Problems & Solutions in Quantum Computing & Quantum Information PDF
Preview Problems & Solutions in Quantum Computing & Quantum Information
PROBLEMS ti SOLUTIONS IN QUANTUM COMPUTING & QUANTUM INFORMATION This page intentionally left blank PROBLEMS 8 SOLUTIONS IN QUANTUM COMPUTING & QUANTUM INFORMATION Willi-Hans Steeb Yorick Hardy Rand Afrikaans University, South Africa r p World Scientific - NEW JERSEY * LONDON * SINGAPORE * BElJlNG SHANGHAI HONG KONG * TAIPEI CHENNAI Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA ofice: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-PublicationD ata A catalogue record for this book is available from the British Library. PROBLEMS AND SOLUTIONS IN QUANTUM COMPUTING AND QUANTUM INFORMATION Copyright 0 2004 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book or parts thereof; may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923,U SA. In this case permission to photocopy is not required from the publisher. ISBN 981-238-789-7 ISBN 981-238-790-(0p bk) Printed in Singapore. Preface The purpose of this book is to supply a collection of problems in quantum computing and quantum information together with their detailed solutions which will prove to be valuable to graduate students as well as to research workers in these fields. All the important concepts and topics such as quan- tum gates and quantum circuits, entanglement, teleportation, Bell states, Bell inequality, Schmidt decomposition, quantum Fourier transform, magic gate, von Neumann entropy, quantum cryptography, quantum error correc- tion, coherent states, squeezed states, POVM measurement, beam splitter and Kerr Hamilton operator are included. The topics range in difficulty from elementary to advanced. Almost all problems are solved in detail and most of the problems are self-contained. All relevant definitions are given. Students can learn important principles and strategies required for problem solving. Teachers will also find this text useful as a supplement, since im- portant concepts and techniques are developed in the problems. The book can also be used as a text or a supplement for linear and multilinear algebra or matrix theory. The material was tested in our lectures given around the world. Any useful suggestions and comments are welcome. The International School for Scientific Computing (ISSC) provides certifi- cate courses for this subject. Please contact the authors if you want to do this course. e-mail addresses of the authors: [email protected] whsQna.rau.ac.za YorickHardyQyahoo.com yhaQna.rau.ac.za Home pages of the authors: http://issc.rau.ac.zaa http://zeus.rau.ac.zaa V This page intentionally left blank Contents I Finite-Dimensional Hilbert Spaces 1 Qubits 3 2 Kronecker Product and Tensor Product 14 3 Matrix Properties 24 4 Density Operators 49 5 Partial Trace 58 6 Unitary Transforms and Quantum Gates 66 7 Measurement 88 8 Entanglement 98 9 Teleportation 132 10 Cloning 141 11 Quantum Algorithms 143 12 Quantum Error Correction 158 13 Quantum Cryptography 162 I1 Infinite-Dimensional Hilbert Spaces 14 Harmonic Oscillator and Bose Operators 169 15 Coherent States 193 16 Squeezed States 205 vii Cont ents viii 17 Entanglement 212 18 Teleportation 225 19 Swapping and Cloning 227 20 Hamilton Operators 232 Bibliography 241 Index 247 Notation 0 empty set N natural numbers Z integers Q rational numbers R real numbers R+ nonnegative real numbers C complex numbers R" n-dimensional Euclidian space C" n-dimensional complex linear space Fl Hilbert space a i %Z real part of the complex number z 3.2 imaginary part of the complex number z A c B subset A of set B A n B the intersection of the sets A and B A U B the union of the sets A and B f 09 composition of two mappings (f o g)(z) = f (g(z)) t time variable X column vector in C" XT transpose of x (row vector) II . II norm scalar product (inner product) in C" scalar product in Hilbert space vector product X X Y A @ B Kronecker product of matrices A and B f @g tensor product of elements f and g of Hilbert spaces det(A) determinant of a square matrix A tr(4 trace of a square matrix A rank(A) rank of matrix A AT transpose of matrix A - A conjugate of matrix A A* conjugate transpose of matrix A ix
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