1st Edition

Quantum Computing From Linear Algebra to Physical Realizations

By Mikio Nakahara, Tetsuo Ohmi Copyright 2008
    438 Pages 134 B/W Illustrations
    by CRC Press

    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.

    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


    Mikio Nakahara, Tetsuo Ohmi

    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