Quantum Error Correction and Fault Tolerant Quantum Computing

Quantum Error Correction and Fault Tolerant Quantum Computing

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Features

  • Provides a careful explanation of quantum error correction and the accuracy threshold theorem
  • Demonstrates how to construct the most important class of codes: quantum stabilizer codes
  • Explains how fault-tolerant quantum operations can be implemented on encoded quantum information
  • Includes an explanation of how accuracy threshold theorem is proved for particular noise models

Summary

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.

Table of Contents

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

 
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