量子计算与量子信息
ContentsIntroduction to the Tenth Anniversary Edition page xviiAfterword to the Tenth Anniversary Edition xixPreface xxiAcknowledgements xxviiNomenclature and notation xxixPart I Fundamental concepts 11 Introduction and overview 11.1 Global perspectives 11.1.1 History of quantum computation and quantuminformation 21.1.2 Future directions 121.2 Quantum bits 131.2.1 Multiple qubits 161.3 Quantum computation 171.3.1 Single qubit gates 171.3.2 Multiple qubit gates 201.3.3 Measurements in bases other than the computational basis 221.3.4 Quantum circuits 221.3.5 Qubit copying circuit? 241.3.6 Example: Bell states 251.3.7 Example: quantum teleportation 261.4 Quantum algorithms 281.4.1 Classical computations on a quantum computer 291.4.2 Quantum parallelism 301.4.3 Deutsch’s algorithm 321.4.4 The Deutsch–Jozsa algorithm 341.4.5 Quantum algorithms summarized 361.5 Experimental quantum information processing 421.5.1 The Stern–Gerlach experiment 431.5.2 Prospects for practical quantum information processing 461.6 Quantum information 501.6.1 Quantum information theory: example problems 521.6.2 Quantum information in a wider context 58x Contents2 Introduction to quantum mechanics 602.1 Linear algebra 612.1.1 Bases and linear independence 622.1.2 Linear operators and matrices 632.1.3 The Pauli matrices 652.1.4 Inner products 652.1.5 Eigenvectors and eigenvalues 682.1.6 Adjoints and Hermitian operators 692.1.7 Tensor products 712.1.8 Operator functions 752.1.9 The commutator and anti-commutator 762.1.10 The polar and singular value decompositions 782.2 The postulates of quantum mechanics 802.2.1 State space 802.2.2 Evolution 812.2.3 Quantum measurement 842.2.4 Distinguishing quantum states 862.2.5 Projective measurements 872.2.6 POVM measurements 902.2.7 Phase 932.2.8 Composite systems 932.2.9 Quantum mechanics: a global view 962.3 Application: superdense coding 972.4 The density operator 982.4.1 Ensembles of quantum states 992.4.2 General properties of the density operator 1012.4.3 The reduced density operator 1052.5 The Schmidt decomposition and purifications 1092.6 EPR and the Bell inequality 1113 Introduction to computer science 1203.1 Models for computation 1223.1.1 Turing machines 1223.1.2 Circuits 1293.2 The analysis of computational problems 1353.2.1 How to quantify computational resources 1363.2.2 Computational complexity 1383.2.3 Decision problems and the complexity classes P and NP 1413.2.4 A plethora of complexity classes 1503.2.5 Energy and computation 1533.3 Perspectives on computer science 161Part II Quantum computation 1714 Quantum circuits 1714.1 Quantum algorithms 1724.2 Single qubit operations 174Contents xi4.3 Controlled operations 1774.4 Measurement 1854.5 Universal quantum gates 1884.5.1 Two-level unitary gates are universal 1894.5.2 Single qubit and CNOT gates are universal 1914.5.3 A discrete set of universal operations 1944.5.4 Approximating arbitrary unitary gates is generically hard 1984.5.5 Quantum computational complexity 2004.6 Summary of the quantum circuit model of computation 2024.7 Simulation of quantum systems 2044.7.1 Simulation in action 2044.7.2 The quantum simulation algorithm 2064.7.3 An illustrative example 2094.7.4 Perspectives on quantum simulation 2115 The quantum Fourier transform and its applications 2165.1 The quantum Fourier transform 2175.2 Phase estimation 2215.2.1 Performance and requirements 2235.3 Applications: order-finding and factoring 2265.3.1 Application: order-finding 2265.3.2 Application: factoring 2325.4 General applications of the quantum Fouriertransform 2345.4.1 Period-finding 2365.4.2 Discrete logarithms 2385.4.3 The hidden subgroup problem 2405.4.4 Other quantum algorithms? 2426 Quantum search algorithms 2486.1 The quantum search algorithm 2486.1.1 The oracle 2486.1.2 The procedure 2506.1.3 Geometric visualization 2526.1.4 Performance 2536.2 Quantum search as a quantum simulation 2556.3 Quantum counting 2616.4 Speeding up the solution of NP-complete problems 2636.5 Quantum search of an unstructured database 2656.6 Optimality of the search algorithm 2696.7 Black box algorithm limits 2717 Quantum computers: physical realization 2777.1 Guiding principles 2777.2 Conditions for quantum computation 2797.2.1 Representation of quantum information 2797.2.2 Performance of unitary transformations 281xii Contents7.2.3 Preparation of fiducial initial states 2817.2.4 Measurement of output result 2827.3 Harmonic oscillator quantum computer 2837.3.1 Physical apparatus 2837.3.2 The Hamiltonian 2847.3.3 Quantum computation 2867.3.4 Drawbacks 2867.4 Optical photon quantum computer 2877.4.1 Physical apparatus 2877.4.2 Quantum computation 2907.4.3 Drawbacks 2967.5 Optical cavity quantum electrodynamics 2977.5.1 Physical apparatus 2987.5.2 The Hamiltonian 3007.5.3 Single-photon single-atom absorption andrefraction 3037.5.4 Quantum computation 3067.6 Ion traps 3097.6.1 Physical apparatus 3097.6.2 The Hamiltonian 3177.6.3 Quantum computation 3197.6.4 Experiment 3217.7 Nuclear magnetic resonance 3247.7.1 Physical apparatus 3257.7.2 The Hamiltonian 3267.7.3 Quantum computation 3317.7.4 Experiment 3367.8 Other implementation schemes 343Part III Quantum information 3538 Quantum noise and quantum operations 3538.1 Classical noise and Markov processes 3548.2 Quantum operations 3568.2.1 Overview 3568.2.2 Environments and quantum operations 3578.2.3 Operator-sum representation 3608.2.4 Axiomatic approach to quantum operations 3668.3 Examples of quantum noise and quantum operations 3738.3.1 Trace and partial trace 3748.3.2 Geometric picture of single qubit quantumoperations 3748.3.3 Bit flip and phase flip channels 3768.3.4 Depolarizing channel 3788.3.5 Amplitude damping 3808.3.6 Phase damping 383Contents xiii8.4 Applications of quantum operations 3868.4.1 Master equations 3868.4.2 Quantum process tomography 3898.5 Limitations of the quantum operations formalism 3949 Distance measures for quantum information 3999.1 Distance measures for classical information 3999.2 How close are two quantum states? 4039.2.1 Trace distance 4039.2.2 Fidelity 4099.2.3 Relationships between distance measures 4159.3 How well does a quantum channel preserve information? 41610 Quantum error-correction 42510.1 Introduction 42610.1.1 The three qubit bit flip code 42710.1.2 Three qubit phase flip code 43010.2 The Shor code 43210.3 Theory of quantum error-correction 43510.3.1 Discretization of the errors 43810.3.2 Independent error models 44110.3.3 Degenerate codes 44410.3.4 The quantum Hamming bound 44410.4 Constructing quantum codes 44510.4.1 Classical linear codes 44510.4.2 Calderbank–Shor–Steane codes 45010.5 Stabilizer codes 45310.5.1 The stabilizer formalism 45410.5.2 Unitary gates and the stabilizer formalism 45910.5.3 Measurement in the stabilizer formalism 46310.5.4 The Gottesman–Knill theorem 46410.5.5 Stabilizer code constructions 46410.5.6 Examples 46710.5.7 Standard form for a stabilizer code 47010.5.8 Quantum circuits for encoding, decoding, andcorrection 47210.6 Fault-tolerant quantum computation 47410.6.1 Fault-tolerance: the big picture 47510.6.2 Fault-tolerant quantum logic 48210.6.3 Fault-tolerant measurement 48910.6.4 Elements of resilient quantum computation 49311 Entropy and information 50011.1 Shannon entropy 50011.2 Basic properties of entropy 50211.2.1 The binary entropy 50211.2.2 The relative entropy 504xiv Contents11.2.3 Conditional entropy and mutual information 50511.2.4 The data processing inequality 50911.3 Von Neumann entropy 51011.3.1 Quantum relative entropy 51111.3.2 Basic properties of entropy 51311.3.3 Measurements and entropy 51411.3.4 Subadditivity 51511.3.5 Concavity of the entropy 51611.3.6 The entropy of a mixture of quantum states 51811.4 Strong subadditivity 51911.4.1 Proof of strong subadditivity 51911.4.2 Strong subadditivity: elementary applications 52212 Quantum information theory 52812.1 Distinguishing quantum states and the accessible information 52912.1.1 The Holevo bound 53112.1.2 Example applications of the Holevo bound 53412.2 Data compression 53612.2.1 Shannon’s noiseless channel coding theorem 53712.2.2 Schumacher’s quantum noiseless channel coding theorem 54212.3 Classical information over noisy quantum channels 54612.3.1 Communication over noisy classical channels 54812.3.2 Communication over noisy quantum channels 55412.4 Quantum information over noisy quantum channels 56112.4.1 Entropy exchange and the quantum Fano inequality 56112.4.2 The quantum data processing inequality 56412.4.3 Quantum Singleton bound 56812.4.4 Quantum error-correction, refrigeration and Maxwell’s demon 56912.5 Entanglement as a physical resource 57112.5.1 Transforming bi-partite pure state entanglement 57312.5.2 Entanglement distillation and dilution 57812.5.3 Entanglement distillation and quantum error-correction 58012.6 Quantum cryptography 58212.6.1 Private key cryptography 58212.6.2 Privacy amplification and information reconciliation 58412.6.3 Quantum key distribution 58612.6.4 Privacy and coherent information 59212.6.5 The security of quantum key distribution 593Appendices 608Appendix 1: Notes on basic probability theory 608Appendix 2: Group theory 610A2.1 Basic definitions 610A2.1.1 Generators 611A2.1.2 Cyclic groups 611A2.1.3 Cosets 612Contents xvA2.2 Representations 612A2.2.1 Equivalence and reducibility 612A2.2.2 Orthogonality 613A2.2.3 The regular representation 614A2.3 Fourier transforms 615Appendix 3: The Solovay--Kitaev theorem 617Appendix 4: Number theory 625A4.1 Fundamentals 625A4.2 Modular arithmetic and Euclid’s algorithm 626A4.3 Reduction of factoring to order-finding 633A4.4 Continued fractions 635Appendix 5: Public key cryptography and the RSA cryptosystem 640Appendix 6: Proof of Lieb’s theorem 645Bibliography 649Index 665
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