Quantum Computation and Quantum Error Prevention

Table of Contents

1. Chapter 1 - Introduction
   1. An Introduction to Quantum Computation
   2. Bits and qubits: An Introduction
   3. Obstacles to Building a Reliable Quantum Computer
2. Chapter 2 - Qubits and Collections of Qubits
   1. Qubit States
   2. Qubit Gates
   3. The Pauli Matrices
   4. States of Many Qubits
   5. Quantum Gates for Many Qubits
   6. Measurement
3. Chapter 3 - Physics of Quantum Information
   1. Schrodinger’s Equation
   2. Density Matrix for Pure States
   3. Measurements Revisited
   4. Density Matrix for a Mixed State
   5. Expectation Values
   6. Types of Two-state Systems
   7. Entangled States
   8. Entanglement: Extentions and Open Problems
4. Chapter 4 - Quantum Information: Basic Principles and Simple Examples
   1. Uncertainty Principle
   2. Quantum Dense Coding
   3. Teleporting a Quantum State
   4. No Cloning!
   5. QKD: BB84
5. Chapter 5 - Quantum Computation
   1. Quantum Computation Basics
   2. Deutsch-Josza Algorithm
   3. Simon’s Algorithm
   4. Shor’s Algorithm
   5. Grover’s Algorithm
6. Chapter 6 - Experiments
7. Chapter 7 - Noise in Quantum Systems
   1. Operator-Sum Decomposition
   2. Sudarshan Representation
   3. Superoperators: (more or less) Standard representation
   4. Notes
8. Chapter 8 - Conclusions
   1. What have we learned?

1. Appendix B - Basic Probability Concepts
2. Appendix C - Complex Numbers
   1. Complex Numbers
3. Appendix D - Vectors and Linear Algebra
   1. Vectors
   2. Linear Algebra
   3. More Dirac Notation
   4. Transformations
   5. Eigenvalues, and Eigenvectors
   6. Tensor Products
4. Appendix E - Group Theory
   1. Definitions and Examples
   2. Properties of Groups with Finite Order
   3. Infinite Order Groups: Lie Groups
5. Appendix F - Density Operator: Extensions
   1. The Coherence Vector
   2. The Coherence Vector: Other Conventions
   3. The Density Matrix for Two Qubits

Preface

These are notes to accompany the course on quantum computing taught at Southern Illinois University. Until otherwise noted these notes are a work in progress. Therefore, if there are any suggestions, questions, comments, errors, etc. please let me know so that appropriate modifications can be made.

There are several good books on quantum computing. This is not an attempt to displace them or replace them. The concentration on error prevention and noise is likely different than what has been done before and the desire is to have them rather self-contained so that few, if any, other resources are absolutely required. However, it is strongly recommended that other resources are consulted along with these notes since they are unlikely to be a complete resource any time soon. Furthermore, the are not likely to be a better resource for many topics which are better and more thoroughly treated elsewhere.

The objective to provide course material which will be introductory enough to enable an undergraduate science major with some background in linear algebra to follow the course. This includes physics, mathematics, computer science, and engineering majors. A good place to start is N. David Mermin’s book [11].

N. David Mermin’s book [11], David J. Giffiths’s book [8], and (of course) Michael Nielsen and Isaac Chuang’s book [13] have all greatly influenced these notes. They have influenced many parts even if they are not explicitly cited. In the case of Griffiths’s book, I taught an undergraduate quantum mechanics course the semester before I taught this course. Therefore many of the examples, pedagogy, and exposition were influenced by his book, which I very much appreciate.

By the time these notes, and the course, are finished, several people will have contributed. These are listed here:

• Mark S. Byrd
• C. Allen Bishop
• Nayeli Zuniga-Hansen
• Seyoum Tsige
• Max Herlache
• Philip Feinsilver, itex2mml, java script, etc.
• Jalal Alowibdi
• Sarah Harvey
• Kevin Reuter