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Quantum Computer Science

Book Description

In the 1990's it was realized that quantum physics has some spectacular applications in computer science. This book is a concise introduction to quantum computation, developing the basic elements of this new branch of computational theory without assuming any background in physics. It begins with an introduction to the quantum theory from a computer-science perspective. It illustrates the quantum-computational approach with several elementary examples of quantum speed-up, before moving to the major applications: Shor's factoring algorithm, Grover's search algorithm, and quantum error correction. The book is intended primarily for computer scientists who know nothing about quantum theory, but will also be of interest to physicists who want to learn the theory of quantum computation, and philosophers of science interested in quantum foundational issues. It evolved during six years of teaching the subject to undergraduates and graduate students in computer science, mathematics, engineering, and physics, at Cornell University.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Contents
  5. Preface
  6. A note on references
  7. 1 Cbits and Qbits
    1. 1.1 What is a quantum computer?
    2. 1.2 Cbits and their states
    3. 1.3 Reversible operations on Cbits
    4. 1.4 Manipulating operations on Cbits
    5. 1.5 Qbits and their states
    6. 1.6 Reversible operations on Qbits
    7. 1.7 Circuit diagrams
    8. 1.8 Measurement gates and the Born rule
    9. 1.9 The generalized Born rule
    10. 1.10 Measurement gates and state preparation
    11. 1.11 Constructing arbitrary 1- and 2-Qbit states
    12. 1.12 Summary: Qbits versus Cbits
  8. 2 General features and some simple examples
    1. 2.1 The general computational process
    2. 2.2 Deutsch’s problem
    3. 2.3 Why additional Qbits needn’t mess things up
    4. 2.4 The Bernstein–Vazirani problem
    5. 2.5 Simon’s problem
    6. 2.6 Constructing Toffoli gates
  9. 3 Breaking RSA encryption
    1. 3.1 Period finding, factoring, and cryptography
    2. 3.2 Number-theoretic preliminaries
    3. 3.3 RSA encryption
    4. 3.4 Quantum period finding: preliminary remarks
    5. 3.5 The quantum Fourier transform
    6. 3.6 Eliminating the 2-Qbit gates
    7. 3.7 Finding the period
    8. 3.8 Calculating the periodic function
    9. 3.9 The unimportance of small phase errors
    10. 3.10 Period finding and factoring
  10. 4 Searching with a quantum computer
    1. 4.1 The nature of the search
    2. 4.2 The Grover iteration
    3. 4.3 How to constructW
    4. 4.4 Generalization to several special numbers
    5. 4.5 Searching for one out of four items
  11. 5 Quantum error correction
    1. 5.1 The miracle of quantum error correction
    2. 5.2 A simplified example
    3. 5.3 The physics of error generation
    4. 5.4 Diagnosing error syndromes
    5. 5.5 The 5-Qbit error-correcting code
    6. 5.6 The 7-Qbit error-correcting code
    7. 5.7 Operations on 7-Qbit codewords
    8. 5.8 A 7-Qbit encoding circuit
    9. 5.9 A 5-Qbit encoding circuit
  12. 6 Protocols that use just a few Qbits
    1. 6.1 Bell states
    2. 6.2 Quantum cryptography
    3. 6.3 Bit commitment
    4. 6.4 Quantum dense coding
    5. 6.5 Teleportation
    6. 6.6 The GHZ puzzle
  13. Appendices
    1. A. Vector spaces: basic properties and Dirac notation
    2. B. Structure of the general 1-Qbit unitary transformation
    3. C. Structure of the general 1-Qbit state
    4. D. Spooky action at a distance
    5. E. Consistency of the generalized Born rule
    6. F. Other aspects of Deutsch’s problem
    7. G. The probability of success in Simon’s problem
    8. H. One way to make a cNOT gate
    9. I. A little elementary group theory
    10. J. Some simple number theory
    11. K. Period finding and continued fractions
    12. L. Better estimates of success in period finding
    13. M. Factoring and period finding
    14. N. Shor’s 9-Qbit error-correcting code
    15. O. A circuit-diagrammatic treatment of the 7-Qbit code
    16. P. On bit commitment
  14. Index