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
鄂公网安备 42010302001612号 