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Contemporary Mathematics 305 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 http://dx.doi.org/10.1090/conm/305 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 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 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permissien0ams. erg. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) © 2002 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. § The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http: I /www. ams. erg/ 10 9 8 7 6 5 4 3 2 1 07 06 05 04 03 02 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 245 Pauli exchange and quantum error correction MARY BETH RUSKAI 251 Relative entropy in quantum information theory BENJAMIN SCHUMACHER AND MICHAEL D. WESTMORELAND 265 An unentangled Gleason's theorem NOLAN R. WALLACH 291 Entangled chains WILLIAM K. WOOTTERS 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. Howard E. Brandt Univ. of Maryland Baltimore County Army Research Lab Baltimore, MD 21250 and Adelphi, MD 20783 [email protected] [email protected] www .csee. umbc.edu;-lomonaco June, 2002 v 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 January 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 January 19-21, 2000 Paul Benioff Gianfranco F. Mascari Charles H. Bennett David A. Meyer Gennady P. Berman John M. Myers Howard E. Brandt Jiannis Pachos Gilles Brassard Mary Beth Ruskai Christopher A. Fuchs Benjamin W. Schumacher Daniel Gottesman Peter W. Shor Lov K. Grover John A. Smolin Lucien Hardy Nolan R. Wallach Timothy F. Havel Michael D. Westmoreland Louis H. Kauffman Umesh V. Vazirani Alexei Y. Kitaev William K. Wootters Samuel J. Lomonaco Wojceich H. Zurek viii 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.
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