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量子计算与量子信息(10周年版)

作者:Michael.A.Nielsen

出版社:清华大学出版社

出版年:2015-10-01

页数:676

豆瓣评分:8.8 去购买

内容简介

  《量子计算与量子信息(10周年版)》介绍了量子计算与量子信息(10周年版)相关知识。


目录

Part I 

Fundamental concepts 1 

1 Introduction and overview 1 

1.1 Global perspectives 1 

1.1.1 History of quantum computation and quantum information 2 

1.1.2 Future directions 12 

1.2 Quantum bits 13 

1.2.1 Multiple qubits 16 

1.3 Quantum computation 17 

1.3.1 Single qubit gates 17 

1.3.2 Multiple qubit gates 20 

1.3.3 Measurements in bases other than the computational basis 22 

1.3.4 Quantum circuits 22 

1.3.5 Qubit copying circuit? 24 

1.3.6 Example: Bell states 25 

1.3.7 Example: quantum teleportation 26 

1.4 Quantum algorithms 28 

1.4.1 Classical computations on a quantum computer 29 

1.4.2 Quantum parallelism 30 

1.4.3 Deutsch’s algorithm 32 

1.4.4 The Deutsch–Jozsa algorithm 34 

1.4.5 Quantum algorithms summarized 36 

1.5 Experimental quantum information processing 42 

1.5.1 The Stern–Gerlach experiment 43 

1.5.2 Prospects for practical quantum information processing 46 

1.6 Quantum information 50 

1.6.1 Quantum information theory: example problems 52 

1.6.2 Quantum information in a wider context 58 

2 Introduction to quantum mechanics 60 

2.1 Linear algebra 61 

2.1.1 Bases and linear independence 62

2.1.2 Linear operators and matrices 63 

2.1.3 The Pauli matrices 65 

2.1.4 Inner products 65 

2.1.5 Eigenvectors and eigenvalues 68 

2.1.6 Adjoints and Hermitian operators 69 

2.1.7 Tensor products 71 

2.1.8 Operator functions 75 

2.1.9 The commutator and anti-commutator 76 

2.1.10 The polar and singular value decompositions 78 

2.2 The postulates of quantum mechanics 80 

2.2.1 State space 80 

2.2.2 Evolution 81 

2.2.3 Quantum measurement 84 

2.2.4 Distinguishing quantum states 86 

2.2.5 Projective measurements 87 

2.2.6 POVM measurements 90 

2.2.7 Phase 93 

2.2.8 Composite systems 93 

2.2.9 Quantum mechanics: a global view 96 

2.3 Application: superdense coding 97 

2.4 The density operator 98 

2.4.1 Ensembles of quantum states 99

2.4.2 General properties of the density operator 101 

2.4.3 The reduced density operator 105 

2.5 The Schmidt decomposition and purifications 109 

2.6 EPR and the Bell inequality 111 

3 Introduction to computer science 120 

3.1 Models for computation 122 

3.1.1 Turing machines 122 

3.1.2 Circuits 129 

3.2 The analysis of computational problems 135 

3.2.1 How to quantify computational resources 136 

3.2.2 Computational complexity 138 

3.2.3 Decision problems and the complexity classes P and NP 141 

3.2.4 A plethora of complexity classes 150 

3.2.5 Energy and computation 153 

3.3 Perspectives on computer science 161 

Part II Quantum computation 171 

4 Quantum circuits 171 

4.1 Quantum algorithms 172 

4.2 Single qubit operations 174 

4.3 Controlled operations 177 

4.4 Measurement 185 

4.5 Universal quantum gates 188 

4.5.1 Two-level unitary gates are universal 189 

4.5.2 Single qubit and CNOT gates are universal 191 

4.5.3 A discrete set of universal operations 194 

4.5.4 Approximating arbitrary unitary gates is generically hard 198 

4.5.5 Quantum computational complexity 200 

4.6 Summary of the quantum circuit model of computation 202 

4.7 Simulation of quantum systems 204 

4.7.1 Simulation in action 204 

4.7.2 The quantum simulation algorithm 206 

4.7.3 An illustrative example 209 

4.7.4 Perspectives on quantum simulation 211 

5 The quantum Fourier transform and its applications 216 

5.1 The quantum Fourier transform 217 

5.2 Phase estimation 221 

5.2.1 Performance and requirements 223 

5.3 Applications: order-finding and factoring 226 

5.3.1 Application: order-finding 226 

5.3.2 Application: factoring 232 

5.4 General applications of the quantum Fourier transform 234 

5.4.1 Period-finding 236 

5.4.2 Discrete logarithms 238 

5.4.3 The hidden subgroup problem 240 

5.4.4 Other quantum algorithms? 242 

6 Quantum search algorithms 248 

6.1 The quantum search algorithm 248 

6.1.1 The oracle 248 

6.1.2 The procedure 250 

6.1.3 Geometric visualization 252 

6.1.4 Performance 253 

6.2 Quantum search as a quantum simulation 255 

6.3 Quantum counting 261 

6.4 Speeding up the solution of NP-complete problems 263 

6.5 Quantum search of an unstructured database 265 

6.6 Optimality of the search algorithm 269 

6.7 Black box algorithm limits 271 

7 Quantum computers: physical realization 277 

7.1 Guiding principles 277 

7.2 Conditions for quantum computation 279 

7.2.1 Representation of quantum information 279 

7.2.2 Performance of unitary transformations 281 

7.2.3 Preparation of fiducial initial states 281 

7.2.4 Measurement of output result 282 

7.3 Harmonic oscillator quantum computer 283 

7.3.1 Physical apparatus 283 

7.3.2 The Hamiltonian 284 

7.3.3 Quantum computation 286 

7.3.4 Drawbacks 286 

7.4 Optical photon quantum computer 287 

7.4.1 Physical apparatus 287 

7.4.2 Quantum computation 290 

7.4.3 Drawbacks 296 

7.5 Optical cavity quantum electrodynamics 297 

7.5.1 Physical apparatus 298 

7.5.2 The Hamiltonian 300 

7.5.3 Single-photon single-atom absorption and refraction 303 

7.5.4 Quantum computation 306 

7.6 Ion traps 309 

7.6.1 Physical apparatus 309 

7.6.2 The Hamiltonian 317 

7.6.3 Quantum computation 319 

7.6.4 Experiment 321 

7.7 Nuclear magnetic resonance 324 

7.7.1 Physical apparatus 325 

7.7.2 The Hamiltonian 326 

7.7.3 Quantum computation 331 

7.7.4 Experiment 336 

7.8 Other implementation schemes 343 

Part III Quantum information 353 

8 Quantum noise and quantum operations 353 

8.1 Classical noise and Markov processes 354 

8.2 Quantum operations 356 

8.2.1 Overview 356 

8.2.2 Environments and quantum operations 357 

8.2.3 Operator-sum representation 360 

8.2.4 Axiomatic approach to quantum operations 366 

8.3 Examples of quantum noise and quantum operations 373 

8.3.1 Trace and partial trace 374 

8.3.2 Geometric picture of single qubit quantum operations 374 

8.3.3 Bit flip and phase flip channels 376 

8.3.4 Depolarizing channel 378 

8.3.5 Amplitude damping 380 

8.3.6 Phase damping 383 

8.4 Applications of quantum operations 386 

8.4.1 Master equations 386 

8.4.2 Quantum process tomography 389 

8.5 Limitations of the quantum operations formalism 394 

9 Distance measures for quantum information 399 

9.1 Distance measures for classical information 399 

9.2 How close are two quantum states? 403 

9.2.1 Trace distance 403 

9.2.2 Fidelity 409 

9.2.3 Relationships between distance measures 415 

9.3 How well does a quantum channel preserve information? 416 

10 Quantum error-correction 425 

11 Entropy and information 500 

12 Quantum information theory 528 

Appendices 608 

Appendix 1: Notes on basic probability theory 608 

Appendix 2: Group theory 610 

Appendix 3: The Solovay--Kitaev theorem 617 

Appendix 4: Number theory 625 

Appendix 5: Public key cryptography and the RSA cryptosystem 640 

Appendix 6: Proof of Lieb’s theorem 645 

Bibliography 649 

Index 

665 



前言/序言

量子信息处理可以带来许多意想不到的具有特殊优势的结果。量子算法可以有效地进行大数质因数分解。在量子算法背景下,很多经典保密通信协议的安全性受到威胁,然而量子保密通信可以抵抗来自包括量子计算机在内的任何针对通道的攻击。由于存在诸多特殊应用,量子计算与量子信息科学近年来得到蓬勃发展。

对于这一相对年轻但又具有广阔发展前景的学科,优秀的系统化的教材比较缺乏。本书是本领域公认的最权威的系统化教材之一,也几乎是本学科研究人员的必备基本材料,有学者曾将之称为量子计算与量子信息学科教材中的“圣经”。

本书分三大部分。

第 1部分为总论和量子物理学基础,还包括少量计算机科学基础知识。量子物理学基础部分主要介绍学习量子计算与量子信息所必需的量子力学基础知识,这一部分采用了侧重于数学框架和公理化体系的讲述方法,从而更便于为非物理专业的读者所理解。

第 2部分讲述量子计算,包括量子算法和实现量子算法的一些物理系统的基础内容。这部分首先介绍了实现普适量子计算所需要的基本逻辑门 ——单量子比特门与条件非门(第4章)。之后较详细地介绍了现有的两个主要量子算法问题,即质因数分解问题(第5章)和搜索问题(第6章)。第7章讲物理实现,介绍了几个主要的实现条件非门的物理系统;这些物理系统虽然目前尚未实现大规模量子计算,但是大多数已经实现或基本实现了普适量子计算所需要的基本逻辑门。

第 3部分为量子信息论,主要介绍量子纠错码和量子信息论的数学框架。这里包括了非常重要的量子密码的基础内容,即理想条件下的安全性证明(第12.6节)。第3部分未包含物理实现内容。

本书的特点是全面包含量子计算和量子信息的核心内容,且系统性强,结构清晰,深入浅出。这使其很适合作为相关专业的本科生和研究生教材,也适合于用作本领域研究人员的基础参考资料;对于那些准备从其他研究领域转行投入本领域研究的具有物理学、信息科学或数学等专业背景的研究人员,本书也是一本非常合适的入门书籍。

如同任何其他书籍一样,本书的内容也不可能面面俱到。本书几乎未涉及连续变量量子信息处理的有关内容。这方面,目前有多篇综述论文和一些专门教材可供参考。另外,尽管同多数同类教材比较起来,本书已经较深入地介绍了量子密码内容,但是相比于其重要性,量子密码方面的内容量还是有些偏少,对这方面感兴趣的读者可参阅相关专著。

王向斌

清华大学物理系

2015年9月

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