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Table of Contents

Part I Fundamental concepts

  • 1 Introduction and overview
  • [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?]
  • [6 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 282]
    • [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]
      • [12.5.1 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 Orthogonality]
      • [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 cryptosystem]
  • [Appendix 6: Proof of Lieb’s theorem]