Quantum computer science: an introduction

Preface

A note on references

1. Cbits and Qbits

1.1. What is a quantum computer?

1.2. Cbits and their states

1.3. Reversible operations on Cbits

1.4. Manipulating operations on Cbits

1.5. Qbits and their states

1.6. Reversible operations on Qbits

1.7. Circuit diagrams

1.8. Measurement gates and the Born rule

1.9. The generalized Born rule

1.10. Measurement gates and state preparation

1.11. Constructing arbitrary 1- and 2-Qbit states

1.12. Summary : Obits versus Cbits

2. General features and some simple examples

2.1. The general computational process

2.2. Deutsch's problems

2.3. Why additional Qbits needn't mess things up

2.4. The Bernstein-Vazirani problem

2.5. Simon's problem

2.6. Constructing Toffoli gates

3. Breaking RSA encryption

3.1. Period finding, factoring, and cryptography

3.2. Number-theoretic preliminaries

3.3. RSA encryption

3.4. Quantum period finding : preliminary remarks

3.5. The quantum Fourier transform

3.6. Eliminating the 2-Qbit gates

3.7. Finding the period

3.8. Calculating the periodic function

3.9. The unimportance of small phase errors

3.10. Period finding and factoring

4. Searching with a quantum computer

4.1. The nature of the search

4.2. The Grover iteration

4.3. How to construct W

4.4. Generalization to several special numbers

4.5. Searching for one out of four items

5. Quantum error correction

5.1. The miracle of quantum error correction

5.2. A simplified example

5.3. The physics of error generation

5.4. Diagnosing error syndromes

5.5. The 5-Qbit error-correcting code

5.6. The 7-Qbit error-correcting code

5.7. Operations on 7-Qbit codewords

5.8. A 7-Qbit encoding circuit

5.9. A 5-Qbit encoding circuit

6. Protocols that use just a few Qbits

6.1. Bell states

6.2. Quantum cryptography

6.3. Bit commitment

6.4. Quantum dense coding

6.5. Teleportation

6.6. The GHZ puzzle

Appendices

A. Vector spaces : basic properties and Dirac notation

B. Structure of the general 1-Qbit unitary transformation

C. Structure of the general 1-Qbit state

D. Spooky action at a distance

E. Consistency of the generalized Born rule

F. Other aspects of Deutsch's problem

G. The probability of success in Simon's problem

H. One way to make a cNOT gate

I. A little elementary group theory

J. Some simple number theory

K. Period finding and continued fractions

L. Better estimates of success in period finding

M. Factoring and period finding

N. Shor's 9-Qbit error-correcting code

O. A circuit-diagrammatic treatment of the 7-Qbit code

P. On bit commitment

Index.