This book provides an introduction to the main ideas and techniques of the field of quantum computation and quantum information. The rapid rate of progress in this field and its cross-disciplinary nature have made it difficult for newcomers to obtain a broad overview of the most important techniques and results of the field. Our purpose in this book is therefore twofold. First， we introduce the background material in computer science， mathematics and physics necessary to understand quantum computation and quantum information. This is done at a level comprehensible to readers with a background at least the equal of a beginning graduate student in one or more of these three disciplines; the most important requirements are a certain level of mathematical maturity， and the desire to learn about quantum computation and quantum information. The second purpose of the book is to develop in detail the central results of quantum computation and quantum information. With thorough study the reader should develop a working understanding of the fundamental tools and results of this exciting field， either as part of their general education， or as a prelude to independent research in quantum computation and quantum information.
作者簡介
Michael nielsen is a Postdoctoral research fellow at the university of Queensland.he was born in Brisbane，Austalia，and received his education at the University of Queensland，obtaining postgraducate degrees in mathematics and physics before receiving his Ph.D.in Physics as a Fulbright Scholar at the University of new mexico.he has held a Vistiong position at the los Alamos national Laboratory，and was the Tolman postdoctoral Fellow at the California Institute of technology.
圖書目錄
Preface Acknowledgements Nomenclature and notation Part I Fundamental concepts Introduction and overview 1.1 Global perspectives 1.1.1 History of quantum computation and quantum information 1.1.2 Future directions 1.2 Quantum bits 1.2.1 Multiple qubits 1.3 Quantum computation 1.3.1 Single qubit gates 1.3.2 Multiple qubit gates 1.3.3 Measurements in bases other than the computational basis 1.3.4 Quantum circuits 1.3.5 Qubit copying circuit 1.3.6 Example: Bell states 1.3.7 Example: quantum teleportation 1.4 Quantum algorithms 1.4.1 Classical computations on a quantum computer 1.4.2 Quantum parallelism 1.4.3 Deutsch''s algorithm 1.4.4 The Deutsch-Jozsa algorithm 1.4.5 Quantum algorithms summarized 1.5 Experimental quantum information processing 1.5.1 The Stern-Gerlach experiment 1.5.2 Prospects for practical quantum information processing 1.6 Quantum information 1.6.1 Quantum information theory: example problems 1.6.2 Quantum information in a wider context 2 Introduction to quantum mechanics 2.1 Linear algebra 2.1.1 Bases and linear independence 2.1.2 Linear operators and matrices 2.1.3 The Pauli matrices 2.1.4 Inner products 2.1.5 Eigenvectors and eigenvalues 2.1.6 Adjoints and Hermitian operators 2.1.7 Tensor products 2.1.8 Operator functions 2.1.9 The commutator and anti-commutator 2.1.10 The polar and singular value decompositions 2.2 The postulates of quantum mechanics 2.2.1 State space 2.2.2 Evolution 2.2.3 Quantum measurement 2.2.4 Distinguishing quantum states 2.2.5 Projective measurements 2.2.6 POVM measurements 2.2.7 Phase 2.2.8 Composite systems 2.2.9 Quantum mechanics: a global view 2.3 Application: superdense coding 2.4 The density operator 2.4.1 Ensembles of quantum states 2.4.2 General properties of the density operator 2.4.3 The reduced density operator 2.5 The Schmidt decomposition and purifications 2.6 EPR and the Bell inequality 3 Introduction to computer science 3.1 Models for computation 3.1.1 Turing machines 3.1.2 Circuits 3.2 The analysis of computational problems 3.2.1 How to quantify computational resources 3.2.2 Computational complexity 3.2.3 Decision problems and the complexity classes P and NP 3.2.4 A plethora of complexity classes 3.2.5 Energy and computation 3.3 Perspectives on computer science Part II Quantum computation 4 Quantum circuits 4.1 Quantum algorithms 4.2 Single qubit operations 4.3 Controlled operations 4.4 Measurement 4.5 Universal quantum gates 4.5.1 Two-level unitary gates are universal 4.5.2 Single qubit and CNOT gates are universal 4.5.3 A discrete set of universal operations 4.5.4 Approximating arbitrary unitary gates is generically hard 4.5.5 Quantum computational complexity 4.6 Summary of the quantum circuit model of computation 4.7 Simulation of quantum systems 4.7.1 Simulation in action 4.7.2 The quantum simulation algorithm 4.7.3 An illustrative example 4.7.4 Perspectives on quantum simulation 5 The quantum Fourier transform and its applications 5.1 The quantum Fourier transform 5.2 Phase estimation 5.2.1 Performance and requirements 5.3 Applications: order-finding and factoring 5.3.1 Application: order-finding 5.3.2 Application: factoring 5.4 General applications of the quantum Fourier transform 5.4.1 Period-finding 5.4.2 Discrete logarithms 5.4.3 The hidden subgroup problem 5.4.4 Other quantum algorithms Quantum search algorithms 6.1 The quantum search algorithm 6.1.1 The oracle 6.1.2 The procedure 6.1.3 Geometric visualization 6.1.4 Performance 6.2 Quantum search as a quantum simulation 6.3 Quantum counting 6.4 Speeding up the solution of NP--complete problems 6.5 Quantum search of an unstructured database 6.6 Optimality of the search algorithm 6.7 Black box algorithm limits 7 Quantum computers: physical realization 7.1 Guiding principles 7.2 Conditions for quantum computation 7.2.1 Representation of quantum information 7.2.2 Performance of unitary transformations 7.2.3 Preparation of fiducial initial states 7.2.4 Measurement of output result 7.3 Harmonic oscillator quantum computer 7.3.1 Physical apparatus 7.3.2 The Hamiltonian 7.3.3 Quantum computation 7.3.4 Drawbacks 7.4 Optical photon quantum computer 7.4.1 Physical apparatus 7.4.2 Quantum computation 7.4.3 Drawbacks 7.5 Optical cavity quantum electrodynamics 7.5.1 Physical apparatus 7.5.2 The Hamiltonian 7.5.3 Single-photon single-atom absorption and refraction 7.5.4 Quantum computation 7.6 Ion traps 7.6.1 Physical apparatus 7.6.2 The Hamiltonian 7.6.3 Quantum computation 7.6.4 Experiment 7.7 Nuclear magnetic resonance 7.7.1 Physical apparatus 7.7.2 The Hamiltonian 7.7.3 Quantum computation 7.7.4 Experiment 7.8 Other implementation schemes Part III Quantum information 8 Quantum noise and quantum operations 8.1 Classical noise and Markov processes 8.2 Quantum operations 8.2.1 Overview 8.2.2 Environments and quantum operations 8.2.3 Operator-sum representation 8.2.4 Axiomatic approach to quantum operations 8.3 Examples of quantum noise and quantum operations 8.3.1 Trace and partial trace 8.3.2 Geometric picture of single qubit quantum operations 8.3.3 Bit flip and phase flip channels 8.3.4 Depolarizing channel 8.3.5 Amplitude damping 8.3.6 Phase damping 8.4 Applications of quantum operations 8.4.1 Master equations 8.4.2 Quantum process tomography 8.5 Limitations of the quantum operations formalism 9 Distance measures for quantum information 9.1 Distance measures for classical information 9.2 How close are two quantum states 9.2.1 Trace distance 9.2.2 Fidelity 9.2.3 Relationships between distance measures 9.3 How well does a quantum channel preserve information 10 Quantum error-correction 10.1 Introduction 10.1.1 .The three qubit bit,flip code 10.1.2 Three qubit phase flip code 10.2 The Shor code 10.3 Theory of quantum error-correction 10.3.1 Discretization of, the errors 10.3.2 Independent error models 10.3.3 Degenerate codes 10.3.4 The quantum Hamming bound 10.4 Constructing quantum codes 10.4.1 Classical linear codes 10.4.2 Calderbank-Shor-Steane codes 10.5 Stabilizer codes 10.5.1 The stabilizer formalism 10.5.2 Unitary gates and the stabilizer formalism 10.5.3 Measurement in the stabilizer formalism 10.5.4 The Gottesman-Knill theorem 10.5.5 Stabilizer code constructions 10.5.6 Examples 10.5.7 Standard form for a stabilizer code 10.5.8 Quantum circuits for encoding, decoding, and correction 10.6 Fault-tolerant quantum computation 10.6.1 Fault-tolerance: the big picture 10.6.2 Fault-tolerant quantum logic 10.6.3 Fault-tolerant measurement 10.6.4 Elements of resilient quantum computation 11 Entropy and information 11.1 Shannon entropy 11.2 Basic properties of entropy 11.2.1 The binary entropy 11.2.2 The relative entropy 11.2.3 Conditional entropy and mutual information 11.2.4 The data processing inequality 11.3 Von Neumann entropy 11.3.1 Quantum relative entropy 11.3.2 Basic properties of entropy 11.3.3 Measurements and entropy 11.3.4 Subadditivity 11.3.5 Concavity of the entropy 11.3.6 The entropy of a mixture of quantum states 11.4 Strong subadditivity 11.4.1 Proof of strong subadditivity 11.4.2 Strong subadditivity: elementary applications 12 Quantum information theory 12.1 Distinguishing quantum states and the accessible information 12.1.1 The Holevo bound 12.1.2 Example applications of the Holevo bound 12.2 Data compression 12.2.1 Shannon''s noiseless channel coding theorem 12.2.2 Schumacher''s quantum noiseless channel coding theorem 12.3 Classical information over noisy quantum channels 12.3.1 Communication over noisy classical channels 12.3.2 Communication over noisy quantum channels 12.4 Quantum information over noisy quantum channels 12.4.1 Entropy exchange and the quantum Fano inequality 12.4.2 The quantum data processing inequality 12.4.3 Quantum Singleton bound 12.4.4 Quantum error-correction, refrigeration and Maxwell''s demon 12.5 Entanglement as a physical resource 12.5.i Transforming bi-partite pure state entanglement 12.5.2 Entanglement distillation and dilution 12.5.3 Entanglement distillation and quantum error-correction 12.6 Quantum cryptography 12.6.1 Private key cryptography 12.6.2 Privacy amplification and information reconciliation 12.6.3 Quantum key distribution 12.6.4 Privacy and coherent information 12.6.5 The security of quantum key distribution Appendices Appendix 1: Notes on basic probability theory Appendix 2: Group theory A2.1 Basic definitions A2.1.1 Generators A2.1.2 Cyclic groups A2.1.3 Cosets A2.2 Representations A2.2.1 Equivalence and reducibility A2.2.2 0rthogonality A2.2.3 The regular representation A2.3 Fourier transforms Appendix 3: The Solovay-Kitaev theorem Appendix 4: Number theory A4.1 Fundamentals A4.2 Modular arithmetic and Euclid''s algorithm A4.3 Reduction of factoring to order-finding A4.4 Continued fractions Appendix 5: Public key cryptography and the RSA cryp~ Appendix 6: Proof of Lieb''s theorem Bibliography Index
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