Quantum Computing : From Linear Algebra to Physical Realizations book cover
1st Edition

Quantum Computing
From Linear Algebra to Physical Realizations

ISBN 9780750309837
Published March 11, 2008 by CRC Press
438 Pages 134 B/W Illustrations

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Book Description

Covering both theory and progressive experiments, Quantum Computing: From Linear Algebra to Physical Realizations explains how and why superposition and entanglement provide the enormous computational power in quantum computing. This self-contained, classroom-tested book is divided into two sections, with the first devoted to the theoretical aspects of quantum computing and the second focused on several candidates of a working quantum computer, evaluating them according to the DiVincenzo criteria.

Topics in Part I

  • Linear algebra
  • Principles of quantum mechanics
  • Qubit and the first application of quantum information processing—quantum key distribution
  • Quantum gates
  • Simple yet elucidating examples of quantum algorithms
  • Quantum circuits that implement integral transforms
  • Practical quantum algorithms, including Grover’s database search algorithm and Shor’s factorization algorithm
  • The disturbing issue of decoherence
  • Important examples of quantum error-correcting codes (QECC)

Topics in Part II

  • DiVincenzo criteria, which are the standards a physical system must satisfy to be a candidate as a working quantum computer
  • Liquid state NMR, one of the well-understood physical systems
  • Ionic and atomic qubits
  • Several types of Josephson junction qubits
  • The quantum dots realization of qubits

Looking at the ways in which quantum computing can become reality, this book delves into enough theoretical background and experimental research to support a thorough understanding of this promising field.

Table of Contents

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
Quantum Key Distribution (BB84 Protocol)
Quantum Gates, Quantum Circuit, and Quantum Computer
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
Open Quantum System
Measurements as Quantum Operations
Lindblad Equation
Quantum Error-Correcting Codes (QECC)
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
DiVincenzo Criteria
Physical Realizations
Beyond DiVincenzo Criteria
NMR Quantum Computer
NMR Spectrometer
Implementation of Gates and Algorithms
Time-Optimal Control of NMR Quantum Computer
Preparation of Pseudopure State
DiVincenzo Criteria
Trapped Ions
Electronic States of Ion as Qubit
Ions in Paul Trap
Ion Qubit
Quantum Gates
DiVincenzo Criteria
Quantum Computing with Neutral Atoms
Trapping Neutral Atoms
1-Qubit Gate
Quantum State Engineering of Neutral Atoms
Preparation of Entangled Neutral Atoms
DiVincenzo Criteria
Josephson Junction Qubits
Nanoscale Josephson Junctions and SQUIDs
Charge Qubit
Flux Qubit
Current-Biased Qubit
Coupled Qubits
DiVincenzo Criteria
Quantum Computing with Quantum Dots
Mesoscopic Semiconductors
Electron Charge Qubit
Electron Spin Qubit
DiVincenzo Criteria
Appendix: Solutions to Selected Exercises

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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