Holdings: Mathematics of quantum computation and quantum technology

Mathematics of quantum computation and quantum technology:

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Bibliographic Details
Format:	Book
Language:	English
Published:	Boca Raton [u.a.] Chapman & Hall 2008
Series:	Chapman & Hall CRC applied mathematics and nonlinear science series 14
Subjects:	Ordinateurs quantiques - Mathématiques Théorie quantique - Mathématiques Mathematik Quantentheorie Quantum computers > Mathematics Quantum theory > Mathematics Quantencomputer
Online Access:	Inhaltsverzeichnis Beschreibung für Leser Table of contents only Inhaltsverzeichnis
Item Description:	Includes bibliographical references and index
Physical Description:	XVIII, 601 S.
ISBN:	9781584888994 1584888997

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245	1	0	\|a Mathematics of quantum computation and quantum technology \|c ed. by Goong Chen ...
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adam_text	CHAPMAN & HALL/CRC APPLIED MATHEMATICS AND NONLINEAR SCIENCE SERIES MATHEMATICS OF QUANTUM COMPUTATION AND QUANTUM TECHNOLOGY EDITED BY GOONG CHEN LOUIS KAUFFMAN SAMUEL J. LOMONACO CHAPMAN & HALL/CRC TAYLOR & FRANCIS GROUP BOCA RATON LONDON NEW YORK CHAPMAN & HALL/CRC IS AN IMPRINT OF THE TAYLOR & FRANCIS GROUP, AN INFORMA BUSINESS CONTENTS PREFACE V QUANTUM COMPUTATION 1 1 QUANTUM HIDDEN SUBGROUP ALGORITHMS: AN ALGORITHMIC TOOLKIT 3 SAMUEL J. LOMONACO AND LOUIS H. KAUFFMAN 1.1 INTRODUCTION 4 1.2 AN EXAMPLE OF SHOR S QUANTUM FACTORING ALGORITHM 4 1.3 DEFINITION OF QUANTUM HIDDEN SUBGROUP (QHS) ALGORITHMS . . 9 1.4 THE GENERIC QHS ALGORITHM 10 1.5 PUSHING AND LIFTING HIDDEN SUBGROUP PROBLEMS (HSPS) 12 1.6 SHOR S ALGORITHM REVISITED 14 ,..1.7 WANDERING SHOR ALGORITHMS, A.K.A. VINTAGE SHOR ALGORITHMS . . 16 1.8 CONTINUOUS (VARIABLE) SHOR ALGORITHMS 22 1.9 THE QUANTUM CIRCLE AND THE DUAL SHOR ALGORITHMS 23 1.10 A QHS ALGORITHM FOR FEYNMAN INTEGRALS 28 . .1.11 QHS ALGORITHMS ON FREE GROUPS 31 1.12 GENERALIZING SHOR S ALGORITHM TO FREE GROUPS 34 1.13 IS GROVER S ALGORITHM A QHS ALGORITHM? 36 1.14 BEYOND QHS ALGORITHMS: A SUGGESTION OF A META-SCHEME;FOR CREATING NEW QUANTUM ALGORITHMS , * * * 42 2 A REALIZATION SCHEME FOR QUANTUM MULTI-OBJECT SEARCH 47 ZIJIAN DIAO, GOONG CHEN, AND PETER SHIUE 2.1 INTRODUCTION 47 XI XLL 2.2 CIRCUIT DESIGN FOR THE MULTI-OBJECT SIGN-FLIPPING OPERATOR ... 50 2.3 ADDITIONAL DISCUSSION 57 2.4 COMPLEXITY ISSUES 58 3 ON INTERPOLATING BETWEEN QUANTUM AND CLASSICAL COMPLEXITY CLASSES 67 J. MAURICE ROJAS 3.1 INTRODUCTION AND MAIN RESULTS 67 3.1.1 OPEN QUESTIONS AND THE RELEVANCE OF ULTRAMETRIC COM- PLEXITY 72 3.2 BACKGROUND AND ANCILLARY RESULTS 73 3.2.1 REVIEW OF RIEMANN HYPOTHESES 77 3.3 THE PROOFS OF OUR MAIN RESULTS 80 3.3.1 THE UNIVARIATE THRESHOLD OVER Q P : PROVING THE MAIN THEOREM 80 3.3.2 DETECTING SQUARE-FREENESS: PROVING COROLLARY 3.1 ... 83 4 QUANTUM ALGORITHMS FOR HAMILTONIAN SIMULATION 89 DOMINIC W. BERRY, GRAEME AHOKAS, RICHARD CLEVE, AND BARRY C. SANDERS 4.1 INTRODUCTION 89 4.2 SIMULATION METHOD OF LLOYD 91 4.3 SIMULATION METHOD OF ATS 92 4.4 HIGHER ORDER INTEGRATORS 95 4.5 LINEAR LIMIT ON SIMULATION TIME 101 4.6 EFFICIENT DECOMPOSITION OF HAMILTONIAN 105 4.7 CONCLUSIONS 110 QUANTUM TECHNOLOGY 113 5 NEW MATHEMATICAL TOOLS FOR QUANTUM TECHNOLOGY 115 C. BRACHER, M. KLEBER, AND T. KRAMER 5.1 PHYSICS IN SMALL DIMENSIONS 115 5.2 PROPAGATORS AND GREEN FUNCTIONS 117 5.3 QUANTUM SOURCES 121 5.3.1 PHOTOELECTRONS EMITTED FROM A QUANTUM SOURCE . . . . 121 5.3.2 CURRENTS GENERATED BY QUANTUM SOURCES 122 5.3.3 RECOVERING FERMI S GOLDEN RULE 124 5.3.4 PHOTODETACHMENT AND WIGNER S THRESHOLD LAWS . . . . 124 5.4 SPATIALLY EXTENDED SOURCES: THE ATOM LASER 126 5.5 BALLISTIC TUNNELING: STM 130 XLLL 5.6 ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS: THE QUANTUM HALL EFFECT 134 5.7 THE SEMICLASSICAL METHOD 138 6 THE PROBABILISTIC NATURE OF QUANTUM MECHANICS 149 LEON COHEN 6.1 INTRODUCTION 150 6.2 ARE THERE WAVE FUNCTIONS IN STANDARD PROBABILITY THEORY? ... 150 6.2.1 THE KHINCHIN THEOREM 153 6.3 TWO VARIABLES 154 6.3.1 GENERALIZED CHARACTERISTIC FUNCTION 156 6.4 VISUALIZATION OF QUANTUM WAVE FUNCTIONS 160 6.5 LOCAL KINETIC ENERGY 165 6.6 CONCLUSION 167 7 SUPERCONDUCTING QUANTUM COMPUTING DEVICES 171 ZHIGANG ZHANG AND GOONG CHEN 7.1 INTRODUCTION 171 7.2 SUPERCONDUCTIVITY 172 7.3 MORE ON COOPER PAIRS AND JOSEPHSON JUNCTIONS 176 7.4 SUPERCONDUCTING CIRCUITS: CLASSICAL 178 7.4.1 CURRENT-BIASED JJ 179 7.4.2 SINGLE COOPER-PAIR BOX (SCB) 182 7.4.3 RF-OR AC-SQUID 183 : 7.4.4 DC-SQUID 184 7.5 SUPERCONDUCTING CIRCUITS: QUANTUM 186 7.6 QUANTUM GATES 188 7.6.1 SOME BASIC FACTS ABOUT SU(2) AND SO(3) 191 7.6.2 ONE QUBIT OPERATIONS (I): CHARGE-QUBIT . 192 7.6.3 ONE QUBIT OPERATIONS (II): FLUX-QUBIT 197 7.6.4 CHARGE-FLUX QUBIT AND PHASE QUBIT 205 7.6.5 TWO QUBIT OPERATIONS: CHARGE AND FLUX QUBITS 206 * 7.6.6 MEASUREMENT OF CHARGE QUBIT. . 215 8 NONDETERMINISTIC LOGIC GATES IN OPTICAL QUANTUM COMPUTING 223 FEDERICO M. SPEDALIERI, JONATHAN P. DOWLING, AND HWANG LEE 8.1 INTRODUCTION 223 8.2 PHOTON AS A QUBIT 225 ; -: 8.3 LINEAR OPTICAL QUANTUM COMPUTING 228 8.4 NONDETERMINISTIC TWO-QUBIT GATE 237 8.5 ANCILLA-STATE PREPARATION 241 XIV 8.6 CLUSTER-STATE APPROACH AND GATE FIDELITY 245 8.7 APPENDICES 250 QUANTUM INFORMATION 257 9 EXPLOITING ENTANGLEMENT IN QUANTUM CRYPTOGRAPHIC PROBES 259 HOWARD E. BRANDT 9.1 INTRODUCTION 260 9.2 PROBE DESIGNS BASED ON F/ (1) 263 9.3 PROBE DESIGNS BASED ON T/ (2) 271 9.4 PROBE DESIGNS BASED ON U^ 273 9.5 CONCLUSION 276 APPENDIX A RENYI INFORMATION GAIN 277 APPENDIX B MAXIMUM RENYI INFORMATION GAIN 281 10 NONBINARY STABILIZER CODES 287 PRADEEP KIRAN SARVEPALLI, SALAH A. ALY, AND ANDREAS KLAPPENECKER 10.1 INTRODUCTION 287 10.2 STABILIZER CODES 289 10.2.1 ERRORBASES 290 10.2.2 STABILIZER CODES 290 10.2.3 STABILIZER AND ERROR CORRECTION 291 10.2.4 MINIMUM DISTANCE 292 10.2.5 PURE AND IMPURE CODES , 292 10.2.6 ENCODING QUANTUM CODES 293 10.3 QUANTUM CODES AND CLASSICAL CODES 293 10.3.1 CODES OVER Q 293 10.3.2 CODES OVER F 2 295 10.4 BOUNDS ON QUANTUM CODES 296 10.5 FAMILIES OF QUANTUM CODES 298 10.5.1 QUANTUM M-ADIC RESIDUE CODES 299 10.5.2 QUANTUM PROJECTIVE REED-MULLER CODES 300 10.5.3 PUNCTURING QUANTUM CODES 302 10.6 CONCLUSION 303 11 ACCESSIBLE INFORMATION ABOUT QUANTUM STATES: AN OPEN OPTIMIZATION PROBLEM 309 JUN SUZUKI, SYED M. ASSAD, AND BERTHOLD-GEORG ENGLERT 11.1 INTRODUCTION 309 XV 11.2 PRELIMINARIES 310 11.2.1 STATES AND MEASUREMENTS . 310 11.2.2 ENTROPY AND INFORMATION 313 , 11.3 THE OPTIMIZATION PROBLEM 316 11.4 THEOREMS 317 11.4.1 CONCAVITY AND CONVEXITY 317 11.4.2 NECESSARY CONDITION 318 11.4.3 SOME BASIC THEOREMS 319 11.4.4 GROUP-COVARIANT CASE 321 11.5 NUMERICAL SEARCH 324 11.6 EXAMPLES 326 11.6.1 TWO QUANTUM STATES IN TWO DIMENSIONS 326 11.6.2 TRINE: Z 3 SYMMETRY IN TWO DIMENSIONS 329 11.6.3 SIX-STATES PROTOCOL: SYMMETRIC GROUP S 3 332 11.6.4 FOUR-GROUP IN FOUR DIMENSIONS 339 11.7 SUMMARY AND OUTLOOK 345 12 QUANTUM ENTANGLEMENT: CONCEPTS AND CRITERIA 349 FU-LI LI AND M. SUHAIL ZUBAIRY 12.1 INTRODUCTION 349 12.2 EPR CORRELATIONS AND QUANTUM ENTANGLEMENT. . . * 353 12.3 ENTANGLEMENT OF PURE STATES . . 356 12.4 CRITERIA ON ENTANGLEMENT OF MIXED STATES 358 12.4.1 PERES-HORODECKI CRITERION 358 12.4.2 SIMON CRITERION 361 12.4.3 DUAN-GIEDKE-CIRAC-ZOLLER CRITERION 363 12.4.4 HILLERY-ZUBAIRY CRITERION 366 12.4.5 SHCHUKIN-VOGEL CRITERION 367 12.5 COHERENCE-INDUCED ENTANGLEMENT 369 12.6 CORRELATED SPONTANEOUS EMISSION LASER AS AN ENTANGLEMENT AMPLIFIER 376 12.6 REMARKS 380 13 PARAMETRIZATIONS OF POSITIVE MATRICES WITH APPLICATIONS 387 M. TSENG, H. ZHOU, AND V. RAMAKRISHNA 13.1 INTRODUCTION 387 13.2 SOURCES OF POSITIVE MATRICES IN QUANTUM THEORY 388 13.3 CHARACTERIZATIONS OF POSITIVE MATRICES 390 13.4 A DIFFERENT PARAMETRIZATION OF POSITIVE MATRICES 393 XVI 13.5 TWO FURTHER APPLICATIONS 397 13.5.1 TOEPLITZ STATES . . .:». . : . . 397 13.5.2 CONSTRAINTS ON RELAXATION RATES 400 13.6 CONCLUSIONS . 403 QUANTUM TOPOLOGY, CATEGORICAL ALGEBRA, AND LOGIC 407 14 QUANTUM COMPUTING AND QUANTUM TOPOLOGY 409 . LOUIS H. KAUFFMAN AND SAMUEL J. LOMONACO , 14.1 INTRODUCTION 409 14.2 KNOTS AND BRAIDS 414 14.3 QUANTUM MECHANICS AND QUANTUM COMPUTATION 419 14.3.1 WHAT IS A QUANTUM COMPUTER? 421 14.4 BRAIDING OPERATORS AND UNIVERSAL QUANTUM GATES 423 14.4.1 UNIVERSAL GATES 426 14.5 A REMARK ABOUT EPR, ENTANGLEMENT AND BELL S INEQUALITY . . . 430 14.6 THE ARAVIND HYPOTHESIS 433 14.7 SU(2) REPRESENTATIONS OF THE ARTIN BRAID GROUP 434 14.8 THE BRACKET POLYNOMIAL AND THE JONES POLYNOMIAL 440 14.8.1 QUANTUM COMPUTATION OF THE JONES POLYNOMIAL .... 445 14.9 QUANTUM TOPOLOGY, COBORDISM CATEGORIES, TEMPERLEY-LIEB ALGEBRA, AND TOPOLOGICAL QUANTUM FIELD THEORY 450 14.10BRAIDING AND TOPOLOGICAL QUANTUM FIELD THEORY 461 14.11 SPIN NETWORKS AND TEMPERLEY-LIEB RECOUPLING THEORY 473 14.11.1 EVALUATIONS 477 14.11.2 SYMMETRY AND UNITARITY 480 14.12FIBONACCI PARTICLES 484 14.13THE FIBONACCI RECOUPLING MODEL 491 14.14QUANTUM COMPUTATION OF COLORED JONES POLYNOMIALS AND THE WITTEN-RESHETIKHIN-TURAEV INVARIANT 504 15 TEMPERLEY-LIEB ALGEBRA: FROM KNOT THEORY TO LOGIC AND COMPUTATION VIA QUANTUM MECHANICS 515 SAMSON ABRAMSKY 15.1 INTRODUCTION 516 15.1.1 KNOT THEORY . . .- 516 15.1.2 CATEGORICAL QUANTUM MECHANICS 516 15.1.3 LOGIC AND COMPUTATION . . 517 15.1.4 OUTLINE OF THE PAPER ; . . . 517 XVII 15.2 THE TEMPERLEY-LIEB ALGEBRA 518 15.2.1 TEMPERLEY-LIEB ALGEBRA: GENERATORS AND RELATIONS . . . 518 15.2.2 DIAGRAM MONOIDS 519 15.2.3 EXPRESSIVENESS OF THE GENERATORS 520 15.2.4 THETRACE . 521 15.2.5 THE CONNECTION TO KNOTS 522 15.3 THE TEMPERLEY-LIEB CATEGORY 523 15.3.1 PIVOTAL CATEGORIES 524 15.3.2 PIVOTAL DAGGER CATEGORIES 527 15.4 FACTORIZATION AND IDEMPOTENTS 528 15.5 CATEGORICAL QUANTUM MECHANICS 530 15.5.1 OUTLINE OF THE APPROACH 531 15.5.2 QUANTUM NON-LOGIC VS. QUANTUM HYPER-LOGIC 533 15.5.3 REMARKS 536 15.6 PLANAR GEOMETRY OF INTERACTION AND THE TEMPERLEY-LIEB ALGEBRA 537 15.6.1 SOME PRELIMINARY NOTIONS 538 15.6.2 FORMALIZING DIAGRAMS 539 15.6.3 CHARACTERIZING PLANARITY 540 15.6.4 THE TEMPERLEY-LIEB CATEGORY 544 15.7 PLANAR A-CALCULUS 548 15.7.1 THE A-CALCULUS 549 15.7.2 TYPES 549 15.7.3 INTERPRETATION IN PIVOTAL CATEGORIES 551 15.7.4 AN EXAMPLE 552 15.7.5 DISCUSSION 553 15.8 FURTHER DIRECTIONS 553 16 QUANTUM MEASUREMENTS WITHOUT SUMS 559 BOB COECKE AND DUSKO PAVLOVIC 16.1 INTRODUCTION 559 16.2 CATEGORICAL SEMANTICS 562 16.2.1 F-COMPACT CATEGORIES 562 16.2.2 GRAPHICAL CALCULUS 564 16.2.3 SCALARS, TRACE, AND PARTIAL TRANSPOSE 566 16.3 SUMS AND BASES IN HILBERT SPACES 568 16.3.1 SUMS IN QUANTUM MECHANICS 568 16.3.2 NO-CLONING AND EXISTENCE OF A NATURAL DIAGONAL .... 569 16.3.3 MEASUREMENT AND BASES 570 16.3.4 VANISHING OF NON-DIAGONAL ELEMENTS AND DELETION . . . 571 16.3.5 CANONICAL BASES 572 XV111 16.4 CLASSICAL OBJECTS 572 16.4.1 SPECIAL F-COMPACT FROBENIUS ALGEBRAS 573 16.4.2 SELF-ADJOINTNESS RELATIVE TO A CLASSICAL OBJECT 576 16.4.3 GHZ STATES AS CLASSICAL OBJECTS . 578 16.4.4 EXTRACTING THE CLASSICAL WORLD 579 16.5 QUANTUM SPECTRA 580 16.5.1 COALGEBRAIC CHARACTERIZATION OF SPECTRA 581 16.5.2 CHARACTERIZATION OF X-CONCEPTS 582 16.6 QUANTUM MEASUREMENTS 582 16.6.1 THE CPM-CONSTRUCTION 584 16.6.2 FORMAL DECOHERENCE 585 16.6.3 DEMOLITION MEASUREMENTS 586 16.7 QUANTUM TELEPORTATION 587 16.8 DENSE CODING 592 APPENDIX PANEL REPORT ON THE FORWARD LOOKING DISCUSSION 597 INDEX 603
adam_txt	CHAPMAN & HALL/CRC APPLIED MATHEMATICS AND NONLINEAR SCIENCE SERIES MATHEMATICS OF QUANTUM COMPUTATION AND QUANTUM TECHNOLOGY EDITED BY GOONG CHEN LOUIS KAUFFMAN SAMUEL J. LOMONACO CHAPMAN & HALL/CRC TAYLOR & FRANCIS GROUP BOCA RATON LONDON NEW YORK CHAPMAN & HALL/CRC IS AN IMPRINT OF THE TAYLOR & FRANCIS GROUP, AN INFORMA BUSINESS CONTENTS PREFACE V QUANTUM COMPUTATION 1 1 QUANTUM HIDDEN SUBGROUP ALGORITHMS: AN ALGORITHMIC TOOLKIT 3 SAMUEL J. LOMONACO AND LOUIS H. KAUFFMAN 1.1 INTRODUCTION 4 1.2 AN EXAMPLE OF SHOR'S QUANTUM FACTORING ALGORITHM 4 1.3 DEFINITION OF QUANTUM HIDDEN SUBGROUP (QHS) ALGORITHMS . . 9 1.4 THE GENERIC QHS ALGORITHM 10 1.5 PUSHING AND LIFTING HIDDEN SUBGROUP PROBLEMS (HSPS) 12 1.6 SHOR'S ALGORITHM REVISITED 14 ,.1.7 WANDERING SHOR ALGORITHMS, A.K.A. VINTAGE SHOR ALGORITHMS . . 16 1.8 CONTINUOUS (VARIABLE) SHOR ALGORITHMS 22 1.9 THE QUANTUM CIRCLE AND THE DUAL SHOR ALGORITHMS 23 1.10 A QHS ALGORITHM FOR FEYNMAN INTEGRALS 28 . .1.11 QHS ALGORITHMS ON FREE GROUPS 31 1.12 GENERALIZING SHOR'S ALGORITHM TO FREE GROUPS 34 1.13 IS GROVER'S ALGORITHM A QHS ALGORITHM? 36 1.14 BEYOND QHS ALGORITHMS: A SUGGESTION OF A META-SCHEME;FOR CREATING NEW QUANTUM ALGORITHMS , * * * 42 2 A REALIZATION SCHEME FOR QUANTUM MULTI-OBJECT SEARCH 47 ZIJIAN DIAO, GOONG CHEN, AND PETER SHIUE 2.1 INTRODUCTION 47 XI XLL 2.2 CIRCUIT DESIGN FOR THE MULTI-OBJECT SIGN-FLIPPING OPERATOR . 50 2.3 ADDITIONAL DISCUSSION 57 2.4 COMPLEXITY ISSUES 58 3 ON INTERPOLATING BETWEEN QUANTUM AND CLASSICAL COMPLEXITY CLASSES 67 J. MAURICE ROJAS 3.1 INTRODUCTION AND MAIN RESULTS 67 3.1.1 OPEN QUESTIONS AND THE RELEVANCE OF ULTRAMETRIC COM- PLEXITY 72 3.2 BACKGROUND AND ANCILLARY RESULTS 73 3.2.1 REVIEW OF RIEMANN HYPOTHESES 77 3.3 THE PROOFS OF OUR MAIN RESULTS 80 3.3.1 THE UNIVARIATE THRESHOLD OVER Q P : PROVING THE MAIN THEOREM 80 3.3.2 DETECTING SQUARE-FREENESS: PROVING COROLLARY 3.1 . 83 4 QUANTUM ALGORITHMS FOR HAMILTONIAN SIMULATION 89 DOMINIC W. BERRY, GRAEME AHOKAS, RICHARD CLEVE, AND BARRY C. SANDERS 4.1 INTRODUCTION 89 4.2 SIMULATION METHOD OF LLOYD 91 4.3 SIMULATION METHOD OF ATS 92 4.4 HIGHER ORDER INTEGRATORS 95 4.5 LINEAR LIMIT ON SIMULATION TIME 101 4.6 EFFICIENT DECOMPOSITION OF HAMILTONIAN 105 4.7 CONCLUSIONS 110 QUANTUM TECHNOLOGY 113 5 NEW MATHEMATICAL TOOLS FOR QUANTUM TECHNOLOGY 115 C. BRACHER, M. KLEBER, AND T. KRAMER 5.1 PHYSICS IN SMALL DIMENSIONS 115 5.2 PROPAGATORS AND GREEN FUNCTIONS 117 5.3 QUANTUM SOURCES 121 5.3.1 PHOTOELECTRONS EMITTED FROM A QUANTUM SOURCE . . . . 121 5.3.2 CURRENTS GENERATED BY QUANTUM SOURCES 122 5.3.3 RECOVERING FERMI'S GOLDEN RULE 124 5.3.4 PHOTODETACHMENT AND WIGNER'S THRESHOLD LAWS . . . . 124 5.4 SPATIALLY EXTENDED SOURCES: THE ATOM LASER 126 5.5 BALLISTIC TUNNELING: STM 130 XLLL 5.6 ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS: THE QUANTUM HALL EFFECT 134 5.7 THE SEMICLASSICAL METHOD 138 6 THE PROBABILISTIC NATURE OF QUANTUM MECHANICS 149 LEON COHEN 6.1 INTRODUCTION 150 6.2 ARE THERE WAVE FUNCTIONS IN STANDARD PROBABILITY THEORY? . 150 6.2.1 THE KHINCHIN THEOREM 153 6.3 TWO VARIABLES 154 6.3.1 GENERALIZED CHARACTERISTIC FUNCTION 156 6.4 VISUALIZATION OF QUANTUM WAVE FUNCTIONS 160 6.5 LOCAL KINETIC ENERGY 165 6.6 CONCLUSION 167 7 SUPERCONDUCTING QUANTUM COMPUTING DEVICES 171 ZHIGANG ZHANG AND GOONG CHEN 7.1 INTRODUCTION 171 7.2 SUPERCONDUCTIVITY 172 7.3 MORE ON COOPER PAIRS AND JOSEPHSON JUNCTIONS 176 7.4 SUPERCONDUCTING CIRCUITS: CLASSICAL 178 7.4.1 CURRENT-BIASED JJ 179 7.4.2 SINGLE COOPER-PAIR BOX (SCB) 182 7.4.3 RF-OR AC-SQUID 183 : 7.4.4 DC-SQUID 184 7.5 SUPERCONDUCTING CIRCUITS: QUANTUM 186 7.6 QUANTUM GATES 188 7.6.1 SOME BASIC FACTS ABOUT SU(2) AND SO(3) 191 7.6.2 ONE QUBIT OPERATIONS (I): CHARGE-QUBIT . 192 7.6.3 ONE QUBIT OPERATIONS (II): FLUX-QUBIT 197 7.6.4 CHARGE-FLUX QUBIT AND PHASE QUBIT 205 7.6.5 TWO QUBIT OPERATIONS: CHARGE AND FLUX QUBITS 206 ' '* 7.6.6 MEASUREMENT OF CHARGE QUBIT. . 215 8 NONDETERMINISTIC LOGIC GATES IN OPTICAL QUANTUM COMPUTING 223 FEDERICO M. SPEDALIERI, JONATHAN P. DOWLING, AND HWANG LEE 8.1 INTRODUCTION 223 8.2 PHOTON AS A QUBIT 225 ; -: 8.3 LINEAR OPTICAL QUANTUM COMPUTING 228 8.4 NONDETERMINISTIC TWO-QUBIT GATE 237 8.5 ANCILLA-STATE PREPARATION 241 XIV 8.6 CLUSTER-STATE APPROACH AND GATE FIDELITY 245 8.7 APPENDICES 250 QUANTUM INFORMATION 257 9 EXPLOITING ENTANGLEMENT IN QUANTUM CRYPTOGRAPHIC PROBES 259 HOWARD E. BRANDT 9.1 INTRODUCTION 260 9.2 PROBE DESIGNS BASED ON F/ (1) 263 9.3 PROBE DESIGNS BASED ON T/ (2) 271 9.4 PROBE DESIGNS BASED ON U^ 273 9.5 CONCLUSION 276 APPENDIX A RENYI INFORMATION GAIN 277 APPENDIX B MAXIMUM RENYI INFORMATION GAIN 281 10 NONBINARY STABILIZER CODES 287 PRADEEP KIRAN SARVEPALLI, SALAH A. ALY, AND ANDREAS KLAPPENECKER 10.1 INTRODUCTION 287 10.2 STABILIZER CODES 289 10.2.1 ERRORBASES 290 10.2.2 STABILIZER CODES 290 10.2.3 STABILIZER AND ERROR CORRECTION 291 10.2.4 MINIMUM DISTANCE 292 10.2.5 PURE AND IMPURE CODES , 292 10.2.6 ENCODING QUANTUM CODES 293 10.3 QUANTUM CODES AND CLASSICAL CODES 293 10.3.1 CODES OVER Q 293 10.3.2 CODES OVER F 2 295 10.4 BOUNDS ON QUANTUM CODES 296 10.5 FAMILIES OF QUANTUM CODES 298 10.5.1 QUANTUM M-ADIC RESIDUE CODES 299 10.5.2 QUANTUM PROJECTIVE REED-MULLER CODES 300 10.5.3 PUNCTURING QUANTUM CODES 302 10.6 CONCLUSION 303 11 ACCESSIBLE INFORMATION ABOUT QUANTUM STATES: AN OPEN OPTIMIZATION PROBLEM 309 JUN SUZUKI, SYED M. ASSAD, AND BERTHOLD-GEORG ENGLERT 11.1 INTRODUCTION 309 XV 11.2 PRELIMINARIES 310 11.2.1 STATES AND MEASUREMENTS . 310 11.2.2 ENTROPY AND INFORMATION 313 , 11.3 THE OPTIMIZATION PROBLEM 316 11.4 THEOREMS 317 11.4.1 CONCAVITY AND CONVEXITY 317 11.4.2 NECESSARY CONDITION 318 11.4.3 SOME BASIC THEOREMS 319 11.4.4 GROUP-COVARIANT CASE 321 11.5 NUMERICAL SEARCH 324 11.6 EXAMPLES 326 11.6.1 TWO QUANTUM STATES IN TWO DIMENSIONS 326 11.6.2 TRINE: Z 3 SYMMETRY IN TWO DIMENSIONS 329 11.6.3 SIX-STATES PROTOCOL: SYMMETRIC GROUP S 3 332 11.6.4 FOUR-GROUP IN FOUR DIMENSIONS 339 11.7 SUMMARY AND OUTLOOK 345 12 QUANTUM ENTANGLEMENT: CONCEPTS AND CRITERIA 349 FU-LI LI AND M. SUHAIL ZUBAIRY 12.1 INTRODUCTION 349 12.2 EPR CORRELATIONS AND QUANTUM ENTANGLEMENT. . . * 353 12.3 ENTANGLEMENT OF PURE STATES . . 356 12.4 CRITERIA ON ENTANGLEMENT OF MIXED STATES 358 12.4.1 PERES-HORODECKI CRITERION 358 12.4.2 SIMON CRITERION 361 12.4.3 DUAN-GIEDKE-CIRAC-ZOLLER CRITERION 363 12.4.4 HILLERY-ZUBAIRY CRITERION 366 12.4.5 SHCHUKIN-VOGEL CRITERION 367 12.5 COHERENCE-INDUCED ENTANGLEMENT 369 12.6 CORRELATED SPONTANEOUS EMISSION LASER AS AN ENTANGLEMENT AMPLIFIER 376 12.6 REMARKS 380 13 PARAMETRIZATIONS OF POSITIVE MATRICES WITH APPLICATIONS 387 M. TSENG, H. ZHOU, AND V. RAMAKRISHNA 13.1 INTRODUCTION 387 13.2 SOURCES OF POSITIVE MATRICES IN QUANTUM THEORY 388 13.3 CHARACTERIZATIONS OF POSITIVE MATRICES 390 13.4 A DIFFERENT PARAMETRIZATION OF POSITIVE MATRICES 393 XVI 13.5 TWO FURTHER APPLICATIONS 397 13.5.1 TOEPLITZ STATES . . '.:». .':'. . 397 13.5.2 CONSTRAINTS ON RELAXATION RATES 400 13.6 CONCLUSIONS '." 403 QUANTUM TOPOLOGY, CATEGORICAL ALGEBRA, AND LOGIC 407 14 QUANTUM COMPUTING AND QUANTUM TOPOLOGY 409 . LOUIS H. KAUFFMAN AND SAMUEL J. LOMONACO , 14.1 INTRODUCTION 409 14.2 KNOTS AND BRAIDS 414 14.3 QUANTUM MECHANICS AND QUANTUM COMPUTATION 419 14.3.1 WHAT IS A QUANTUM COMPUTER? 421 14.4 BRAIDING OPERATORS AND UNIVERSAL QUANTUM GATES 423 14.4.1 UNIVERSAL GATES 426 14.5 A REMARK ABOUT EPR, ENTANGLEMENT AND BELL'S INEQUALITY . . . 430 14.6 THE ARAVIND HYPOTHESIS 433 14.7 SU(2) REPRESENTATIONS OF THE ARTIN BRAID GROUP 434 14.8 THE BRACKET POLYNOMIAL AND THE JONES POLYNOMIAL 440 14.8.1 QUANTUM COMPUTATION OF THE JONES POLYNOMIAL . 445 14.9 QUANTUM TOPOLOGY, COBORDISM CATEGORIES, TEMPERLEY-LIEB ALGEBRA, AND TOPOLOGICAL QUANTUM FIELD THEORY 450 14.10BRAIDING AND TOPOLOGICAL QUANTUM FIELD THEORY 461 14.11 SPIN NETWORKS AND TEMPERLEY-LIEB RECOUPLING THEORY 473 14.11.1 EVALUATIONS 477 14.11.2 SYMMETRY AND UNITARITY 480 14.12FIBONACCI PARTICLES 484 14.13THE FIBONACCI RECOUPLING MODEL 491 14.14QUANTUM COMPUTATION OF COLORED JONES POLYNOMIALS AND THE WITTEN-RESHETIKHIN-TURAEV INVARIANT 504 15 TEMPERLEY-LIEB ALGEBRA: FROM KNOT THEORY TO LOGIC AND COMPUTATION VIA QUANTUM MECHANICS 515 SAMSON ABRAMSKY 15.1 INTRODUCTION 516 15.1.1 KNOT THEORY . . .- 516 15.1.2 CATEGORICAL QUANTUM MECHANICS 516 15.1.3 LOGIC AND COMPUTATION . '. 517 15.1.4 OUTLINE OF THE PAPER ; . . . 517 XVII 15.2 THE TEMPERLEY-LIEB ALGEBRA 518 15.2.1 TEMPERLEY-LIEB ALGEBRA: GENERATORS AND RELATIONS . . . 518 15.2.2 DIAGRAM MONOIDS 519 15.2.3 EXPRESSIVENESS OF THE GENERATORS 520 15.2.4 THETRACE . 521 15.2.5 THE CONNECTION TO KNOTS 522 15.3 THE TEMPERLEY-LIEB CATEGORY 523 15.3.1 PIVOTAL CATEGORIES 524 15.3.2 PIVOTAL DAGGER CATEGORIES 527 15.4 FACTORIZATION AND IDEMPOTENTS 528 15.5 CATEGORICAL QUANTUM MECHANICS 530 15.5.1 OUTLINE OF THE APPROACH 531 15.5.2 QUANTUM NON-LOGIC VS. QUANTUM HYPER-LOGIC 533 15.5.3 REMARKS 536 15.6 PLANAR GEOMETRY OF INTERACTION AND THE TEMPERLEY-LIEB ALGEBRA 537 15.6.1 SOME PRELIMINARY NOTIONS 538 15.6.2 FORMALIZING DIAGRAMS 539 15.6.3 CHARACTERIZING PLANARITY 540 15.6.4 THE TEMPERLEY-LIEB CATEGORY 544 15.7 PLANAR A-CALCULUS 548 15.7.1 THE A-CALCULUS 549 15.7.2 TYPES 549 15.7.3 INTERPRETATION IN PIVOTAL CATEGORIES 551 15.7.4 AN EXAMPLE 552 15.7.5 DISCUSSION 553 15.8 FURTHER DIRECTIONS 553 16 QUANTUM MEASUREMENTS WITHOUT SUMS 559 BOB COECKE AND DUSKO PAVLOVIC 16.1 INTRODUCTION 559 16.2 CATEGORICAL SEMANTICS 562 16.2.1 F-COMPACT CATEGORIES 562 16.2.2 GRAPHICAL CALCULUS 564 16.2.3 SCALARS, TRACE, AND PARTIAL TRANSPOSE 566 16.3 SUMS AND BASES IN HILBERT SPACES 568 16.3.1 SUMS IN QUANTUM MECHANICS 568 16.3.2 NO-CLONING AND EXISTENCE OF A NATURAL DIAGONAL . 569 16.3.3 MEASUREMENT AND BASES 570 16.3.4 VANISHING OF NON-DIAGONAL ELEMENTS AND DELETION . . . 571 16.3.5 CANONICAL BASES 572 XV111 16.4 CLASSICAL OBJECTS 572 16.4.1 SPECIAL F-COMPACT FROBENIUS ALGEBRAS 573 16.4.2 SELF-ADJOINTNESS RELATIVE TO A CLASSICAL OBJECT 576 16.4.3 GHZ STATES AS CLASSICAL OBJECTS . 578 16.4.4 EXTRACTING THE CLASSICAL WORLD 579 16.5 QUANTUM SPECTRA 580 16.5.1 COALGEBRAIC CHARACTERIZATION OF SPECTRA 581 16.5.2 CHARACTERIZATION OF X-CONCEPTS 582 16.6 QUANTUM MEASUREMENTS 582 16.6.1 THE CPM-CONSTRUCTION 584 16.6.2 FORMAL DECOHERENCE 585 16.6.3 DEMOLITION MEASUREMENTS 586 16.7 QUANTUM TELEPORTATION 587 16.8 DENSE CODING 592 APPENDIX PANEL REPORT ON THE FORWARD LOOKING DISCUSSION 597 INDEX 603
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spellingShingle	Mathematics of quantum computation and quantum technology Chapman & Hall CRC applied mathematics and nonlinear science series Ordinateurs quantiques - Mathématiques Théorie quantique - Mathématiques Mathematik Quantentheorie Quantum computers Mathematics Quantum theory Mathematics Quantentheorie (DE-588)4047992-4 gnd Mathematik (DE-588)4037944-9 gnd Quantencomputer (DE-588)4533372-5 gnd
subject_GND	(DE-588)4047992-4 (DE-588)4037944-9 (DE-588)4533372-5
title	Mathematics of quantum computation and quantum technology
title_auth	Mathematics of quantum computation and quantum technology
title_exact_search	Mathematics of quantum computation and quantum technology
title_exact_search_txtP	Mathematics of quantum computation and quantum technology
title_full	Mathematics of quantum computation and quantum technology ed. by Goong Chen ...
title_fullStr	Mathematics of quantum computation and quantum technology ed. by Goong Chen ...
title_full_unstemmed	Mathematics of quantum computation and quantum technology ed. by Goong Chen ...
title_short	Mathematics of quantum computation and quantum technology
title_sort	mathematics of quantum computation and quantum technology
topic	Ordinateurs quantiques - Mathématiques Théorie quantique - Mathématiques Mathematik Quantentheorie Quantum computers Mathematics Quantum theory Mathematics Quantentheorie (DE-588)4047992-4 gnd Mathematik (DE-588)4037944-9 gnd Quantencomputer (DE-588)4533372-5 gnd
topic_facet	Ordinateurs quantiques - Mathématiques Théorie quantique - Mathématiques Mathematik Quantentheorie Quantum computers Mathematics Quantum theory Mathematics Quantencomputer
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