Principles of Quantum Computation and Information 2nd Edition by Giuliano Benenti, Giulio Casati, Davide Rossini, Giuliano Strini ISBN 9789813237223 9813237228

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ISBN 10: 9813237228
ISBN 13: 9789813237223
Author: Giuliano Benenti, Giulio Casati, Davide Rossini, Giuliano Strini

Principles of Quantum Computation and Information 2nd Edition Table of contents:

1. Introduction to classical computation

1.1 The Turing machine

1.1.1 Addition on a Turing machine

1.1.2 The Church–Turing thesis

1.1.3 The universal Turing machine

1.1.4 The probabilistic Turing machine

1.1.5 ⋆ The halting problem

1.2 The circuit model of computation

1.2.1 Binary arithmetics

1.2.2 Elementary logic gates

1.2.3 Universal classical computation

1.3 Computational complexity

1.3.1 Tractable vs

1.3.2 Complexity classes

1.3.3 ⋆ The Chernoff bound

1.4 ⋆ Computing dynamical systems

1.4.1 ⋆ Deterministic chaos

1.4.2 ⋆ Algorithmic complexity

1.5 Energy and information

1.5.1 Maxwell’s demon

1.5.2 Landauer’s principle

1.5.3 Extracting work from information

1.6 Reversible computation

1.6.1 Toffoli and Fredkin gates

1.6.2 ⋆ The billiard-ball computer

1.7 ⋆ Energy dissipation in computation

1.7.1 ⋆ Experimental realization of a Maxwell’s demon

1.7.2 ⋆ Experimental verification of Landauer’s principle

1.7.3 ⋆ Energy dissipation in real classical computer

1.7.4 ⋆ Experimental realization of reversible computers

1.7.5 ⋆ Neuromorfic computing

1.8 A guide to the bibliography

2. Introduction to quantum mechanics

2.1 The Stern–Gerlach experiment

2.2 Young’s double-slit experiment

2.3 The postulates of quantum mechanics

2.3.1 Dynamical evolution

2.3.2 Outcomes of a measurement

2.3.3 The post-measurement state

2.3.4 Heisenberg’s uncertainty principle

2.4 The EPR paradox

2.5 Bell’s inequalities

2.6 The density matrix

2.6.1 Composite systems

2.7 The Schmidt decomposition

2.8 Purification

2.9 Generalized measurements

2.9.1 POVM measurements

2.10 A guide to the bibliography

3. Quantum computation

3.1 The qubit

3.1.1 Pure qubit states: The Bloch sphere

3.1.2 Mixed qubit states: The Bloch ball

3.2 Measuring the state of a qubit

3.2.1 Pure qubit states

3.2.2 Mixed qubit states

3.3 The circuit model of quantum computation

3.4 Single-qubit gates

3.4.1 Rotations of the Bloch sphere

3.5 Controlled gates and entanglement generation

3.5.1 The Bell basis

3.6 Hamiltonian model for one- and two-qubit gates

3.7 Universal quantum gates

3.7.1 ⋆ Preparation of the initial state

3.8 Unitary errors

3.9 Function evaluation

3.10 ⋆ The quantum adder

3.11 Adiabatic theorem

3.11.1 Adiabatic condition

3.11.2 Berry phase

3.12 ⋆ Non-Abelian geometric phase

3.13 Adiabatic quantum computation

3.14 ⋆ Maximum speed of quantum gates

3.14.1 ⋆ Speed limit of an autonomous time evolution

3.14.2 ⋆ Speed limit of single-qubit gates

3.15 ⋆ Holonomic quantum computation

3.16 A guide to the bibliography

4. Quantum algorithms

4.1 Deutsch’s algorithm

4.1.1 The Deutsch–Jozsa problem

4.1.2 ⋆ An extension of Deutsch’s algorithm

4.2 Quantum search

4.2.1 Searching one item out of four

4.2.2 Searching one item out of N

4.2.3 Geometric visualization

4.2.4 Searching by adiabatic quantum evolution

4.3 The quantum Fourier transform

4.4 Quantum phase estimation

4.5 ⋆ Finding eigenvalues and eigenvectors

4.6 Period finding and Shor’s algorithm

4.7 Quantum computation of dynamical systems

4.7.1 Quantum simulation of the Schrödinger equation

4.7.2 ⋆ The quantum baker’s map

4.7.3 ⋆ The quantum sawtooth map

4.7.4 Information extraction for dynamical quantum systems

4.8 Universal quantum simulation

4.9 A guide to the bibliography

5. Quantum communication

5.1 Classical cryptography

5.1.1 The Vernam cypher

5.1.2 The public-key cryptosystem

5.1.3 The RSA protocol

5.2 The no-cloning theorem

5.2.1 Faster-than-light transmission of information?

5.2.2 ⋆ The no-signalling condition

5.2.3 ⋆ Universal quantum cloning

5.2.4 ⋆ The universal-NOT gate

5.3 Quantum cryptography

5.3.1 The BB84 protocol

5.3.2 The E91 protocol

5.4 Dense coding

5.5 Quantum teleportation

5.5.1 ⋆ Conclusive teleportation

5.6 Quantum mechanics with continuous variables

5.6.1 ⋆ General framework for Gaussian states

5.7 Quantum cryptography with continuous variables

5.8 A guide to the bibliography

6. Entanglement and non-classical correlations

6.1 Definition of entanglement

6.1.1 Basic properties

6.2 Bipartite separability criteria

6.2.1 The Peres separability criterion

6.2.2 Positive maps

6.2.3 Entanglement witnesses

6.2.4 Positive maps and witnesses

6.3 The Shannon entropy

6.3.1 Mutual information

6.4 The von Neumann entropy

6.4.1 Example 1: source of orthogonal pure states

6.4.2 Example 2: source of non-orthogonal pure states

6.5 Entanglement concentration

6.5.1 ⋆ Entanglement of a random state

6.6 Requirements for bipartite entanglement measures

6.7 Other entanglement measures

6.7.1 ⋆ Concurrence

6.7.2 ⋆ Negativity

6.8 ⋆ Multipartite entanglement

6.8.1 ⋆ Monogamy of entanglement and tangle measures

6.9 Quantum discord

6.9.1 Definition

6.9.2 Basic properties

6.9.3 Examples

6.9.4 ⋆ Other measures of quantum correlations

6.10 ⋆ Quantum discord in continuous systems

6.10.1 ⋆ Entropy of a Gaussian state

6.10.2 ⋆ Discord of a Gaussian state

6.11 ⋆ Entropies in physics

6.11.1 ⋆ Thermodynamic entropy

6.11.2 ⋆ Statistical entropy

6.11.3 ⋆ Dynamical Kolmogorov–Sinai entropy

6.12 A guide to the bibliography

7. Decoherence

7.1 The Kraus representation

7.2 Decoherence models for a single qubit

7.2.1 The quantum black box

7.2.2 Measuring a quantum operation acting on a qubit

7.2.3 Quantum circuits simulating noise channels

7.2.4 The bit-flip channel

7.2.5 The phase-flip channel

7.2.6 The bit-phase-flip channel

7.2.7 The depolarizing channel

7.2.8 Amplitude damping

7.2.9 Phase damping

7.2.10 De-entanglement

7.3 ⋆ The Bloch-Fano representation

7.3.1 ⋆ Bloch-Fano representation of a state

7.3.2 ⋆ Bloch-Fano representation of a quantum operation

7.4 The master equation

7.4.1 ⋆ Derivation of the master equation

7.4.2 The master equation and quantum operations

7.4.3 The master equation for a single qubit

7.5 ⋆ Non-Markovian quantum dynamics

7.6 Quantum to classical transition

7.6.1 Schrödinger’s cat

7.6.2 Decoherence and destruction of cat states

7.7 Decoherence and quantum measurements

7.7.1 ⋆ Weak measurements

7.7.2 ⋆ Decoherence and quantum trajectories

7.8 A guide to the bibliography

8. Quantum information theory

8.1 Classical data compression

8.1.1 Shannon’s noiseless coding theorem

8.1.2 Examples of data compression

8.1.3 Capacity of classical channels

8.2 Quantum data compression

8.2.1 Schumacher’s quantum noiseless coding theorem

8.2.2 Compression of an n-qubit message

8.2.3 Example 1: two-qubit messages

8.2.4 Example 2: three-qubit messages

8.3 Accessible information

8.3.1 The Holevo bound

8.3.2 Example 1: two non-orthogonal pure states

8.3.3 ⋆ Example 2: three non-orthogonal pure states

8.4 Capacities of quantum channels

8.4.1 Classical capacity

8.4.2 Quantum capacity

8.5 ⋆ Quantum memory channels

8.6 A guide to the bibliography

9. Quantum error correction

9.1 The three-qubit bit-flip code

9.2 The three-qubit phase-flip code

9.3 The nine-qubit Shor code

9.4 General properties of quantum error correction

9.4.1 The quantum Hamming bound

9.5 Stabilizer coding

9.5.1 The nine-qubit Shor code revisited

9.5.2 ⋆ General formalism for stabilizer codes

9.5.3 ⋆ Logical operators for stabilizer codes

9.6 ⋆ The five-qubit code

9.7 Decoherence-free subspaces

9.7.1 ⋆ Conditions for decoherence-free dynamics

9.7.2 ⋆ The spin-boson model

9.8 ⋆ Dynamical decoupling

9.8.1 ⋆ Explicit form of control Hamiltonian

9.9 ⋆ The Zeno effect

9.10 Fault-tolerant quantum computation

9.10.1 Avoidance of error propagation

9.10.2 Fault-tolerant quantum gates

9.10.3 The noise threshold for quantum computation

9.11 A guide to the bibliography

10. Principles of experimental implementations of quantum protocols

10.1 Cavity quantum electrodynamics

10.1.1 Interaction of a two-level atom with a classical field

10.1.2 The Jaynes–Cummings model

10.1.3 Rabi oscillations

10.1.4 Entanglement generation

10.2 The ion-trap quantum computer

10.2.1 The Paul trap

10.2.2 Laser pulses

10.3 Solid-state qubits

10.3.1 Spins in semiconductors

10.3.2 Quantum dots

10.3.3 Superconducting qubit circuits

10.4 Quantum communication with photons

10.4.1 Linear optics

10.4.2 Non-linear optics and probabilistic gates

10.4.3 Experimental quantum-key distribution

10.5 Problems and prospects

10.6 A guide to the bibliography

11. Quantum information in many-body systems

11.1 Quantum simulators

11.1.1 Ultracold atoms

11.1.2 Arrays of coupled QED cavities

11.2 Emergence of quantum correlations

11.2.1 The Hubbard model

11.3 The spin-1/2 quantum Ising chain

11.3.1 Jordan–Wigner transformation

11.3.2 Diagonalization of the Ising chain

11.3.3 Two-spin concurrence

11.3.4 Entanglement block entropy

11.3.5 The Ising model revisited: Kitaev chain

11.4 Area-law scaling of the entanglement

11.5 Matrix product states

11.5.1 Examples of MPS wave functions

11.6 Graphical representation of matrix product states

11.6.1 Expectation values of observables

11.6.2 ⋆ Scaling of correlation functions with the distance

11.6.3 Gauge freedom

11.6.4 Schmidt decomposition of a MPS

11.7 Ground-state search in the Hilbert space corner

11.7.1 Density-matrix renormalization group

11.7.2 ⋆ DMRG as a variational optimization over the MPS class

11.8 Time evolution of matrix product states

11.8.1 Finite-temperature calculations

11.8.2 Mixed-state time evolution

11.9 ⋆ General tensor-network structures

11.9.1 ⋆ Projected entangled pair states

11.9.2 ⋆ Hierarchical tensor networks

11.10 A guide to the bibliography

Conclusions and prospects

Appendix A Elements of linear algebra

A.1 Finite-dimensional vector spaces

A.1.1 Basic properties of vector spaces

A.1.2 Inner product and norm of a vector

A.1.3 Linear independence and the notion of basis

A.1.4 Linear operators

A.1.5 Tensor product

A.1.6 Matrix decompositions

A.1.7 Symplectic decompositions

A.2 Infinite-dimensional vector spaces

A.2.1 Discrete and continuous bases

A.2.2 The Dirac delta function

A.2.3 Orthonormality and completeness relations

A.2.4 Position and momentum representations

A.2.5 Position and momentum operators

Appendix B Solutions to the exercises

B.1 Chapter 1

B.2 Chapter 2

B.3 Chapter 3

B.4 Chapter 4

B.5 Chapter 5

B.6 Chapter 6

B.7 Chapter 7

B.8 Chapter 8

B.9 Chapter 9

B.10 Chapter 10

B.11 Chapter 11

B.12 Appendix A

Bibliography

Index

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Tags: Giuliano Benenti, Giulio Casati, Davide Rossini, Giuliano Strini, Quantum Computation, Information