Quantum Computing: From Linear Algebra to Physical Realizations

ISBN-13: 9780750309837 / Angielski / Twarda / 2008 / 438 str.

M. Nakahara; M. Salomaa; Nakahara Nakahara

Opis

With explicit derivations, exercises, and selected solutions, this self-contained, classroom-tested book examines the theoretical aspects of quantum computing and focuses on several candidates of a working quantum computer, evaluating them according to the DiVincenzo criteria. It covers theoretical tools, such as vectors, matrices, quantum gates, and integral and Fourier transforms. The authors discuss how decoherence is an obstacle to the physical realization of a working quantum computer and how quantum error-correcting codes can sometimes overcome this. They also explore applications, including the ion trap quantum computer and the solid state realization of a quantum computer, at the forefront of the field.

Recenzje

The book is very well structured and offers good theoretical explanations reinforced by examples. As the authors mention in the Preface, the book can be used for a quantum computing course. It is also recommended to advanced undergraduate students, postgraduate students and researchers in physics, mathematics and computer science.
—Zentralblatt MATH 1185

Spis treści

From linear algebra to quantum computing
Basics of Vectors and Matrices
Vector Spaces
Linear Dependence and Independence of Vectors
Dual Vector Spaces
Basis, Projection Operator, and Completeness Relation
Linear Operators and Matrices
Eigenvalue Problems
Pauli Matrices
Spectral Decomposition
Singular Value Decomposition (SVD)
Tensor Product (Kronecker Product)
Framework of Quantum Mechanics
Fundamental Postulates
Some Examples
Multipartite System, Tensor Product, and Entangled State
Mixed States and Density Matrices
Qubits and Quantum Key Distribution
Qubits
Quantum Key Distribution (BB84 Protocol)
Quantum Gates, Quantum Circuit, and Quantum Computer
Introduction
Quantum Gates
Correspondence with Classical Logic Gates
No-Cloning Theorem
Dense Coding and Quantum Teleportation
Universal Quantum Gates
Quantum Parallelism and Entanglement
Simple Quantum Algorithms
Deutsch Algorithm
Deutsch–Jozsa Algorithm and Bernstein–Vazirani Algorithm
Simon’s Algorithm
Quantum Integral Transforms
Quantum Integral Transforms
Quantum Fourier Transform (QFT)
Application of QFT: Period-Finding
Implementation of QFT
Walsh–Hadamard Transform
Selective Phase Rotation Transform
Grover’s Search Algorithm
Searching for a Single File
Searching for d Files
Shor’s Factorization Algorithm
The RSA Cryptosystem
Factorization Algorithm
Quantum Part of Shor’s Algorithm
Probability Distribution
Continued Fractions and Order-Finding
Modular Exponential Function
Decoherence
Open Quantum System
Measurements as Quantum Operations
Examples
Lindblad Equation
Quantum Error-Correcting Codes (QECC)
Introduction
3-Qubit Bit-Flip Code and Phase-Flip Code
Shor’s 9-Qubit Code
Calderbank–Shor–Steane (CSS) 7-Qubit QECC
DiVincenzo–Shor 5-Qubit QECC
Physical realizations of quantum computing
DiVincenzo Criteria
Introduction
DiVincenzo Criteria
Physical Realizations
Beyond DiVincenzo Criteria
NMR Quantum Computer
Introduction
NMR Spectrometer
Hamiltonian
Implementation of Gates and Algorithms
Time-Optimal Control of NMR Quantum Computer
Measurements
Preparation of Pseudopure State
DiVincenzo Criteria
Trapped Ions
Introduction
Electronic States of Ion as Qubit
Ions in Paul Trap
Ion Qubit
Quantum Gates
Readout
DiVincenzo Criteria
Quantum Computing with Neutral Atoms
Introduction
Trapping Neutral Atoms
1-Qubit Gate
Quantum State Engineering of Neutral Atoms
Preparation of Entangled Neutral Atoms
DiVincenzo Criteria
Josephson Junction Qubits
Introduction
Nanoscale Josephson Junctions and SQUIDs
Charge Qubit
Flux Qubit
Quantronium
Current-Biased Qubit
Readout
Coupled Qubits
DiVincenzo Criteria
Quantum Computing with Quantum Dots
Introduction
Mesoscopic Semiconductors
Electron Charge Qubit
Electron Spin Qubit
DiVincenzo Criteria
Appendix: Solutions to Selected Exercises
Index

O autorze

Mikio Nakahara, Tetsuo Ohmi