دانلود کتاب Mathematics of Quantum Computing: an Introduction

فهرست مطالب :

Preface......Page 6
Contents......Page 8
Glossary......Page 11
Symbols......Page 13
1.1 Some History......Page 18
1.2 Reader\'s Manual......Page 21
1.3 What is not in this Book......Page 24
1.4 Notation and References......Page 25
2.1 Generalities......Page 27
2.2 Mathematical Notions: Hilbert Spaces and Operators......Page 28
2.3.1 Pure States......Page 45
2.3.2 Mixed States......Page 59
2.4 Qubits......Page 72
2.5 Operators on Qubits......Page 79
2.6 Further Reading......Page 90
3.1 Towards Qbytes......Page 92
3.2.1 Definition......Page 93
3.2.2 Computational Basis......Page 100
3.3 States and Observables for Composite Systems......Page 105
3.4 Schmidt Decomposition......Page 121
3.5 Quantum Operations......Page 124
3.6 Further Reading......Page 140
4.1 Generalities......Page 141
4.2 Definition and Characterization......Page 142
4.3 Entanglement Swapping......Page 147
4.4 Einstein–Podolsky–Rosen-Paradox......Page 149
4.5.1 Original Bell Inequality......Page 155
4.5.2 CHSH Generalization of the Bell Inequality......Page 160
4.6.1 Bell Telephone......Page 168
4.6.2 Perfect Quantum Copier......Page 172
4.7 Further Reading......Page 174
5.1 Classical Gates......Page 175
5.2 Quantum Gates......Page 182
5.2.1 Unary Quantum Gates......Page 183
5.2.2 Binary Quantum Gates......Page 185
5.2.3 General Quantum Gates......Page 186
5.3 Quantum Circuits......Page 215
5.4 On the Process of Quantum Algorithms......Page 220
5.4.1 Preparation of Input and Use of Auxiliary Registers......Page 221
5.4.2 Implementation of Functions and Quantum Parallelism......Page 222
5.4.3 Reading the Output Register......Page 226
5.5.1 Quantum Adder......Page 227
5.5.2 Quantum Adder Modulo N......Page 240
5.5.3 Quantum Multiplier Modulo N......Page 243
5.5.4 Quantum Circuit for Exponentiation Modulo N......Page 247
5.5.5 Quantum Fourier Transform......Page 251
5.6 Further Reading......Page 259
6.1 Early Promise: Deutsch–Jozsa Algorithm......Page 261
6.2 Dense Quantum Coding......Page 265
6.3 Teleportation......Page 267
6.4.1 Ciphers in Cryptography......Page 269
6.4.2 Quantum Key Distribution without Entanglement......Page 272
6.4.3 Quantum Key Distribution with Entanglement......Page 276
6.4.4 RSA Public Key Distribution......Page 280
6.5.1 Generalities......Page 285
6.5.2 The Algorithm......Page 287
6.5.3 Step 1: Selection of b and Calculation of `3́9`42`\"̇613A``45`47`\"603Agcd(b,N)......Page 289
6.5.4 Step 2: Determining the Period with a Quantum Computer......Page 290
6.5.5 Step 3: Probability of Selecting a Suitable b......Page 304
6.5.6 Balance Sheet of Steps......Page 310
6.6 Generalizing: The Abelian Hidden Subgroup Problem......Page 315
6.7 Finding the Discrete Logarithm as a Hidden Subgroup Problem......Page 324
6.8 Breaking Bitcoin Signatures......Page 331
6.9.1 Search Algorithm for Known Number of Objects......Page 338
6.9.2 Search Algorithm for Unknown Number of Objects......Page 351
6.10 Further Reading......Page 355
7.1 What Can Go Wrong?......Page 356
7.2 Classical Error Correction......Page 358
7.3.1 Correctable Errors......Page 368
7.3.2 Detection and Correction......Page 397
7.3.3 Stabilizer Formalism......Page 403
7.4 Further Reading......Page 415
8.1 Introduction......Page 416
8.2 Starting Point and Assumptions......Page 417
8.3 Generic Adiabatic Algorithm......Page 426
8.4 Adiabatic Quantum Search......Page 432
8.5 Replicating a Circuit Based by an Adiabatic Computation......Page 453
8.6 Replicating an Adiabatic by a Circuit Based Computation......Page 497
8.7 Further Reading......Page 510
9 Epilogue......Page 511
Appendix A Elementary Probability Theory......Page 513
Appendix B Elementary Arithmetic Operations......Page 517
Appendix C Landau Symbols......Page 524
Appendix D Modular Arithmetic......Page 526
Appendix E Continued Fractions......Page 556
F.1 Groups, Subgroups and Quotient Groups......Page 569
F.2 Homomorphisms, Characters and Dual Groups......Page 586
F.3 Quantum Fourier Transform on Groups......Page 609
F.4 Elliptic Curves......Page 610
F.5 The Pauli Group......Page 618
G.1 Resolvents and Projections......Page 630
G.2 Adiabatic Generator and Intertwiner......Page 636
G.3 Reduced Resolvent, Bounds on Time Derivatives and an Adiabatic Theorem......Page 638
Appendix Solutions to Exercises......Page 650
Appendix References......Page 764
Index......Page 768