1st Edition

Quantum Error Correction and Fault Tolerant Quantum Computing

By Frank Gaitan Copyright 2013
    312 Pages 24 B/W Illustrations
    by CRC Press

    It was once widely believed that quantum computation would never become a reality. However, the discovery of quantum error correction and the proof of the accuracy threshold theorem nearly ten years ago gave rise to extensive development and research aimed at creating a working, scalable quantum computer. Over a decade has passed since this monumental accomplishment yet no book-length pedagogical presentation of this important theory exists.

    Quantum Error Correction and Fault Tolerant Quantum Computing offers the first full-length exposition on the realization of a theory once thought impossible. It provides in-depth coverage on the most important class of codes discovered to date—quantum stabilizer codes. It brings together the central themes of quantum error correction and fault-tolerant procedures to prove the accuracy threshold theorem for a particular noise error model. The author also includes a derivation of well-known bounds on the parameters of quantum error correcting code.

    Packed with over 40 real-world problems, 35 field exercises, and 17 worked-out examples, this book is the essential resource for any researcher interested in entering the quantum field as well as for those who want to understand how the unexpected realization of quantum computing is possible.

    Introduction
    Historical Background
    Classical Error Correcting Codes
    Using Quantum Systems to Store and Process Data
    Quantum Error Correcting Codes-First Pass
    Quantum Error Correcting Codes
    Quantum Operations
    Quantum Error Correcting Codes: Definitions
    Example: Calderbank-Shor-Steane [7, 1, 3] Code
    Quantum Stabilizer Codes
    General Framework
    Examples
    Alternate Formulation: Finite Geometry
    Concatenated Codes
    Quantum Stabilizer Codes: Efficient Encoding and Decoding
    Standard Form
    Encoding
    Decoding
    Fault-Tolerant Quantum Computing
    Fault-Tolerance
    Error Correction
    Encoded Operations in N(Qn) n N(S)
    Measurement
    Four-Qubit Interlude
    Multi-Qubit Stabilizer Codes
    Operations Outside N(Qn)-Toffoli Gate
    Example: [5, 1, 3J] Code
    Example: [4, 2, 2J] Code
    Accuracy Threshold Theorem
    Preliminaries
    Threshold Analysis
    Bounds on Quantum Error Correcting Codes
    Quantum Hamming Bound
    Quantum Gilbert-Varshamov Bound
    Quantum Singleton Bound
    Linear Programming Bounds for QECCs
    Entanglement Purification and QECCs
    Appendix A: Group Theory
    Fundamental Notions
    Group Action
    Mapping Groups
    Appendix B: Quantum Mechanics
    States
    Composite Systems
    Observables
    Dynamics
    Measurement and State Preparation
    Mixed States
    References
    Appendix

    Biography

    Gaitan, Frank