Quantum Computation and Quantum Error Prevention

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Many people have contributed to these notes which are based on a class taught by Mark Byrd in the Spring of 2009. Please see the Notes and Credits for further information and a list of contributors.

Table of Contents

   Preface

1. Chapter 1 - Introduction
   1. Introduction
   2. An Introduction to Quantum Computation
   3. Bits and qubits: An Introduction
   4. Obstacles to Building a Reliable Quantum Computer
2. Chapter 2 - Qubits and Collections of Qubits
   1. Introduction
   2. Qubit States
   3. Qubit Gates
   4. The Pauli Matrices
   5. States of Many Qubits
   6. Quantum Gates for Many Qubits
   7. Measurement
3. Chapter 3 - Physics of Quantum Information
   1. Introduction
   2. Schrodinger’s Equation
   3. Density Matrix for Pure States
   4. Measurements Revisited
   5. Density Matrix for a Mixed State
   6. Expectation Values
4. Chapter 4 - Entanglement
   1. Introduction
   2. Entangled Pure States
   3. Entangled Mixed States
   4. Extensions and Open Problems
5. Chapter 5 - Quantum Information: Basics and Simple Examples
   1. Introduction
   2. No Cloning!
   3. Uncertainty Principle
   4. Quantum Dense Coding
   5. Teleporting a Quantum State
   6. QKD: BB84
6. Chapter 6 - Noise in Quantum Systems
   1. Introduction
   2. SMR Representation or Operator-Sum Representation
   3. Modelling Open System Evolution
   4. Unitary Degree of Freedom in the OSR
   5. Examples
   6. Notes
7. Chapter 7 - Quantum Error Correcting Codes
   1. Introduction
   2. Shor's Nine-Qubit Quantum Error Correcting Code
   3. Quantum Error Correcting Codes: General Properties
   4. Stabilizer Codes
   5. CSS codes
   6. Steane's Seven Qubit Code
8. Chapter 8 - Decoherence-Free/Noiseless Subsystems
   1. Introduction
   2. General Considerations
   3. DNS Examples
   4. Quantum Computing on a DNS
   5. QC Examples
9. Chapter 9 - Dynamical Decoupling Controls
   1. Introduction
   2. General Conditions
   3. The Magnus Expansion
   4. First-Order Theory
   5. The Single Qubit Case
   6. Extensions
10. Chapter 10 - Fault-Tolerant Quantum Computing
    1. Introduction
    2. Requirements for Fault-Tolerance
    3. Concatenated Codes
    4. Fault-Tolerant Quantum Computing for the Steane Code
11. Chapter 11 - Hybrid Methods of Quantum Error Prevention
    1. Introduction
    2. General Principles for Combining Error Prevention Methods
    3. Examples of Hybrid Error Prevention
12. Chapter 12 - Conclusions and Further Study

Appendices

1. Appendix A - Basic Probability Concepts
2. Appendix B - Complex Numbers
3. Appendix C - Vectors and Linear Algebra
   1. Introduction
   2. Vectors
   3. Linear Algebra: Matrices
   4. More Dirac Notation
   5. Transformations
   6. Eigenvalues and Eigenvectors
   7. Tensor Products
4. Appendix D - Group Theory
   1. Introduction
   2. Definitions and Examples
   3. Comparing Groups: Homomorphisms and Isomorphisms
   4. Discussion
   5. Applications to Physics
   6. A Little Representation Theory
   7. Infinite Order Groups: Lie Groups
   8. More Representation Theory
5. Appendix E - Density Operator: Extensions
   1. Introduction
   2. An N-dimensional Generalization of the Polarization Vector
   3. The Density Matrix for Two Qubits
6. Appendix F - Classical Error Correcting Codes
   1. Introduction
   2. Binary Operations
   3. Definitions and Basics
   4. Linear Codes
   5. Errors
   6. The Disjointness Condition and Correcting Errors
   7. The Hamming Bound
7. Appendix G - NOTES and CREDITS
8. Extensions
9. Testing

Glossary

Glossary

Index

Index

Bibliography

Bibliography

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Much of this material is based upon work supported by the National Science Foundation under Grant No. 0545798. However, any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.