# Quantum Computation and Information

Share this page *Edited by *
*Samuel J. Lomonaco, Jr.; Howard E. Brandt*

This book is a collection of papers given by invited speakers at the AMS
Special Session on Quantum Computation and Information held at the January 2000
Annual Meeting of the AMS in Washington, DC.

The papers in this volume give readers a broad introduction to the many
mathematical research challenges posed by the new and emerging field of quantum
computation and quantum information. Of particular interest is a long paper by
Lomonaco and Kauffman discussing mathematical and computational aspects of the
so-called hidden subgroup algorithm.

This book is intended to help readers recognize that, as a result of this
new field of quantum information science, mathematical research opportunities
abound in such diverse mathematical fields as algebraic coding theory,
algebraic geometry, algebraic topology, communication theory, control theory,
cryptography, differential geometry, differential topology, dynamical systems,
game theory, group theory, information theory, number theory, operator theory,
robotics, theory of computation, mathematical logic, mathematical physics, and
more. It is hoped that this book will act as a catalyst to encourage members of
the mathematical community to take advantage of the many mathematical research
opportunities arising from the “grand challenge” of Quantum
Information Science.

This book is the companion volume to Quantum Computation: A
Grand Mathematical Challenge for the Twenty-First Century and the
Millennium, PSAPM/58,
Volume 58 in the Proceedings of Symposia in Applied Mathematics series.

# Table of Contents

## Quantum Computation and Information

- Contents iii4 free
- Preface v6 free
- Gilles Brassard Awarded Pot de Vin Prize vii8 free
- List of Participants viii9 free
- Space searches with a quantum robot 110 free
- Perturbation theory and numerical modeling of quantum logic operations with a large number of qubits 1322
- Inconclusive rate with a positive operator valued measure 4352
- Quantum amplitude amplification and estimation 5362
- Manipulating the entanglement of one copy of a two-particle pure entangled state 7584
- Geometric algebra in quantum information processing 8190
- Quantum computing and the Jones polynomial 101110
- Quantum hidden subgroup algorithms: A mathematical perspective 139148
- Part 1. Preamble 141150
- Part 2. Algebraic Preliminaries 148157
- Part 3. QRandφ(): The Progenitor of All QHSAs 154163
- Part 4. Vintage Simon Algorithms 157166
- Part 5. Vintage Shor Algorithms 164173
- 11. Vintage Shor quantum hidden subgroup algorithms(QHSAs) 164173
- 12. Direct summand structure 165174
- 13. Vintage Shor QHSAs with group probe Ã= ZQ. 168177
- 14. Finding Shor transversals for vintage ZQ Shor algorithms 169178
- 15. Maximal Shor transversals 172181
- 16. Identifying characters of cyclic groups with points on the unit circle S1 in the complex plane C 173182
- 17. Group norms 174183
- 18. Vintage ZQ Shor QHSAs (Cont.) 176185
- 19. When are characters of Ã= ZQ close to some character of a maximal cyclic subgroup ZP of Hφ? 181190
- 20. Summary of Vintage ZQ Shor QHSAs 183192
- 21. A cursory analysis of complexity 185194
- 22. Two alternative vintage ZQ Shor algorithms 188197

- Part 6. Epilogue 192201
- References 200209

- Improved two-party and multi-party purification protocols 203212
- Quantum games and quantum algorithms 213222
- A proof that measured data and equations of quantum mechanics can be linked only by guesswork 221230
- Quantum computation by geometrical means 245254
- Pauli exchange and quantum error correction 251260
- Relative entropy in quantum information theory 265274
- An unentangled Gleason's theorem 291300
- Entangled chains 299308