CONTEMPORARY MATHEMATICS

305

Quantum Computation and Information

AMS Special Session Quantum Computation and Information

January 19-21, 2000 Washington , D.C.

Samuel J. Lomonaco, Jr. Howard E. Brandt

Editors

CoNTEMPORARY MATHEMATICS

305

Quantum Computation and Information

AMS Special Session Quantum Computation and Information

January 19-21, 2000 Washington, D.C.

Samuel J. Lomonaco, Jr. Howard E. Brandt

Editors

American Mathematical Society Providence, Rhode Island

http://dx.doi.org/10.1090/conm/305

Editorial Board Dennis DeTurck, managing editor

Andreas Blass Andy R. Magid Michael Vogelius

This volume contains the proceedings of an AMS Special Session on Quantum Com-putation and Information held in Washington, D.C., on January 19-21, 2000.

2000 Mathematics Subject Classification. Primary 81P68, 94-02; Secondary 68Q05, 81-01, 81R99, 94A60, 94A99.

Library of Congress Cataloging-in-Publication Data AMS Special Session Quantum Computation and Information (2000: Washington, D.C.)

Quantum computation and information : AMS Special Session Quantum Computation and Information, Washington, D.C., January 19-21, 2000 / Samuel J. Lomonaco, Jr., Howard E. Brandt, editors.

p. em. -(Contemporary mathematics; v. 305) Includes bibliographic references. ISBN 0-8218-2140-7 (alk. paper) 1. Quantum computers-Congresses. I. Lomonaco, Samuel J. II. Brandt, Howard E. Ill. Title.

IV. Contemporary mathematics (American Mathematical Society) ; v. 305.

QA 76.889 .A47 2000 004.1'4-dc21 2002026239

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Contents

Preface v

Gilles Brassard Awarded Pot de Vin Prize vn

List of Participants vm

Space searches with a quantum robot PAUL BENIOFF 1

Perturbation theory and numerical modeling of quantum logic operations with a large number of qubits G. P. BERMAN, G. D. DooLEN, D. I. KAMENEv, G. V. LoPEZ, AND V. I. TSIFRINOVICH 13

Inconclusive rate with a positive operator valued measure HOWARD E. BRANDT 43

Quantum amplitude amplification and estimation GILLES BRASSARD, PETER H0YER, MICHELE MOSCA, AND ALAIN TAPP 53

Manipulating the entanglement of one copy of a two-particle pure entangled state LUCIEN HARDY 75

Geometric algebra in quantum information processing TIMOTHY F. HAVEL AND CHRIS J. L. DORAN 81

Quantum computing and the Jones polynomial LOUIS H. KAUFFMAN 101

Quantum hidden subgroup algorithms: A mathematical perspective SAMUEL J. LOMONACO, JR. AND LOUIS H. KAUFFMAN 139

Improved two-party and multi-party purification protocols ELITZA N. MANEVA AND JOHN A. SMOLIN 203

Quantum games and quantum algorithms DAVID A. MEYER 213

A proof that measured data and equations of quantum mechanics can be linked only by guesswork JOHN M. MYERS AND F. HADI MADJID 221

iii

iv CONTENTS

Quantum computation by geometrical means JIANNIS PACHOS

Pauli exchange and quantum error correction MARY BETH RUSKAI

Relative entropy in quantum information theory BENJAMIN SCHUMACHER AND MICHAEL D. WESTMORELAND

An unentangled Gleason's theorem NOLAN R. WALLACH

Entangled chains WILLIAM K. WOOTTERS

245

251

265

291

299

Preface

This book is a collection of papers given by invited speakers at the AMS Spe-cial Session on Quantum Computation and Information held at the Annual Meeting of the American Mathematical Society in Washington, DC, January 19-21, 2000. This AMS Special Session was held in conjunction with the AMS Short Course on Quantum Computation, January 17-18, 2000, which has been published sepa-rately as the AMS PSAPM Volume 58 entitled "Quantum Computation: A Grand Mathematical Challenge for the Twenty-First Century and the Millennium."

This Special Session, together with its accompanying Short Course, was the first time at an AMS meeting that the new and emerging discipline of quantum information science was formally introduced to the AMS community. It is hoped that this event, together with the two books recording this occasion, 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 was partially supported by the Defense Advanced Research Projects Agency (DARPA) and Air Force Materiel Command USAF under agreement num-ber F30602-01-0522, by Army Research Office (ARO) Grant #P-38804-PH-QC, by the National Institute of Standards and Technology (NIST), and by L-0-0-P Fund No. 2000WADC.

The editors thank UMBC and ARL for support in this endeavor. Thanks are also due to Jekkin Shah for spending many hours transforming the papers contributed to this volume into AMS LaTeX. Moreover, the editors thank the staff of the American Mathematical Society for their editorial support.

Finally, thanks are due to all the AMS Session participants whose efforts and contributions made all this possible.

Samuel J. Lomonaco, Jr. Univ. of Maryland Baltimore County Baltimore, MD 21250 Lomonaco@umbc.edu www .csee. umbc.edu;-lomonaco

v

and

June, 2002

Howard E. Brandt Army Research Lab Adelphi, MD 20783 hbrandt@arl.army.mil

Gilles Brassard Awarded

Pot de Vin Prize

The Millennium 2000 Pot de Vin Prize was awarded to Gilles Brassard for his many outstanding contributions to quantum computation and quantum informa-tion science. This award was presented at the AMS Special Session on Quantum Computation and Information held at the Annual Meeting of the American Math-ematical Society in Washington, DC, in J anuary 2000.

Description of the Pot de Vin Prize The Pot de Vin Prize is an award coveted by Nobel Laureates and Fields

Medalists. It was established to bribe (FR, pot de vin) major contributors to the field of quantum computation and quantum information into giving an invited talk. All Nobel Laureates and Fields Medalists are automatically disqualified from receiving the prize. The Pot de Vin Prize is sponsored by the L-0-0-P Fund.

vii

List of Participants

The First Special Session on Quantum Computation and Information Annual Meeting of the American Mathematical Society

Washington, DC

Paul Benioff Charles H. Bennett Gennady P. Berman Howard E. Brandt Gilles Brassard Christopher A. Fuchs Daniel Gottesman Lov K. Grover Lucien Hardy Timothy F. Havel Louis H. Kauffman Alexei Y. Kitaev Samuel J. Lomonaco

January 19-21, 2000

viii

Gianfranco F. Mascari David A. Meyer John M. Myers Jiannis Pachos Mary Beth Ruskai Benjamin W. Schumacher Peter W. Shor John A. Smolin Nolan R. Wallach Michael D. Westmoreland Umesh V. Vazirani William K. Wootters Wojceich H. Zurek

Titles in This Series

305 Samuel J. Lomonaco, Jr. and Howard E. Brandt, Editors, Quantum computation and information, 2002

304 Jorge Alberto Calvo, Kenneth C. Millett, and Eric J. Rawdon, Editors, Physical knots: Knotting, linking, and folding geometric objects in JR3 , 2002

303 William Cherry and Chung-Chun Yang, Editors, Value distribution theory and complex dynamics, 2002

302 Yi Zhang, Editor, Logic and algebra, 2002 301 Jerry Bona, Roy Choudhury, and David Kaup, Editors, The legacy of the inverse

scattering transform in applied mathematics, 2002 300 Sergei Vostokov and Yuri Zarhin, Editors, Algebraic number theory and algebraic

geometry: Papers dedicated to A. N. Parshin on the occasion of his sixtieth birthday, 2002 299 George Kamberov, Peter Norman, Franz Pedit, and Ulrich Pinkall, Quaternions,

spinors, and surfaces, 2002 298 Robert Gilman, Alexei G. Myasnikov, and Vladimir Shpilrain, Editors,

Computational and statistical group theory, 2002 297 Stephen Berman, Paul Fendley, Yi-Zhi Huang, Kailash Misra, and Brian

Parshall, Editors, Recent developments in infinite-dimensional Lie algebras and conformal field theory, 2002

296 Sean Cleary, Robert Gilman, Alexei G. Myasnikov, and Vladimir Shpilrain, Editors, Combinatorial and geometric group theory, 2002

295 Zhangxin Chen and Richard E. Ewing, Editors, Fluid flow and transport in porous media: Mathematical and numerical treatment, 2002

294 Robert Coquereaux, Ariel Garcia, and Roberto Trinchero, Editors, Quantum symmetries in theoretical physics and mathematics, 2002

293 Donald M. Davis, Jack Morava, Goro Nishida, W. Stephen Wilson, and Nobuaki Yagita, Editors, Recent progress in homotopy theory, 2002

292 A. Chenciner, R. Cushman, C. Robinson, and z. Xia, Editors, Celestial Mechanics, 2002

291 Bruce C. Berndt and Ken Ono, Editors, q-series with applications to combinatorics, number theory, and physics, 2001

290 Michel L. Lapidus and Machiel van Frankenhuysen, Editors, Dynamical, spectral, and arithmetic zeta functions, 2001

289 Salvador Perez-Esteva and Carlos Villegas-Blas, Editors, Second summer school in analysis and mathematical physics: Topics in analysis: Harmonic, complex, nonlinear and quantization, 2001

288 Marisa Fermindez and Joseph A. Wolf, Editors, Global differential geometry: The mathematical legacy of Alfred Gray, 2001

287 Marlos A. G. Viana and Donald St. P. Richards, Editors, Algebraic methods in statistics and probability, 2001

286 Edward L. Green, Serkan Ho§ten, Reinhard C. Laubenbacher, and Victoria Ann Powers, Editors, Symbolic computation: Solving equations in algebra, geometry, and engineering, 2001

285 Joshua A. Leslie and Thierry P. Robart, Editors, The geometrical study of differential equations, 2001

284 Gaston M. N'Guerekata and Asamoah Nkwanta, Editors, Council for African American researchers in the mathematical sciences: Volume IV, 2001

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/.

This book is a collection of papers given by invited speakers at the ftrst 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 the companion volume to Quantum Computation: A Grand Mathematical Challenge for the Twenty-First Century and the Millennium, Volume 58 in the Proceedings of Symposia in Applied Mathematics series.

ISBN 0-8218-2140-7

9 780821 821404

CONM/305