak [36], p. 318.

19

These are a Hadamard gate and n − 1 conditional rotations on the first qubit , followed by a Hadamard gate and n − 2 conditional rotations on the second qubit , and so on. In all there will be $$ n + \left( {n - 1} \right) + \cdots + 1 = n\left( {n + 1} \right)/2 $$ gates. In addition, there are at most n/2 swaps to be done where each swap can be accomplished using three controlled-not gates.

20

This, of course, does not mean that QFT can be used in such applications as speech recognition or other signal processing applications. This is because the amplitudes in a quantum computer cannot be directly accessed by measurement. Thus, there is no way of determining the Fourier transformed amplitudes of the original state. Worse still, there is, in general, no way to efficiently prepare the original state to be Fourier transformed (see Nielsen and Chuang [34], p. 220).

21

Cleve et al. [12], p. 6.

22

Two integers are relatively prime if they do not share common positive factors (divisors) except 1.

23

For applications in quantum computing, it is convenient to allow $$ a_{0} = 0 $$ as well.

24

For a good description of continued fractions see http://​www.​mcs.​surrey.​ac.​uk/​Personal/​R.​Knott/​Fibonacci/​cfINTRO.​html#euclidsAlg.

25

Shor [37]. An expanded version of the paper is available in Shor [38].

26

The RSA system was invented by Ronald Rivest, Adi Shamir, and Leonard Adleman in 1978. It rules e-commerce and pops up in countless security applications. They received the Turing Award (2002) for their contributions to public key cryptography .

27

Coprime: Two integers a and b are coprime (or relatively prime) if the greatest common divisor of a and b is 1. For example, 55555 and 7811 are coprime even though neither number is itself a prime. One may use Euclid’s algorithm to determine if x and N are coprime.

28

Vandersypen et al. [45]. They designed and made a molecule that has seven nuclear spins—the nuclei of five fluorine and two carbon atoms—which can interact with each other as qubits , be programmed by radio frequency pulses and be detected by nuclear magnetic resonance (NMR) instruments. They controlled a vial of a billion-billion (10¹⁸) of these molecules as they executed Shor’s algorithm and correctly identified 3 and 5 as the factors of 15.

29

Lucero et al. [32].

30

Monz et al. [33]. See also: Johnston [27].

31

Smolin et al. [39, 40].

32

See, e.g., Johansson and Larsson [26].

33

See, e.g., Geller and Zhou [18].

34

See, e.g., Nielsen and Chuang [34], pp. 221–226, Cleve et al. [12], Kitaev [30], Jozsa [28] and Vazirani [46].

35

Grover [20, 21]. A popularized version of the algorithm appears in Grover [22]. See also: Grover [23].

36

Boyer et al. [8].

37

Chuang et al. [11]. See also: Gershenfeld and Chuang [19].

38

Vandersypen et al. [44].

39

Figgatt et al. [16].

40

Bennett et al. [4].

41

Modal value: the most frequently occurring value.

42

Bennett and Wiesner [6].

43

Bennett et al. [5].

44

Torlina et al. [42]. See also: ANU [1].

45

Peres [35].

46

ASRC [2].

47

Turing [43].

48

Landauer [31].

49

Brandt et al. [9].

© Springer Nature Singapore Pte Ltd. 2020

R. K. BeraThe Amazing World of Quantum ComputingUndergraduate Lecture Notes in Physicshttps://doi.org/10.1007/978-981-15-2471-4_11

11. Quantum Error Corrections

Rajendra K. Bera1  

(1)

Acadinnet Education Services India, Bangalore, Karnataka, India

Rajendra K. Bera

Email: rajendrabera@yahoo.com

Accept corrections and you’ll improve and increase.

―Israelmore Ayivor

Abstract

This chapter briefly discusses error-correction algorithms. In an ideal quantum computer, these are not needed nor are they needed in algorithm development. They are included to make the reader aware of the nature of the problems quantum computer designers face and how some of them can be solved algorithmically.

11.1 Introduction

In designing quantum algorithms, we assumed an ideal, non-interfering, execution environment. This is far from reality. Superposition and entanglement are very fragile quantum states because of the difficulty of insulating quantum computers from a variety of causes—decoherence, cosmic radiation, and spontaneous emission. Maintaining the state of a qubit for prolonged periods is difficult enough leave alone preserving the states of entangled qubits. Inevitably, the computer and the environment couple to vitiate the computer’s quantum state. Attempts to eliminate or minimize this problem by hardware and software means is an ongoing research area. Here, we restrict ourselves to software means, i.e., build error-correction algorithms by creating information redundancies in an enlarged Hilbert space for error detection and correction.

A classical bit is in either state 0 or 1. Its physical state is defined by a large number of electrons, so a few electrons going astray will not affect its state identity.¹ A physical qubit is often represented by a single or a few quantum particles (e.g., electrons), hence inadvertent errors are serious. Classical digital computers have inbuilt hardware bit-parity checks and self-correcting steps to restore a bit’s state if it inadvertently flips. A qubit, on the other hand, has a continuum of quantum states and it is not obvious that similar corrective measures are generally possible. For example, a likely source of error in setting a qubit’s state is over-rotation by an imperfect unitary gate (unitary gates rotate state vectors without changing the vector’s length). Thus, a state $$ \left( {\alpha \left| 0 \right\rangle + \beta \left| 1 \right\rangle } \right) $$ instead of becoming $$ \left( {\alpha \left| 0 \right\rangle + \beta \,{\text{e}}^{i\varphi } \left| 1 \right\rangle } \right) $$ may become $$ \left( {\alpha \left| 0 \right\rangle + \beta \,{\text{e}}^{i(\varphi + \delta )} \left| 1 \right\rangle } \right) $$. While δ may be very small, it will still be wrong, and if left uncorrected, will amplify to larger errors. Thus, not only must bit flips be corrected but also its phase must be correct, else, quantum parallelism, which depends on coherent interference between the superpositions, will be lost. Another source of error—inadvertent “measurement”—occurs due to unexpected computer-environment interactions where the environment interferes with the Hilbert space reserved for computation in a manner that collapses the evolving state vector. This is generally seen as a hardware design problem.

11.2 Protecting the Computational Hilbert Space

At one time, it appeared the no-cloning theorem and the general impossibility of measuring the exact state of an unknown qubit would make error correction by algorithmic means impossible. In 1995, Peter Shor showed that for a certain class of errors (that also appear in classical computers) it is possible to restore a quantum state using only partial state information.² Since then, several error-correction codes have been devised. They have some similarities with, and some striking differences from classical error-correcting codes. Indeed, classical ideas have seeded the emergence of quantum error-correction codes. A salient feature of these codes is that the original state is recovered by a skillful measurement followed by a unitary operation determined by the measured outcome.

At the algorithmic (logical) level, a qubit is a two-dimensional subspace of a higher dimensional system. At a physical level, qubits are chosen such that it is possible to detect and correct unwanted changes in their state. Programmed manipulation of the information encoded in the qubits generally requires arbitrary and precise control over the entire computing system. Since individual elements in physical devices will always have residual interactions, these must be accounted for when designing logical operations.³ Our intention here is to provide a glimpse of the central ideas that led to the development of error-correction algorithms which eventually strengthened our belief that practical quantum computers can be built without waiting for certain hardware limitations to be resolved by advancing hardware technology.⁴

Since straightforward replication of qubits in an unknown state $$ \left| \psi \right\rangle $$ is impossible due to the no-cloning theorem, i.e., we cannot perform such operations as $$ \left| \psi \right\rangle \to \left| \psi \right\rangle \otimes \left| \psi \right\rangle \otimes \left| \psi \right\rangle $$,⁵ entanglement is used to spread the information held by designated qubits over multiple qubits through an encoding. This encoding is such that under evolution by a suitable operator, the appropriate qubits will recover their original state. In short, we fight entanglement with entanglement. Entanglement has no classical analog. It encodes information in the correlations among the qubits in a given Hilbert space, and this space explodes exponentially with the number of qubits.

Quantum information science explores, not the frontier of short distances as in particle physics, or of long distances as in cosmology, but rather the frontier of highly complex quantum states, the entanglement frontier.⁶

The simplest error conditions (e.g., bit flip, phase flip and their combination) assume that each qubit interacts with the environment independent of any other. This lets interaction operators to be tensor products of 1-qubit interaction operators. Immediately, Pauli matrices (or operators) become relevant because they span the space of 2 × 2 matrices, and the n-qubit Pauli group Pn (i.e., the group generated by tensor products of the four Pauli operators) spans the space of 2n × 2n matrices.

The running of a quantum algorithm must be restricted to a carefully chosen sub-space of some larger Hilbert space. The trick is to detect and undo unintended anomalies as they develop in the chosen sub-space without upsetting intended computations. Fortunately, some useful correction algorithms are now known. Further, a theory of quantum error correction has also developed, which allows quantum computers to compute effectively in the presence of noise and allows communications over noisy quantum channels to take place reliably.

The interaction of the computational Hilbert space with the environment (e.g., ambient heat bath, cosmic rays, stray gas molecules, etc.) can be broadly placed under the headings dissipation and decoherence .⁷

11.2.1 Dissipation

Dissipation is a process by which a qubit loses energy to its environment. For example, if an excited state is used to represent a $$ \left| 1 \right\rangle $$ and a lower energy state as $$ \left| 0 \right\rangle $$, a qubit might spontaneously transition from $$ \left| 1 \right\rangle $$ to $$ \left| 0 \right\rangle $$ emitting a photon in the process. To see that dissipation is non-unitary, consider the following description of the bit-flip process:

Here, the matrix D is

which is clearly not a unitary matrix since

11.2.2 Decoherence

Decoherence is more insidious. It is a coupling between two initially isolated quantum systems (say, the qubits and the environment) that tends to randomize the relative phases of the possible states of memory registers. This destroys the planned interference effects of a computational algorithm and entangles the state of the quantum computer with the environment. Decoherence usually occurs on a faster timescale than dissipation ; it lies at the heart of the quantum-to-classical transition. “Decoherence is a pure quantum effect, to be distinguished from classical dissipation and stochastic fluctuations (noise).”⁸

In decoherence , information encoded in a quantum state “leaks out” to the environment. If the effects of the environment are not explicitly modeled, it would appear as if the logical qubits are no longer evolving in accordance with the Schrödinger equation . In fact, it is this coupling between a quantum system and its environment, and the resulting loss of coherence that prevents quantum effects from being evident at the macroscopic level. When that happens, that state will look locally like a classical state. Therefore, as far as a local observer is concerned, there is no difference between a classical bit and a qubit that has become hopelessly entangled with the rest of the universe.⁹ For example, suppose we have a qubit in the state and further suppose that this qubit gets entangled with a second qubit so that the joint state of the two qubits is . If we now ignore the second qubit, the first qubit will be in the maximally mixed state, i.e., no matter what measurement you make on it, you will get a random output. You will never see interference between the $$ \left| {00} \right\rangle $$ and $$ \left| {11} \right\rangle $$ branches of the wave function, because for interference to occur the two branches must be identical in all aspects and this cannot happen by changing the first qubit alone to make $$ \left| {00} \right\rangle $$ identical to $$ \left| {11} \right\rangle $$. To see an interference pattern, one must perform a joint measurement on the two qubits together.

Decoherence is one of the most pervasive processes in the universe. Indeed, it is precisely because decoherence is so powerful that the quantum fault-tolerance theorem came as a shock to physicists.¹⁰ Decoherence is one more manifestation of the second law of thermodynamics. (Quantum states are very easy to destroy and very hard to put back together.) Decoherence times depend on the physical properties of the qubit , the size of the memory register, the temperature of the environment, the rate of collisions with ambient gas molecules, etc. A very rough estimate of decoherence time can be made based on Heisenberg’s uncertainty principle in energy and time:

where h is the Planck constant (h = 6.626070040(81) × 10⁻³⁴ j s), k is the Boltzmann constant (k = 1.38064852(79) × 10⁻²³ j/°K), and T is the absolute temperature of the environment.¹¹ At room temperature (≈300 °K), this gives a typical decoherence time of about 10⁻¹⁴ s. At lower temperatures, decoherence times are longer. For example, at liquid helium temperatures, it is about 100 times longer than at room temperature. The obvious ways of increasing decoherence times is to chill the computer and seal it in as best a vacuum as we can, apart from choosing qubit materials which provide better decoherence times. It is imperative that any quantum computation be completed before decoherence starts and destroys valid superposition of states. Current decoherence times are typically a few microseconds.¹² For an excellent tutorial on decoherence , see Marquardt and Puttmann (2008) and a recent paper by Schlosshauer [17].¹³

11.2.3 Algorithmic Error Correction Is Possible

Presently available computing times before decoherence are very small and hence permit only a few computational steps. However, computing times can improve if suitable error-correction algorithms are found. In 1996, A. M. Steane¹⁴ and independently A. R. Calderbank and P. W. Shor¹⁵ found that some ideas used in the construction of classical linear codes can be used to correct errors in quantum computing by the clever use of quantum entanglement. The class of quantum error-correction codes they devised are known as the Calderbank–Shor–Steane (CSS) codes. The codes are limited to correcting a group of errors that are unitary—spin and phase flips—which can be described by Pauli matrices; they are called depolarization errors. Such errors are large and discrete and hence the most tractable of all quantum errors.

11.3 Calderbank–Shor–Steane Error Correction

The idea behind the CSS codes is relatively obvious, namely that quantum states can be encoded. In classical information theory, coding just refers to the use of a string of bits to stand in for the value of one bit (or perhaps a smaller block of bits). Embedding redundancy into the encoding allows at least some errors to be caught and repaired. This form of encoding is standard practice in digital communications. However, it was not at all obvious how redundancy could be used in quantum computation. The no-cloning theorem seemed to say that even the simplest kind of redundancy was not possible even in principle. Amidst skepticism, Shor¹⁶ and Steane¹⁷ independently discovered an ingenious way to use entanglement, in the service of redundancy and error correction. In fact, they used entanglement to fight entanglement!

11.3.1 Encoding-Decoding

Quantum error-correction codes work by encoding quantum states in a special way and then decoding when it is required to recover the original state without error. Clearly, this assumption is vulnerable if the quantum gates used in the process are themselves noisy. Fortunately, the quantum fault-tolerance theorem mentioned earlier comes to our rescue. In fact, error correction can be implemented fault-tolerantly, i.e., in such a way that it is insensitive to errors that occur during the error detection operations themselves.

Finally, there is the threshold theorem¹⁸ that says that provided the noise in individual quantum gates is below a certain constant threshold it is possible to efficiently perform an arbitrarily large quantum computation. There are caveats. Nevertheless, it is a remarkable theorem indicating that noise likely poses no fundamental barrier to the performance of large-scale quantum computations. Error-correction codes related to this theorem are called concatenated quantum codes. In these codes, each qubit is itself further encoded in a hierarchical tree of entangled qubits . In this way, concatenated codes allow correctable quantum computations of unlimited duration!

The key idea is that if we wish to protect a message against the effects of noise, then we should encode the message by adding some redundant information to the message. That way, even if some bits of the message get corrupted, there will be enough redundancy in the encoded message to recover the message completely by decoding. This redundancy is essential. The amount of redundancy required depends on the severity of noise.

11.3.2 Steps of Error Correction

The steps of error correction are encoding, error detection, and recovery.

Encoding: Quantum states are encoded by unitary operations into a quantum error-correcting code, formally defined as a subspace C of some larger Hilbert space. This code may subsequently be affected by noise.

Error detection: A syndrome¹⁹ measurement is made to diagnose the type of error which occurred. In effect, the measured value tells us what procedure to use to recover the original state of the code.

Recovery: Post-syndrome measurement, a recovery operation returns the quantum system to the original state of the code.

We assume that all errors are the result of quantum interactions between a set of qubits and the environment. In addition, the possible errors for each single qubit considered are linear combinations of the following: no errors (I), bit flip errors (X), phase errors (Z), and bit flip phase errors (Y). Note that these are describable by Pauli matrices. A general form of a single-bit error is thus

An error correcting code for a set of errors $$ E_{i} $$ consists of a mapping C that embeds n data bits in n + k code bits (without making any error) together with a syndrome extraction operator $$ S_{C} $$ that maps n + k code bits to the set of indices of correctable errors $$ E_{i} $$ such that

The k bits of C(x) provide the desired redundancy in the n bit message.

In the encoding stage, given an error correcting code C with syndrome extraction operator $$ S_{C} $$, an n-bit quantum state $$ \left| \psi \right\rangle $$ is encoded in a n + k-bit quantum state $$ \left| \phi \right\rangle = C\left| \psi \right\rangle $$. Now assume that $$ \left| \phi \right\rangle $$ has been corrupted to $$ \sum\nolimits_{i} {e_{i} E_{i } \left| \phi \right\rangle .} $$ In the error detection stage, apply $$ S_{C} $$ to $$ \sum\nolimits_{i} {e_{i} E_{i } \left| \phi \right\rangle } $$ padded with enough $$ \left| 0 \right\rangle $$ bits,

Quantum parallelism gives a superposition of different errors each associated with their respective error index i. Next, measure the $$ \left| i \right\rangle $$ component of the result. This will yield a (random) value i₀ and project the state to $$ E_{{i_{0} }} \left| {\phi ,i_{0} } \right\rangle . $$ Finally, in the recovery stage, apply the inverse error map $$ E_{{i_{0} }}^{ - 1} $$ to the first n + k qubits of $$ E_{{i_{0} }} \left| {\phi ,i_{0} } \right\rangle $$ to get the corrected state $$ \left| \phi \right\rangle $$.

Example

Consider the simple error correcting code C that maps $$ \left| 0 \right\rangle $$ to $$ \left| {000} \right\rangle $$ and $$ \left| 1 \right\rangle $$ to $$ \left| {111} \right\rangle $$. C can correct single bit flip errors

The syndrome extraction operator is

with the corresponding error-correction operators shown in the table below. In this example $$ E_{i} = E_{i}^{ - 1} . $$ Note that operations like x0 XOR x2 are parity checks.

Bit flipped

Syndrome

Error correction

None

None

0

X ⊗ I ⊗ I

1

I ⊗ X ⊗ I

2

I ⊗ I ⊗ X

Consider the quantum bit $$ \left| \psi \right\rangle = \frac{1}{\sqrt 2 }\left( {\left| 0 \right\rangle - \left| 1 \right\rangle } \right) $$, which is encoded as

and the error

The resulting error state is

Next, apply the syndrome extraction²⁰ to $$ \left( {E\left| \phi \right\rangle \otimes \left| {000} \right\rangle } \right), $$

Measuring the last three bits will yield either $$ \left| {110} \right\rangle $$ or $$ \left| {101} \right\rangle $$. Assume that $$ \left| {110} \right\rangle $$ is measured, then the state becomes

Note that almost magically a part of the error has disappeared. The remaining part of the error can be removed by applying the inverse error operator X ⊗ I ⊗ I, corresponding to the measured value $$ \left| {110} \right\rangle $$, to the first three bits, to produce

What we have demonstrated just now is that it is possible to use several entangled qubits to represent one logical qubit. Such entanglement spreads out the state of the qubit in a way that errors in any “part” of the entangled qubit can be detected, diagnosed, and corrected. Thus, while entangling qubits with the environment may introduce errors, entangling qubits with themselves might immunize them from such errors. A remarkable aspect of the CSS codes is that the process of error correction has an essential digital character to it, even though a qubit can be in a continuum of possible states. Error detection involves the performance of a series of binary-valued quantum measurements. Then these bit values provide an instruction for an error detection step, which involves a discrete rotation of a specific state. This digital character derives from the fact that any error which the environment can cause on a single qubit acts in a subspace orthogonal to the state space of the coded qubit itself. This leaves the complex coefficients, to a very high accuracy, untouched by the error process (error containment) and allows the error detection and correction steps to work in a way which is oblivious to their values.

The need for error correction will, of course, diminish as the technology required to build reliable quantum computers improve. At one time it appeared that building a 1000-qubit computer may be out of reach. With recent advances in fabrication techniques, and improved control and measurement techniques, it is no longer true.

11.4 Decoherence-Free Subspace

There is an interesting situation where it is possible to provide passive protection against errors—as opposed to the active error protection of the quantum error-correction codes discussed above. The model assumes that all qubits in the register are affected by the same error at the same time. This is very different from an independent error model. Does such collective dephasing occur? It does in situations where the physical dimensions of the register are smaller than the shortest wavelength of the field.

Consider the following encoding:

The states $$ \left| {0_{L} } \right\rangle $$ and $$ \left| {1_{L} } \right\rangle $$ form an orthonormal basis. Note also that $$ \left| 0_{L} \right\rangle^{*} = \left| {1_{L} } \right\rangle $$.

Decoherence can be modeled by an operation called collective dephasing. The operation transforms $$ \left| 1 \right\rangle $$ into $$ {\text{e}}^{i\theta } \left| 1 \right\rangle $$ for both physical qubits at the same time and leaves $$ \left| 0 \right\rangle $$ unchanged for both of them. This results in

and

If all operations are carried out on qubits encoded in the same way, collective dephasing only produces a global phase change and hence does not affect measurements. Recall that in quantum mechanics only phase differences between qubits matter.

In February 2001, Kielpinski, Meyer, Rowe, Sackett, Itano, Monroe, and Wineland²¹ reported an experiment in which they encoded a qubit into a decoherence free subspace of a pair of trapped ⁹Be⁺ ions. They used encoding exactly like the one shown above. Then they measured the storage time under ambient conditions and under interaction with an engineered noisy environment and observed that the encoding increased the storage time by up to an order of magnitude.

11.5 Concluding Remarks

Error correction is about ensuring that information is not vitiated in the presence of noise. Quantum error correction (QEC) is now a well-established subject. The basic mathematical methods, as we have seen, are elegant and fairly easy to understand. More advanced methods (e.g., fault-tolerant methods) will need to consider the physics of noise, and the fidelity that can be achieved by QEC. Some level of QEC will always be required because of unavoidable imprecision in building the computer hardware. This has an impact because quantum algorithms use large-scale quantum interference which by its nature is fragile, i.e., highly sensitive to any imprecision in hardware implemention of unitary operators and the fact that absolute isolation of quantum computers from the Universe is impossible. Before QEC algorithms were developed, the fragility of quantum computers appeared so daunting that large-scale quantum computing appeared impossible to achieve. QEC algorithms came as a blessing because large-scale quantum computation finally became possible. Those interested in delving into QEC further would greatly benefit by reading Gottesman [7], Steane [22], and Knight [12] as a starting point.

References

1.

S. Aaronson, PHYS771 Lecture 11: Decoherence and Hidden Variables (University of Waterloo, 2006). http://​www.​scottaaronson.​com/​democritus/​lec11.​html

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D. Aharonov, M. Ben-Or, Fault-tolerant quantum computation with constant error, in Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, El Paso, Texas, USA, 4–6 May 1997, pp. 176–188

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D. Aharonov, M. Ben-Or, Fault-tolerant quantum computation with constant error. arXiv:​quant-ph/​9906129v1, 30 Jun 1999. https://​arxiv.​org/​pdf/​quant-ph/​9906129.​pdf

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P. Aliferis, D. Gottesman, J. Preskill, Quantum accuracy threshold for concatenated distance-3 codes. Quantum Inf. Comput. 6, 97–165 (2006). Also as arXiv:​quant-ph/​0504218v3, 21 Oct 2005. https://​arxiv.​org/​pdf/​quant-ph/​0504218.​pdf

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P. Ball, The era of quantum computing is here. Outlook: cloudy. Quanta Mag. 24 Jan 2018. https://​d2r55xnwy6nx47.​cloudfront.​net/​uploads/​2018/​01/​the-era-of-quantum-computing-is-here-outlook-cloudy-20180124.​pdf

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A.R. Calderbank, P.W. Shor, Good quantum error-correcting codes exist. Phys. Rev. A, 54(2), 1098–1105 (1996). http://​www-math.​mit.​edu/​~shor/​papers/​good-codes.​pdf Also as arXiv:​quant-ph/​9512032. https://​arxiv.​org/​pdf/​quant-ph/​9512032.​pdf

7.

D. Gottesman Class of quantum error-correcting codes saturating the quantum Hamming bound. Phys. Rev. A, 54, 1862–1868 (1996). A version available at https://​arxiv.​org/​pdf/​quant-ph/​9604038.​pdf

8.

D. Gottesman, Stabilizer codes and quantum error correction. Ph.D. thesis, Caltech (1997). Also as arXiv:​quant-ph/​9705052v1, 28 May 1997. https://​arxiv.​org/​pdf/​quant-ph/​9705052.​pdf

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D. Gottesman, Fault-tolerant quantum computation with constant overhead. arXiv:​1310.​2984v3 [quant-ph], 22 July 2014. https://​arxiv.​org/​pdf/​1310.​2984.​pdf

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R.W. Heeres, P. Reinhold, N. Ofek, L. Frunzio, L. Jiang, M.H. Devoret, R.J. Schoelkopf, Implementing a universal gate set on a logical qubit encoded in an oscillator. Nat. Commun. 8, Published online 21 July 2017. https://​www.​nature.​com/​articles/​s41467-017-00045-1

11.

D. Kielpinski, A. Ben-Kish, J. Britton, V. Meyer, M.A. Rowe, C.A. Sackett, W.M. Itano, C. Monroe, D.J. Wineland, Recent results in trapped-ion quantum computing at NIST. arXiv:​quant-ph/​0102086v1, 16 Feb 2001. https://​arxiv.​org/​pdf/​quant-ph/​0102086v1

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W. Knight, Serious quantum computers are finally here. What are we going to do with them? MIT Technol. Rev. 21 Feb 2018. https://​www.​technologyreview​.​com/​s/​610250/​serious-quantum-computers-are-finally-here-what-are-we-going-to-do-with-them/​

13.

E. Knill, R. Laflamme, W.H. Zurek, Resilient quantum computation: error models and thresholds. Proc. R. Soc. Lond. A, 454, 365–384 (1998). http://​rspa.​royalsocietypubl​ishing.​org/​content/​royprsa/​454/​1969/​365.​full.​pdf

14.

J. Preskill, Quantum computing and the entanglement frontier. arXiv:​1203.​5813v3, 10 Nov 2012. https://​arxiv.​org/​abs/​1203.​5813 (Abstract), https://​arxiv.​org/​pdf/​1203.​5813.​pdf (Paper)

15.

R. Raussendorf, J. Harrington, Fault-tolerant quantum computation with high threshold in two dimensions. arXiv:​quant-ph/​0610082v2, 14 May 2007. https://​arxiv.​org/​pdf/​quant-ph/​0610082.​pdf

16.

R. Raussendorf, Key ideas in quantum error correction. Philos. Trans. R. Soc. Lond. Ser. A, 370, 4541–4565 (2012). http://​rsta.​royalsocietypubl​ishing.​org/​content/​roypta/​370/​1975/​4541.​full.​pdf

17.

M. Schlosshauer, The quantum-to-classical transition and decoherence. arXiv:​1404.​2635v1 [quant-ph] 9 Apr 2014. https://​arxiv.​org/​pdf/​1404.​2635.​pdf

18.

P.W. Shor, Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52(4), R2493–R2496 (1995)

19.

A.M. Steane, Multiple particle interference and quantum error correction. Proc. R. Soc. Lond. Ser. A, 452, 2551–2577 (1996a). Also at arXiv (1995). http://​arxiv.​org/​pdf/​quant-ph/​9601029.​pdf

20.

A.M. Steane, Error correcting codes in quantum theory. Phys. Rev. Lett. 77, 793 (1996b). https://​users.​physics.​ox.​ac.​uk/​~Steane/​pubs/​Steane_​PRL95.​pdf

21.

A.M. Steane, Introduction to quantum error correction. Philos. Trans. R. Soc. Lond. Ser. A, 356, 1739–1758 (1998)

22.

A.M. Steane, A tutorial on quantum error correction, in Proceedings of the International School of Physics “Enrico Fermi”, Course CLXII, “Quantum Computers, Algorithms and Chaos”, ed. by G. Casati, D.L. Shepelyansky, P. Zoller (IOS Press, Amsterdam, 2006), pp. 1–32. https://​www2.​physics.​ox.​ac.​uk/​sites/​default/​files/​ErrorCorrectionS​teane06.​pdf

23.

C.P. Williams, S.H. Clearwater, Explorations in Quantum Computing (Springer, New York, 1998)

Footnotes

1

One may view this as a form of repetition code, where each bit is encoded in many electrons (the repetition), and after each machine cycle, it is returned to the value held by the majority of the electrons (the error correction).

2

Shor [18].

3

For some recent advances, see Heeres et al. [10].

4

See, e.g., Preskill [14].

5

In classical computing, 0 → 000 and 1 → 111 is no big deal.

6

Preskill [14].

7

Williams and Clearwater [23], p. 214.

8

Schlosshauer [17].

9

See, e.g., Aaronson [1].

10

The fault-tolerance theorem roughly says that, if the rate of decoherence per qubit per gate operation is below a constant threshold, then it is possible, in principle, to correct errors faster than they occur and thereby perform an arbitrarily long quantum computation. See, e.g., Aharonov and Ben-Or [2, 3], Aliferis et al. [4], Gottesman [8], Knill et al. [13], and Raussendorf and Harrington [15].

11

NIST reference on Constants, Units, and Uncertainty: see https://​physics.​nist.​gov/​cgi-bin/​cuu/​Value?​h for the Planck Constant, and https://​physics.​nist.​gov/​cgi-bin/​cuu/​Value?​k for the Boltzmann Constant. The websites provide updates.

12

Ball [5].

13

Schlosshauer [17].

14

Steane [19]. See also: Steane [20], Steane [21], Raussendorf [16].

15

Calderbank and Shor [6]. See also: Shor [18].

16

Shor [18].

17

Steane [20].

18

See, e.g., Gottesman [9].

19

Syndrome: a group or pattern of symptoms that together is indicative of a particular disease, disorder, or condition.

20

This is the operator, $$ S_{C} :\left| {x0,x1,x2,0,0,0} \right\rangle $$ → $$ \left| {x0,x1,x2,x0\,{\text{XOR}}\,x 1,x0{\text{ XOR }}x2,} \right. $$ $$ \left. {x1{\text{ XOR }}x2} \right\rangle $$.

21

Kielpinski et al. [11].

© Springer Nature Singapore Pte Ltd. 2020

R. K. BeraThe Amazing World of Quantum ComputingUndergraduate Lecture Notes in Physicshttps://doi.org/10.1007/978-981-15-2471-4_12

12. Time-Multiplexed Interpretation of Measurement

Rajendra K. Bera1  

(1)

Acadinnet Education Services India, Bangalore, Karnataka, India

Rajendra K. Bera

Email: rajendrabera@yahoo.com

At the heart of quantum mechanics is a rule that sometimes governs politicians or CEOs - as long as no one is watching, anything goes.

—Lawrence M. Krauss

Abstract

This chapter describes a new interpretation of quantum mechanics by positing that the sub-Planck scale structure of the state vector is such that its eigenstates are dynamically time-division multi-plexed. To this is added a probabilistic measurement model which determines only the instantaneous eigenstate of the system at the instant of measurement. The instant of measurement is chosen randomly by the classical measurement apparatus, once activated, within a small interval. The measured result is regarded as the joint product of the quantum system and the macroscopic classical measuring system. Measurement is complete when the wave function assumes the measured state.

12.1 Introduction

The macroscopic world is classical.

The microscopic world is quantum.

In Chap. 2, Sect. 2.​10, we presented three well-known interpretations of quantum mechanics. In this chapter, we present a new one proposed in 2009 by Bera and Menon ¹ who interpret the terms superposition, entanglement, and measurement differently. Underlying this interpretation is a hypothesized deterministic cyclic structure of the wave function for a quantum system at the sub-Planck scale. The cyclic structure comprises a sequential succession of the eigenstates that comprise a given wave function. Between non-unitary operations of measurement on the wave function, the sequential arrangement of the current eigenstates chosen for the system is immaterial, but once chosen it remains fixed until another measurement changes the wave function. The probabilistic aspect of quantum mechanics is interpreted by hypothesizing a measurement mechanism which acts instantaneously but the instant of measurement is chosen randomly by the classical measurement system over a small but finite interval from the time the measurement system is activated. At the instant the measurement is made, the wave function irrevocably collapses to a new state (erasing some of the past quantum information) and continues from thereon in that state till changed by a unitary operation or a new measurement.

Measurement is the mystery. A quantum object generally offers more encoded, revelatory options for measurement than can be seen in practice. It is this aspect that we are concerned with here. In short, we conjecture about the “underlying reality” that creates measurement probabilities.

Note that a mathematical theory is abstract. It describes any system by some indexed list of properties and their possible values and the rules by which the properties can change. It is devoid of semantics. It is therefore necessary that any interpretation is consistent with any underlying physics measurable at the quantum mechanical level. The conjectured mechanism we now describe for relating inputs with outputs at the subquantum level maintains this consistency. It thus assumes that quantum mechanics approximates a deeper theory. We emphasize that while the formalism of quantum mechanics is widely accepted, it does not have a unique interpretation due to the incompatibility between Postulates 2 and 3 (see Chap. 2, Sect. 2.​7). Indeed, without Postulate 3 telling us what we can observe, the equations of quantum mechanics would be just pure mathematics devoid of physical meaning. Any interpretation can come only after an investigation into the logical structure of the postulates of quantum mechanics is made. For example, Newtonian mechanics does not define the structure of matter. Therefore, how we model the structure of matter is largely an independent issue. However, given the success of Newtonian mechanics, any model we propose is expected to be compatible with Newton’s laws of motion in the realm where it rules. Since Newton, our understanding of the structure of matter has undergone several changes without affecting Newton’s laws of motion. A question such as whether a particular result deduced from Newton’s laws of motion is deducible from a given model of material structure is therefore not relevant.

Likewise, as long as our interpretation (or model) of superposition, entanglement, and measurement does not require the postulates of quantum mechanics to be altered, all the predictions made by quantum mechanics will remain compatible with our interpretation. This assertion is important because no comments are made on the Hamiltonian, which captures the detailed dynamics of a quantum system. Quantum mechanics does not tell us how to construct the Hamiltonian. In fact, real-life problems seeking solutions in quantum mechanics need to be addressed in detail by physical theories built within the framework of quantum mechanics. The postulates of quantum mechanics provide only the scaffolding around which detailed physical theories are to be built. This gives ample opportunities to speculate about the abstract state vector $$ \left| \psi \right\rangle $$ at the sub-Planck level without affecting the postulates of quantum mechanics . The sub-Planck scale provides the freedom to construct mechanisms for interpretations that need not be bound by the laws of quantum mechanics because those laws are not expected to rule in that scale. The high point of the new interpretation described here is that it explains the measurement postulate as the inability of a classical measuring device to measure at a precisely predefined time.

12.2 A Conjectured Sub-planck Mechanism

Following the principle of Occam’s razor that “entities should not be multiplied unnecessarily” or the law of parsimony, the interpretation adopted by Bera and Menon ² posits that the sub-Planck scale structure of the state vector is such that its eigenstates are dynamically time-division multiplexed. To this is added a probabilistic measurement model which determines only the instantaneous eigenstate of the system at the instant of measurement. The instant of measurement is chosen randomly by the classical measurement apparatus, once activated, within a small interval. The measured result is regarded as the joint product of the quantum system and the macroscopic classical measuring system. Measurement is complete when the wave function assumes the measured state.

In the dynamic time-division multiplexing, superposed states appear as time-sliced in a cyclic manner such that the time spent by an eigenstate in a cycle is related to the complex amplitudes appearing in the state vector. Entangled states binding two or more particles appear in this interpretation as the synchronization of the sub-Planck level oscillatory states of the participating qubits . Unlike the Copenhagen interpretation , in this interpretation it is not meaningless to ask about the state of the system in the absence of a measuring system. Indeed, the interpretation can be understood in terms of images and concepts familiar to us from everyday experience in the macroscopic world we live in and hence appears more intuitive to the human mind than the other interpretations in the literature.

The basic sub-Planck model that underpins the new interpretation is as follows. Consider a qubit with the state vector $$ \left| \psi \right\rangle = \alpha \left| 0 \right\rangle + \beta \left| 1 \right\rangle $$. Its hypothesized structure in the sub-Planck scale is illustrated in Fig. 12.1.

Fig. 12.1

A single particle system in superposition state

In a cycle time of $$ T_{\text{c}} $$, the qubit oscillates between the eigenstates $$ \left| 0 \right\rangle $$ and $$ \left| 1 \right\rangle $$. We assume $$ T_{\text{c}} $$ to be much smaller than Planck time (≪10⁻⁴³ s) to allow us to interpret the state vector independently of the Schrödinger equation . (The implicit assumption here is that in some averaged sense, perhaps with some additional information, these oscillations will represent the state vector $$ \left| \psi \right\rangle $$, say, analogous to the case of a volume of gas in classical mechanics, where random molecular motions, appropriately averaged, represent classical pressure, temperature, and density of a volume of gas.) It is not necessary for us to know the value of $$ T_{\text{c}} $$. We only assert that it is a universal constant. Within a cycle, the time spent by the qubit in state $$ \left| 0 \right\rangle $$ is $$ T_{0} = \left| \alpha \right|^{2} T_{\text{c}} $$ and in state $$ \left| 1 \right\rangle $$ is $$ T_{1} = \left| \beta \right|^{2} T_{\text{c}} $$ so that $$ T_{\text{c}} = T_{0} + T_{1} $$. Superposition is interpreted here as the deterministic linear sequential progression of the qubit’s states $$ \left| 0 \right\rangle $$ and $$ \left| 1 \right\rangle $$. A measurement on this qubit will return the instantaneous pre-measurement state of the qubit and collapse the qubit to the measured state.

The measurement device is modeled as follows. Let $$ \Delta t_{\text{m}} $$ be the interval during which the measuring device measures, where $$ \Delta t_{\text{m}} $$ is only assumed to be orders of magnitude greater than Planck time. During $$ \Delta t_{\text{m}} $$, at some random instant, the device measures $$ \left| \psi \right\rangle $$. This avoids temporal bias. Thus, the source of indeterminism built in Postulate 3 is posited as the classical measuring device’s inability to measure with temporal precision. The measurement concurrently induces the state vector to collapse to the measured state.

If the measurement basis differs from $$ \left\{ {\left| 0 \right\rangle ,\left| 1 \right\rangle } \right\} $$, say, it is $$ \left\{ {\left| x \right\rangle ,\left| y \right\rangle } \right\} $$ obtained by rotating the basis $$ \left\{ {\left| 0 \right\rangle ,\left| 1 \right\rangle } \right\} $$ anti-clockwise by the angle θ, then $$ \left| 0 \right\rangle = \cos \theta \left| x \right\rangle - \sin \theta \left| y \right\rangle $$ and $$ \left| 1 \right\rangle = \sin \theta \left| x \right\rangle + \cos \theta \left| y \right\rangle $$ and conversely $$ \left| x \right\rangle = \cos \theta \left| 0 \right\rangle - \sin \theta \left| 1 \right\rangle $$ and $$ \left| y \right\rangle = \sin \theta \left| 0 \right\rangle + \cos \theta \left| 1 \right\rangle $$. Figure 12.2 shows how the vectors $$ \left| 0 \right\rangle ,\left| 1 \right\rangle $$ will be observed by a measuring device in the basis $$ \left\{ {\left| x \right\rangle ,\left| y \right\rangle } \right\} $$ and the probabilities with which it will measure $$ \left| x \right\rangle $$ or $$ \left| y \right\rangle $$. Choosing a basis different from $$ \left\{ {\left| 0 \right\rangle ,\left| 1 \right\rangle } \right\} $$ means changing the values of $$ t_{0} $$ and $$ t_{1} $$ to $$ t_{x} $$ and $$ t_{y} $$ and correspondingly re-labeling the eigenstates to $$ \left| x \right\rangle $$ and $$ \left| y \right\rangle $$.

Fig. 12.2

Projection of a single particle system to {$$ \left| 0 \right\rangle $$, $$ \left| 1 \right\rangle $$} and {$$ \left| x \right\rangle $$, $$ \left| y \right\rangle $$} bases

Thus, we can easily verify that

Further, Tx and Ty corresponding to the time durations the system will be in state $$ \left| x \right\rangle $$ and $$ \left| y \right\rangle $$, respectively, with respect to Tc in {$$ \left| 0 \right\rangle $$, $$ \left| 1 \right\rangle $$} basis is given by

Thus, if the measurement basis is $$ \left\{ {\left| x \right\rangle ,\left| y \right\rangle } \right\} $$, the system, when measured, will randomly collapse to $$ \left| x \right\rangle $$ or $$ \left| y \right\rangle $$ with probability $$ T_{x} /T_{\text{c}} = \left| {\alpha \cos \theta + \beta \sin \theta } \right|^{2} $$ or $$ T_{y} /T_{\text{c}} = \left| {\beta \cos \theta - \alpha \sin \theta } \right|^{2} $$, respectively.

Finally, the posited model of entanglement between qubits requires that any unitary operation that causes entanglement, say, between two qubits , also synchronizes their sub-Planck level oscillations. This is shown in Fig. 12.3 for the 2-qubit Bell states ,

Fig. 12.3

Two-particle entangled systems; $$ \left| \psi_{1} \right\rangle $$ (left) $$ \left| \psi_{2} \right\rangle $$ (right)

Note that while entanglement results in a synchronous state for the two particles, the converse is not necessarily true. When a measurement is made on one of the entangled particles, both will collapse simultaneously. According to our model, the particles will collapse to the state they are in at the instant of measurement (such as τ₁ or τ₂ in Fig. 12.3), which is in accordance with Postulate 3. We do not know how Nature might accomplish the required synchronization. It is, of course, clear that this interpretation cannot violate the uncertainty principle since Postulate 3 is not violated. Everything rests entirely on the notion of external observations because without it there are no means to ascribe a physical interpretation.

12.3 Application of the Basic Model

We now provide a few examples of quantum systems to show that our interpretation is consistent with the outcomes of measurements made on those systems at any instant.

12.3.1 Measurement of a Two-Particle Entangled System

Recall that a measurement made on either particle in an entangled pair will automatically and instantaneously alter the state of the other particle. We are now confronted with two measurement possibilities: (1) measurement using commutating observables (such as of electron spin along the same axis); and (2) measurement using non-commutating observables (such as of electron spins along different axes).

(1)

Commutating observables. Consider an entangled pair of electrons, where $$ \left| 0 \right\rangle $$ and $$ \left| 1 \right\rangle $$ represent spin-up and spin-down, respectively. Then according to our model, if a measurement is made along the spin axis, the electrons will collapse to similar spins, with the spin state determined by the instant of measurement if the entangled pair is described by the Bell state $$ \left| {\psi_{1} } \right\rangle $$ in Fig. 12.3. Likewise, it will collapse to opposite spins if the entangled pair is described by $$ \left| {\psi_{2} } \right\rangle $$ in Fig. 12.3.

(2)

Non-commutating observables. Consider an entangled pair of electrons where the electrons have spin components along two axes, say, x-axis and y-axis (see Fig. 12.4). Note that at any given instant the spin of both electrons will be along only one of the axes. Further, at any instant, such as τ₁, τ₂, τ₃, and τ₄ shown in Fig. 12.4, the two electrons can be in only one of the four states: $$ \left| 00 \right\rangle $$x, $$ \left| 11 \right\rangle $$x, $$ \left| 00 \right\rangle $$y, and $$ \left| 11 \right\rangle $$y, respectively. The suffixes x, y represent the x, y components, respectively, of $$ \left| 00 \right\rangle $$ and $$ \left| 11 \right\rangle $$.

Fig. 12.4

Two-particle entangled system with non-commutating observables

Now, if a measurement is made and the system collapses, say, to the x-component, then a subsequent measurement along the y-axis will return a null result.

In the more general case of a system of n-particles, if a combined measurement is made on m ≤ n of those particles, then those m-particles will collapse to one of their possible group states (the actual number of states at any given instant may vary from 1 to 2m) on measurement while the remaining n − m particles will assume states which are consistent with the collapsed state of the m-particles.

12.3.2 Quantum Adder

The interpretation explains the quantum adder in a consistent manner. Let the initial input state of the required three particle system, where each particle represents a qubit , be given by $$ \left| {\psi_{0} } \right\rangle = \left| {000} \right\rangle $$. Now apply the Hadamard gate to the first two qubits to create the four possible inputs for the addition operation. Thus, we have

To carry out the add operation apply the Toffoli gate to the three qubits with the third qubit as target, followed by the Cnot gate to the first two qubits with the second qubit as target to get

where the second qubit is the sum and the third qubit is the carry bit. Note that the carry bit in the adder is the result of an AND operation. The carry and AND are really the same thing. The sum bit comes from an XOR gate (that is, the Cnot operation). Figure 12.5 captures the four possible eigenstates represented by $$ \left| {\psi_{1} } \right\rangle $$ and $$ \left| {\psi_{2} } \right\rangle $$ at the instants τ₁, τ₂, τ₃, and τ₄.

Fig. 12.5

Quantum adder input and output states; $$ \left| \psi_{1} \right\rangle $$ (left), $$ \left| \psi_{2} \right\rangle $$ (right)

12.4 Teleporting a Qubit of an Unknown State

Suppose Alice wishes to teleport a qubit , labeled by subscript 1, of unknown state

to Bob. In addition, there is an entangled pair of auxiliary qubits designated by subscripts 2 and 3 in the state

Alice holds the qubits with subscripts 1 and 2 while Bob holds the qubit with subscript 3. Thus, the initial state of the 3-qubit system is (see Fig. 12.6a where the qubit subscripts (in this and subsequent Fig. 12.6b, c) have been omitted since they can be inferred from their position in the state $$ \left| \ldots \right\rangle $$) given by

Fig. 12.6

a Initial state $$ \left| \psi_{0} \right\rangle $$ of the teleportation system. b State $$ \left| \psi_{1} \right\rangle $$ of the teleportation system after Cnot operation. c State $$ \left| \psi_{2} \right\rangle $$ of the teleportation system after Hadamard operation

Alice now applies the Cnot gate (with qubit 2 as the target) to the qubits held by her. This changes the state of the 3-qubit system to (see Fig. 12.6b)

Next, Alice applies the Hadamard gate to qubit 1 which puts the 3-qubit system in the state shown in Fig. 12.6c.

Finally, Alice makes a “combined” measurement on the two qubits she holds. Such a measurement gives access to some combined (or global) information on both qubits, but none on a single qubit , i.e., no distinction between the two qubits can be established. Her measurement will lead the pair to collapse to one of the four possible states $$ \left| {0_{1} 0_{2} } \right\rangle $$, $$ \left| {0_{1} 1_{2} } \right\rangle $$, $$ \left| {1_{1} 0_{2} } \right\rangle $$, or $$ \left| {1_{1} 1_{2} } \right\rangle $$, while the third qubit , correspondingly, will immediately collapse to the state $$ \alpha \left| {1_{3} } \right\rangle - \beta \left| {0_{3} } \right\rangle $$, $$ \alpha \left| {0_{3} } \right\rangle - \beta \left| {1_{3} } \right\rangle $$, $$ \alpha \left| {1_{3} } \right\rangle + \beta \left| {0_{3} } \right\rangle $$, or $$ \alpha \left| {0_{3} } \right\rangle + \beta \left| {1_{3} } \right\rangle $$, respectively, since it is also entangled with qubits 1 and 2. Table 12.1 shows the measurement result Alice will get depending upon the instant the measurement actually occurred, along with the post-measurement state of qubit 3 held by Bob.

Table 12.1

Measurement outcomes in the teleportation algorithm

Measurement instant

State of qubits 1 and 2 after Alice’s measurement

State of qubit 3 after Alice’s measurement

Decoder to bring qubit 3 to state $$ \left| \phi \right\rangle $$

τ₁, τ₂

Y

$$ \left| 0_{3} \right\rangle $$ → −$$ \left| 1_{3} \right\rangle $$

$$ \left| 1_{3} \right\rangle $$ → $$ \left| 0_{3} \right\rangle $$

τ₃, τ₄

Z

$$ \left| 0_{3} \right\rangle $$ → $$ \left| 0_{3} \right\rangle $$

$$ \left| 1_{3} \right\rangle $$ → −$$ \left| 1_{3} \right\rangle $$

τ₅, τ₆

X

$$ \left| 0_{3} \right\rangle $$ → $$ \left| 1_{3} \right\rangle $$

$$ \left| 1_{3} \right\rangle $$ → $$ \left| 0_{3} \right\rangle $$

τ₇, τ₈

I

$$ \left| 0_{3} \right\rangle $$ → $$ \left| 0_{3} \right\rangle $$

$$ \left| 1_{3} \right\rangle $$ → $$ \left| 1_{3} \right\rangle $$

Alice communicates the classical result of her “combined” measurement ($$ \left| {0_{1} 0_{2} } \right\rangle $$, $$ \left| {0_{1} 1_{2} } \right\rangle $$, $$ \left| {1_{1} 0_{2} } \right\rangle $$, or $$ \left| {1_{1} 1_{2} } \right\rangle $$) to Bob (using classical means such as telephone, email, etc.). Bob then uses the decoder (a unitary transformation) listed in Table 12.1 corresponding to the state of qubits 1 and 2 conveyed to him by Alice to bring his qubit to state $$ \left| \phi \right\rangle = \alpha \left| {0_{3} } \right\rangle + \beta \left| {1_{3} } \right\rangle $$.

12.5 Concluding Remarks

The time-multiplexed interpretation of quantum measurement has greater appeal to human intuition which is reluctant to abandon concepts from classical physics. As a working strategy, there is no issue with the “shut up and compute” Copenhagen interpretation , but at a psychological level, humans are more comfortable with an interpretation they can intuitively relate to. To this end, the new interpretation has value because it appears to be compatible with measurements at the classical level. Humans are not physically equipped to perceive the quantum world and our measuring devices are classical. We can therefore only speculate what Nature is like and remind ourselves that conjectures are deemed scientifically valid only if there is potential scope of finding an error. “Though it stresses our fallibility it does not resign itself to scepticism, for it also stresses the fact that knowledge can grow, and that science can progress—just because we can learn from our mistakes.”³ The process is criticism controlled.

References

1.

R.K. Bera, V. Menon, A new interpretation of superposition, entanglement, and measurement in quantum mechanics. arXiv:​0908.​0957 [quant-ph], 07 Aug 2009. https://​arxiv.​org/​pdf/​0908.​0957.​pdf

2.

R.K. Bera, V. Menon, The essence of quantum computing. Advanced Computing & Communications, 2(3), 20–32 (2018)

3.

K. Popper, Conjectures and Refutations: The Growth of Scientific Knowledge (Routledge, 1963). Also as K. Popper (1968). Reprint, Harper & Row. Conjectures and Refutations

Footnotes

1

Bera and Menon [1, 2].

2

Bera and Menon [1, 2].

3

Popper [3].

Index

A

Are quantum computers more powerful?

Axioms of quantum mechanics

B

Basic quantum operations

BB84 protocol

Bell inequality

Bell states

Bennett’s solution for junk bits

Bera and Menon

Birth of modern quantum mechanics

Bit copying

Bitwise operators

Born’s probabilistic interpretation

Bra and ket operations

C

Can quantum computers prove theorems?

Cascaded measurements

Cauchy–Schwarz inequality

Causality and determinism

Church–Turing thesis

Classical mechanics

powers our intuition

Classification of complexity

Complementarity (wave-particle duality)

Computational complexity

Controlled-U gate

Cryptography

D

De Broglie formula

Decision problems

Deutsch on the Church–Turing thesis

Dirac notations

E

Einstein, Podolsky, Rosen pose a paradox

Einstein’s formula

Elements of linear algebra

Encryption and key distribution

EPR paradox and Bell inequalities

F

Factorizable states

Feynman, Richard

First-order predicate calculus (first order logic)

Fourier analysis

Fredkin gate

G

Galileo–Newton to Schrödinger–Born

Global phase factor

H

Heisenberg’s uncertainty principle

Hermitian operators

Hilbert space grows rapidly

Hilbert’s second problem

Hilbert’s tenth problem

I

Information is physical

Interpretations of quantum mechanics

Bohm’s interpretation

Copenhagen interpretation

Everett’s many-world interpretation

time-multiplexed interpretation of measurement

K

Knowledge and entropy

L

Linear operators and matrices

commutator and anti-commutator

completeness relation

diagonal representation

eigenvalues and eigenvectors

inner product

normal operators and spectral decomposition

orthonormal decomposition

outer product

Pauli matrices

polar and singular value decompositions

positive operator

tensor product

trace of a matrix

unitary operators

M

Mach–Zehnder interferometer

Maxwell’s demon

Measurement of quantum systems

distinguishing quantum states

effect of phase on measurement

positive operator-valued Measurements (POVM)

Modulo arithmetic

modulo 2 arithmetic

N

No-cloning theorem

consequences of

No-deleting theorem

No hiding theorem

Notations in quantum mechanics

NP-complete problems

stand or fall together

O

Observables and operators

Observer in physics

One-molecule gas

O notation

O , Ω, Θ,o , ω notations

Operator/matrix types

Operators (a summary)

P

Planck’s formula

Postulates of quantum mechanics

Projective measurements

Properties of congruence

congruence classes

Propositional calculus (propositional logic)

Q

Quantum algorithms

computing the period of a sequence

dense coding and teleportation

general remarks

Grover’s search algorithm

phase estimation problem

quantum Fourier transform

Shor’s factoring algorithm

Quantum error corrections

Calderbank–Shor–Steane error correction

decoherence

decoherence-free subspace

dissipation

protecting the computational Hilbert space

Quantum gates

Quantum operators

controlled-not

Hadamard

that act on a qubit

that acts on a qubit pair

Quantum system collapses when measured

Quantum system evolves via unitary transformations

Qubit

R

Recursive set

Relative phase factor

Reversible classical computation

S

Schrödinger equation

Simple quantum algorithms

computing f(x) in parallel

computing x + y

computing x ∧ y

Deutsch algorithm

Deutsch–Jozsa algorithm

Elitzur–Vaidman bomb problem

Hardy’s reprieve

securing banknotes

swapping states

Superposition and indeterminacy

Superposition, measurement, and entanglement

T

Teleportation

Toffoli gate

Turing and the entscheidungsproblem

Turing’s halting problem

Two-layer description of the world

U

Unitary operators

Universal set of gates

UTM, DTM, PTM, QTM

W

Waves

group and phase velocities

probability waves

standing or stationary waves

travelling waves

wave equation

wave packets

Whither causality?