Mathematics of Quantum Computing: an Introduction

773 Pages·2019·4.137 MB·English

by  Scherer, Wolfgang

Wolfgang Scherer
Mathematics
of Quantum
Computing
An Introduction
Mathematics of Quantum Computing
Wolfgang Scherer
Mathematics of Quantum
Computing
An Introduction
123
WolfgangScherer
Kingston, UK
ISBN978-3-030-12357-4 ISBN978-3-030-12358-1 (eBook)
https://doi.org/10.1007/978-3-030-12358-1
Translation from the German edition language edition: Mathematik der Quanteninformatik by
Dr.WolfgangScherer,©Springer-VerlagGmbHGermany2016.PublishedbySpringer-VerlagGmbH
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Para Negri
und für Matthias und Sebastian
Preface
In the last two decades the digitization of our lives has accelerated at a sometimes
breathtaking rate. It isappearing in ever more aspectsof our existence to the point
of becoming all-encompassing. Ever larger amounts of data are generated, stored,
processed and transmitted. This is driven by increases in processing speed and
computational performance. The latter is achieved by ever greater miniaturization
ofcircuitsandphysicalmemoryrequirements.Asthistrendcontinuesrathersooner
than later atomic or even sub-atomic scales will be reached. At the latest at that
pointthelawsofquantummechanicswillberequiredtodescribethecomputational
process along with the handling of memory.
Originating from this anticipation as well as from scientific curiosity, many
researchersoverthelastquartercenturyhavethusinvestigatedhowinformationcan
be stored and processed in systems described by quantum mechanical laws. In
doing so they created the science of quantum computing.
Quantum computing is unique in the sense that nowhere else are fundamental
questionsinphysicssocloselyconnectedtopotentiallyhugepracticalimplications
andbenefits.Ourverybasicunderstandingofwhatconstitutesrealityischallenged
bytheeffectswhichatthesametimeseemtoenableenormousefficiencygainsand
to revolutionize computational power and cryptographic protocols.
What is also quite enticing is that quantum computing draws from many ‘dis-
tinct’branchesofmathematicssuchas,ofcourse,analysisandlinearalgebra,butto
anevengreaterextentfunctionalanalysis,grouptheory,numbertheory,probability
theory and not least computer science.
This book aims to give an introduction to the mathematics of this wide-ranging
and promising field. The reason that despite being an introduction it is so volu-
minousisthatthereaderistakenbythehandandledthroughallargumentsstepby
step.Allresultsareproveninthetextin—forthecognoscentiperhapsexcruciating
—detail. Numerous exercises with their solutions provided allow the reader to test
and develop their understanding. Any requisites from branches, such as number
theory or group theory, are provided with all stated results proven in the book as
well.
vii
viii Preface
For the above reasons this book is eminently suitable for self-study of the
subject.Theattentiveanddiligentreaderdoesnothavetoconsultotherresourcesto
followthearguments.Thelevelofmathematicalknow-howrequiredapproximately
corresponds to second year undergraduate knowledge in mathematics or physics.
It is very much a text in mathematical style in that we follow the pattern of
motivating text—definition—lemma/theorem/corollary—proof—explanatory text
and all over again. In doing so all relevant assumptions are clearly stated. At the
same time, it provides ample opportunities for the reader to become familiar with
standard techniques in quantum computing as well as in the related mathematical
sub-fields. Having mastered this book the reader will be equipped to digest sci-
entific papers on quantum computing.
I enjoyed writing this book. I very much hope it is equally enjoyable to read it.
Acknowledgements Throughout my academic life many people have taught,
motivated, enlightened and inspired me. I am truly grateful to every one of them
and hope they will find this book to their liking.
A very special thanks goes to the organizers and participants of the 2017
Summer Academy of the Studienstiftung des Deutschen Volkes. Their review and
critical feedback for the German version has helped to erase several errors and to
improve the presentation. It also was a pleasure to spend some time with them in
lovely South Tirol. In particular, I am wholeheartedly grateful to Joachim Hilgert
who, in addition to giving feedback on the German version, also very swiftly and
thoroughly proofread a part of this manuscript.
ThelargestdebtofgratitudeisowedtoMaria-Eugenia,MatthiasandSebastian,
whoduringmanyyearsofthisprojecthavetakenthebackseatandtookmyretreat
from family life in their stride but never wavered in their support and shared my
enthusiasm.IamverygratefulinparticulartoSebastian,whoonceagainproofread
large portions of the manuscript. His detailed review detected many errors and his
constructive criticism during numerous enjoyable and lengthy sessions helped
improve precision and clarity of the exposition.
Needless to say that even his diligence will not have detected every error or
shortcoming. Those were still caused by the author.
Kingston Upon Thames, UK Wolfgang Scherer
March 2019
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Some History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Reader’s Manual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 What is not in this Book. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Notation and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Basic Notions of Quantum Mechanics. . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Mathematical Notions: HILBERT Spaces and Operators. . . . . . . . . 12
2.3 Physical Notions: States and Observables. . . . . . . . . . . . . . . . . . 29
2.3.1 Pure States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.2 Mixed States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4 Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.5 Operators on Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.6 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3 Tensor Products and Composite Systems . . . . . . . . . . . . . . . . . . . . . 77
3.1 Towards Qbytes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.2 Tensor Products of HILBERT Spaces . . . . . . . . . . . . . . . . . . . . . . 78
3.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.2.2 Computational Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.3 States and Observables for Composite Systems . . . . . . . . . . . . . 90
3.4 SCHMIDT Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.5 Quantum Operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.6 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.2 Definition and Characterization . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.3 Entanglement Swapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.4 EINSTEIN–PODOLSKY–ROSEN-Paradox . . . . . . . . . . . . . . . . . . . . . . 135
ix
x Contents
4.5 BELL Inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.5.1 Original BELL Inequality. . . . . . . . . . . . . . . . . . . . . . . . . 141
4.5.2 CHSH Generalization of the BELL Inequality. . . . . . . . . . 146
4.6 Two Impossible Devices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.6.1 BELL Telephone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.6.2 Perfect Quantum Copier. . . . . . . . . . . . . . . . . . . . . . . . . 158
4.7 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5 Quantum Gates and Circuits for Elementary Calculations. . . . . . . . 161
5.1 Classical Gates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.2 Quantum Gates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
5.2.1 Unary Quantum Gates . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.2.2 Binary Quantum Gates. . . . . . . . . . . . . . . . . . . . . . . . . . 171
5.2.3 General Quantum Gates . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.3 Quantum Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
5.4 On the Process of Quantum Algorithms. . . . . . . . . . . . . . . . . . . 206
5.4.1 Preparation of Input and Use of Auxiliary Registers . . . . 207
5.4.2 Implementationof Functions and Quantum Parallelism... 208
5.4.3 Reading the Output Register. . . . . . . . . . . . . . . . . . . . . . 212
5.5 Circuits for Elementary Arithmetic Operations . . . . . . . . . . . . . . 213
5.5.1 Quantum Adder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
5.5.2 Quantum Adder Modulo N. . . . . . . . . . . . . . . . . . . . . . . 226
5.5.3 Quantum Multiplier Modulo N . . . . . . . . . . . . . . . . . . . . 229
5.5.4 Quantum Circuit for Exponentiation Modulo N . . . . . . . . 233
5.5.5 Quantum FOURIER Transform . . . . . . . . . . . . . . . . . . . . . 237
5.6 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
6 On the Use of Entanglement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
6.1 Early Promise: DEUTSCH–JOZSA Algorithm . . . . . . . . . . . . . . . . . 247
6.2 Dense Quantum Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
6.3 Teleportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
6.4 Quantum Cryptography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
6.4.1 Ciphers in Cryptography . . . . . . . . . . . . . . . . . . . . . . . . 255
6.4.2 Quantum Key Distribution without Entanglement . . . . . . 258
6.4.3 Quantum Key Distribution with Entanglement. . . . . . . . . 262
6.4.4 RSA Public Key Distribution . . . . . . . . . . . . . . . . . . . . . 266
6.5 SHOR Factorization Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 271
6.5.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
6.5.2 The Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
6.5.3 Step 1: Selection of b and Calculation of gcdðb;NÞ. . . . . 275
6.5.4 Step 2: Determining the Period with a Quantum
Computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
6.5.5 Step 3: Probability of Selecting a Suitable b . . . . . . . . . . 290
6.5.6 Balance Sheet of Steps. . . . . . . . . . . . . . . . . . . . . . . . . . 296
Contents xi
6.6 Generalizing: The Abelian Hidden Subgroup Problem . . . . . . . . 301
6.7 Finding the Discrete Logarithm as a Hidden Subgroup
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
6.8 Breaking Bitcoin Signatures . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
6.9 GROVER Search Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
6.9.1 Search Algorithm for Known Number of Objects . . . . . . 324
6.9.2 Search Algorithm for Unknown Number of Objects. . . . . 337
6.10 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
7 Error Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
7.1 What Can Go Wrong? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
7.2 Classical Error Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
7.3 Quantum Error Correction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
7.3.1 Correctable Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
7.3.2 Detection and Correction . . . . . . . . . . . . . . . . . . . . . . . . 384
7.3.3 Stabilizer Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
7.4 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
8 Adiabatic Quantum Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
8.2 Starting Point and Assumptions. . . . . . . . . . . . . . . . . . . . . . . . . 404
8.3 Generic Adiabatic Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . 413
8.4 Adiabatic Quantum Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
8.5 Replicating a Circuit Based by an Adiabatic Computation. . . . . . 440
8.6 Replicating an Adiabatic by a Circuit Based Computation. . . . . . 484
8.7 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497
9 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
Appendix A: Elementary Probability Theory. .... .... .... ..... .... 501
Appendix B: Elementary Arithmetic Operations .. .... .... ..... .... 505
Appendix C: LANDAU Symbols ..... .... .... .... .... .... ..... .... 513
Appendix D: Modular Arithmetic .. .... .... .... .... .... ..... .... 515
Appendix E: Continued Fractions.. .... .... .... .... .... ..... .... 545
Appendix F: Some Group Theory.. .... .... .... .... .... ..... .... 559
Appendix G: Proof of a Quantum Adiabatic Theorem . .... ..... .... 621
Solutions to Exercises... .... ..... .... .... .... .... .... ..... .... 641
References.... .... .... .... ..... .... .... .... .... .... ..... .... 755
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 759