Holdings: Explorations in quantum computing

Explorations in quantum computing:

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Bibliographic Details
Main Author:	Williams, Colin P. (Author)
Format:	Book
Language:	English
Published:	London Springer 2011
Edition:	second edition
Series:	Texts in computer science
Subjects:	Quantenphysik Quantencomputer Computerphysik
Online Access:	Inhaltstext Inhaltsverzeichnis
Physical Description:	XXII, 717 Seiten Diagramme
ISBN:	9781846288869

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adam_text	IMAGE 1 CONTENTS PART I WHAT IS QUANTUM COMPUTING? 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 TRENDS IN COMPUTER MINIATURIZATION . . . . . . . . . . . . . . . . 4 1.2 IMPLICIT ASSUMPTIONS IN THE THEORY OF COMPUTATION . . . . . . . . 7 1.3 QUANTIZATION: FROM BITS TO QUBITS . . . . . . . . . . . . . . . . . 8 1.3.1 KET VECTOR REPRESENTATION OF A QUBIT . . . . . . . . . . . 9 1.3.2 SUPERPOSITION STATES OF A SINGLE QUBIT . . . . . . . . . . 9 1.3.3 BLOCH SPHERE PICTURE OF A QUBIT . . . . . . . . . . . . . . 11 1.3.4 READING THE BIT VALUE OF A QUBIT . . . . . . . . . . . . . 15 1.4 MULTI-QUBIT QUANTUM MEMORY REGISTERS . . . . . . . . . . . . . . 17 1.4.1 THE COMPUTATIONAL BASIS . . . . . . . . . . . . . . . . . 17 1.4.2 DIRECT PRODUCT FOR FORMING MULTI-QUBIT STATES . . . . . . 19 1.4.3 INTERFERENCE EFFECTS . . . . . . . . . . . . . . . . . . . . 20 1.4.4 ENTANGLEMENT . . . . . . . . . . . . . . . . . . . . . . . 21 1.5 EVOLVING A QUANTUM MEMORY REGISTER: SCHROEDINGER'S EQUATION . 23 1.5.1 SCHROEDINGER'S EQUATION . . . . . . . . . . . . . . . . . . 24 1.5.2 HAMILTONIANS . . . . . . . . . . . . . . . . . . . . . . . . 24 1.5.3 SOLUTION AS A UNITARY EVOLUTION OF THE INITIAL STATE . . . . 25 1.5.4 COMPUTATIONAL INTERPRETATION . . . . . . . . . . . . . . . 26 1.6 EXTRACTING ANSWERS FROM QUANTUM COMPUTERS . . . . . . . . . . 26 1.6.1 OBSERVABLES IN QUANTUM MECHANICS . . . . . . . . . . . 26 1.6.2 OBSERVING IN THE COMPUTATIONAL BASIS . . . . . . . . . . 29 1.6.3 ALTERNATIVE BASES . . . . . . . . . . . . . . . . . . . . . 30 1.6.4 CHANGE OF BASIS . . . . . . . . . . . . . . . . . . . . . . 32 1.6.5 OBSERVING IN AN ARBITRARY BASIS . . . . . . . . . . . . . . 34 1.7 QUANTUM PARALLELISM AND THE DEUTSCH-JOZSA ALGORITHM . . . . . . 35 1.7.1 THE PROBLEM: IS F (X) CONSTANT OR BALANCED? . . . . . . . 36 1.7.2 EMBEDDING F (X) IN A QUANTUM BLACK-BOX FUNCTION . . . 37 1.7.3 MOVING FUNCTION VALUES BETWEEN KETS AND PHASE FACTORS 38 1.7.4 INTERFERENCE REVEALS THE DECISION . . . . . . . . . . . . . 39 1.7.5 GENERALIZED DEUTSCH-JOZSA PROBLEM . . . . . . . . . . . 40 XI IMAGE 2 XII CONTENTS 1.8 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 1.9 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2 QUANTUM GATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.1 CLASSICAL LOGIC GATES . . . . . . . . . . . . . . . . . . . . . . . . 52 2.1.1 BOOLEAN FUNCTIONS AND COMBINATIONAL LOGIC . . . . . . . 52 2.1.2 IRREVERSIBLE GATES: AND AND OR . . . . . . . . . . . . . 53 2.1.3 UNIVERSAL GATES: NAND AND NOR . . . . . . . . . . . . 55 2.1.4 REVERSIBLE GATES: NOT, SWAP, AND CNOT . . . . . . . . 57 2.1.5 UNIVERSAL REVERSIBLE GATES: FREDKIN AND TOFFOLI . . 60 2.1.6 REVERSIBLE GATES EXPRESSED AS PERMUTATION MATRICES . . . 61 2.1.7 WILL FUTURE CLASSICAL COMPUTERS BE REVERSIBLE? . . . . . 63 2.1.8 COST OF SIMULATING IRREVERSIBLE COMPUTATIONS REVERSIBLY 64 2.1.9 ANCILLAE IN REVERSIBLE COMPUTING . . . . . . . . . . . . 66 2.2 UNIVERSAL REVERSIBLE BASIS . . . . . . . . . . . . . . . . . . . . . 67 2.2.1 CAN ALL BOOLEAN CIRCUITS BE SIMULATED REVERSIBLY? . . . 68 2.3 QUANTUM LOGIC GATES . . . . . . . . . . . . . . . . . . . . . . . . 69 2.3.1 FROM QUANTUM DYNAMICS TO QUANTUM GATES . . . . . . . 70 2.3.2 PROPERTIES OF QUANTUM GATES ARISING FROM UNITARITY . . . 71 2.4 1-QUBIT GATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.4.1 SPECIAL 1-QUBIT GATES . . . . . . . . . . . . . . . . . . . 71 2.4.2 ROTATIONS ABOUT THE X -, Y -, AND Z -AXES . . . . . . . . . . 76 2.4.3 ARBITRARY 1-QUBIT GATES: THE PAULI DECOMPOSITION . . . . 81 2.4.4 DECOMPOSITION OF R X GATE . . . . . . . . . . . . . . . . 83 2.5 CONTROLLED QUANTUM GATES . . . . . . . . . . . . . . . . . . . . . 83 2.5.1 MEANING OF A "CONTROLLED" GATE IN THE QUANTUM CONTEXT 85 2.5.2 SEMI-CLASSICAL CONTROLLED GATES . . . . . . . . . . . . . 86 2.5.3 MULTIPLY-CONTROLLED GATES . . . . . . . . . . . . . . . . . 87 2.5.4 CIRCUIT FOR CONTROLLED- U . . . . . . . . . . . . . . . . . . 87 2.5.5 FLIPPING THE CONTROL AND TARGET QUBITS . . . . . . . . . . 90 2.5.6 CONTROL-ON- \| 0 * QUANTUM GATES . . . . . . . . . . . . . . 90 2.5.7 CIRCUIT FOR CONTROLLED-CONTROLLED- U . . . . . . . . . . . 91 2.6 UNIVERSAL QUANTUM GATES . . . . . . . . . . . . . . . . . . . . . . 92 2.7 SPECIAL 2-QUBIT GATES . . . . . . . . . . . . . . . . . . . . . . . . 94 2.7.1 CSIGN, SWAP * , ISWAP, BERKELEY B . . . . . . . . . . . 95 2.7.2 INTERRELATIONSHIPS BETWEEN TYPES OF 2-QUBIT GATES . . . . 97 2.8 ENTANGLING POWER OF QUANTUM GATES . . . . . . . . . . . . . . . . 100 2.8.1 "TANGLE" AS A MEASURE OF THE ENTANGLEMENT WITHIN A STATE . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 2.8.2 "ENTANGLING POWER" AS THE MEAN TANGLE GENERATED BY A GATE . . . . . . . . . . . . . . . . . . . . . . . . . . 103 2.8.3 CNOT FROM ANY MAXIMALLY ENTANGLING GATE . . . . . . . 106 2.8.4 THE MAGIC BASIS AND ITS EFFECT ON ENTANGLING POWER . . . 106 2.9 ARBITRARY 2-QUBIT GATES: THE KRAUSS-CIRAC DECOMPOSITION . . . . 107 2.9.1 ENTANGLING POWER OF AN ARBITRARY 2-QUBIT GATE . . . . . 109 IMAGE 3 CONTENTS XIII 2.9.2 CIRCUIT FOR AN ARBITRARY REAL 2-QUBIT GATE . . . . . . . . 110 2.9.3 CIRCUIT FOR AN ARBITRARY COMPLEX 2-QUBIT GATE . . . . . . 111 2.9.4 CIRCUIT FOR AN ARBITRARY 2-QUBIT GATE USING SWAP * . . . 111 2.10 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 2.11 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 3 QUANTUM CIRCUITS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 3.1 QUANTUM CIRCUIT DIAGRAMS . . . . . . . . . . . . . . . . . . . . . 123 3.2 COMPUTING THE UNITARY MATRIX FOR A GIVEN QUANTUM CIRCUIT . . . 124 3.2.1 COMPOSING QUANTUM GATES IN SERIES: THE DOT PRODUCT . 126 3.2.2 COMPOSING QUANTUM GATES IN PARALLEL: THE DIRECT PRODUCT . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.2.3 COMPOSING QUANTUM GATES CONDITIONALLY: THE DIRECT SUM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 3.2.4 MEASURES OF QUANTUM CIRCUIT COMPLEXITY . . . . . . . . 130 3.3 QUANTUM PERMUTATIONS . . . . . . . . . . . . . . . . . . . . . . . 131 3.3.1 QUBIT REVERSAL PERMUTATION: P 2 N . . . . . . . . . . . . . 131 3.3.2 QUBIT CYCLIC LEFT SHIFT PERMUTATION: * 2 N . . . . . . . . . 135 3.3.3 AMPLITUDE DOWNSHIFT PERMUTATION: Q 2 N . . . . . . . . . 137 3.3.4 QUANTUM PERMUTATIONS FOR CLASSICAL MICROPROCESSOR DESIGN? . . . . . . . . . . . . . . . . . . . . . . . . . . 139 3.4 QUANTUM FOURIER TRANSFORM: QFT . . . . . . . . . . . . . . . . . 140 3.4.1 CONTINUOUS SIGNALS AS SUMS OF SINES AND COSINES . . . . 141 3.4.2 DISCRETE SIGNALS AS SAMPLES OF CONTINUOUS SIGNALS . . . . 142 3.4.3 DISCRETE SIGNALS AS SUPERPOSITIONS . . . . . . . . . . . . 144 3.4.4 QFT OF A COMPUTATIONAL BASIS STATE . . . . . . . . . . . 145 3.4.5 QFT OF A SUPERPOSITION . . . . . . . . . . . . . . . . . . 147 3.4.6 QFT MATRIX . . . . . . . . . . . . . . . . . . . . . . . . 148 3.4.7 QFT CIRCUIT . . . . . . . . . . . . . . . . . . . . . . . . 150 3.5 QUANTUM WAVELET TRANSFORM: QWT . . . . . . . . . . . . . . . . 151 3.5.1 CONTINUOUS VERSUS DISCRETE WAVELET TRANSFORMS . . . . . 152 3.5.2 DETERMINING THE VALUES OF THE WAVELET FILTER COEFFICIENTS 154 3.5.3 FACTORIZATION OF DAUBECHIES D ( 4 ) 2 N WAVELET KERNEL . . . . 157 3.5.4 QUANTUM CIRCUIT FOR D ( 4 ) 2 N WAVELET KERNEL . . . . . . . . 158 3.5.5 QUANTUM CIRCUIT FOR THE WAVELET PACKET ALGORITHM . . . 158 3.5.6 QUANTUM CIRCUIT WAVELET PYRAMIDAL ALGORITHM . . . . . 160 3.6 QUANTUM COSINE TRANSFORM: QCT . . . . . . . . . . . . . . . . . 162 3.6.1 SIGNALS AS SUMS OF COSINES ONLY . . . . . . . . . . . . . 163 3.6.2 DISCRETE COSINE TRANSFORM DCT-II AND ITS RELATION TO DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 3.6.3 QCT II N TRANSFORMATION . . . . . . . . . . . . . . . . . . 165 3.6.4 QCT II N MATRIX . . . . . . . . . . . . . . . . . . . . . . . 165 3.6.5 QCT II N CIRCUIT . . . . . . . . . . . . . . . . . . . . . . . 166 3.7 CIRCUITS FOR A ARBITRARY UNITARY MATRICES . . . . . . . . . . . . . . 172 3.7.1 USES OF QUANTUM CIRCUIT DECOMPOSITIONS . . . . . . . . 173 IMAGE 4 XIV CONTENTS 3.7.2 CHOICE OF WHICH GATE SET TO USE . . . . . . . . . . . . . 173 3.7.3 CIRCUIT COMPLEXITY TO IMPLEMENT ARBITRARY UNITARY MATRICES . . . . . . . . . . . . . . . . . . . . . . . . . . 173 3.7.4 ALGEBRAIC METHOD . . . . . . . . . . . . . . . . . . . . . 174 3.7.5 SIMPLIF ICATION VIA REWRITE RULES . . . . . . . . . . . . . 178 3.7.6 NUMERICAL METHOD . . . . . . . . . . . . . . . . . . . . . 180 3.7.7 RE-USE METHOD . . . . . . . . . . . . . . . . . . . . . . . 184 3.8 PROBABILISTIC NON-UNITARY QUANTUM CIRCUITS . . . . . . . . . . . . 190 3.8.1 HAMILTONIAN BUILT FROM NON-UNITARY OPERATOR . . . . . . 191 3.8.2 UNITARY EMBEDDING OF THE NON-UNITARY OPERATOR . . . . . 191 3.8.3 NON-UNITARILY TRANSFORMED DENSITY MATRIX . . . . . . . . 191 3.8.4 SUCCESS PROBABILITY . . . . . . . . . . . . . . . . . . . . 193 3.8.5 FIDELITY WHEN SUCCESSFUL . . . . . . . . . . . . . . . . . 193 3.9 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 3.10 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 4 QUANTUM UNIVERSALITY, COMPUTABILITY, & COMPLEXITY . . . . . . . . . 201 4.1 MODELS OF COMPUTATION . . . . . . . . . . . . . . . . . . . . . . . 202 4.1.1 THE INSPIRATION BEHIND TURING'S MODEL OF COMPUTATION: THE ENTSCHEIDUNGSPROBLEM . . . . . . . . . . . . . . . . 202 4.1.2 DETERMINISTIC TURING MACHINES . . . . . . . . . . . . . . 204 4.1.3 PROBABILISTIC TURING MACHINES . . . . . . . . . . . . . . . 205 4.1.4 THE ALTERNATIVE GOEDEL, CHURCH, AND POST MODELS . . . . . 207 4.1.5 EQUIVALENCE OF THE MODELS OF COMPUTATION . . . . . . . . 208 4.2 UNIVERSALITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 4.2.1 THE STRONG CHURCH-TURING THESIS . . . . . . . . . . . . . 208 4.2.2 QUANTUM CHALLENGE TO THE STRONG CHURCH-TURING THESIS . 209 4.2.3 QUANTUM TURING MACHINES . . . . . . . . . . . . . . . . 210 4.3 COMPUTABILITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 4.3.1 DOES QUANTUM COMPUTABILITY OFFER ANYTHING NEW? . . . 214 4.3.2 DECIDABILITY: RESOLUTION OF THE ENTSCHEIDUNGSPROBLEM . . 215 4.3.3 PROOF VERSUS TRUTH: GOEDEL'S INCOMPLETENESS THEOREM . . 217 4.3.4 PROVING VERSUS PROVIDING PROOF . . . . . . . . . . . . . . 218 4.4 COMPLEXITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 4.4.1 POLYNOMIAL VERSUS EXPONENTIAL GROWTH . . . . . . . . . . 223 4.4.2 BIG O , * AND * NOTATION . . . . . . . . . . . . . . . . . 225 4.4.3 CLASSICAL COMPLEXITY ZOO . . . . . . . . . . . . . . . . . 225 4.4.4 QUANTUM COMPLEXITY ZOO . . . . . . . . . . . . . . . . . 229 4.5 WHAT ARE POSSIBLE "KILLER-APS" FOR QUANTUM COMPUTERS? . . . . 233 4.6 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 4.7 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 PART II WHAT CAN YOU DO WITH A QUANTUM COMPUTER? 5 PERFORMING SEARCH WITH A QUANTUM COMPUTER . . . . . . . . . . . . 241 5.1 THE UNSTRUCTURED SEARCH PROBLEM . . . . . . . . . . . . . . . . . 242 IMAGE 5 CONTENTS XV 5.1.1 MEANING OF THE ORACLE . . . . . . . . . . . . . . . . . . . 243 5.2 CLASSICAL SOLUTION: GENERATE-AND-TEST . . . . . . . . . . . . . . . 244 5.3 QUANTUM SOLUTION: GROVER'S ALGORITHM . . . . . . . . . . . . . . 245 5.4 HOW DOES GROVER'S ALGORITHM WORK? . . . . . . . . . . . . . . . 247 5.4.1 HOW MUCH AMPLITUDE AMPLIFICATION IS NEEDED TO ENSURE SUCCESS? . . . . . . . . . . . . . . . . . . . . 248 5.4.2 AN EXACT ANALYSIS OF AMPLITUDE AMPLIFICATION . . . . . 249 5.4.3 THE ORACLE IN AMPLITUDE AMPLIF ICATION . . . . . . . . . . 250 5.5 QUANTUM SEARCH WITH MULTIPLE SOLUTIONS . . . . . . . . . . . . . 251 5.5.1 AMPLITUDE AMPLIFICATION IN THE CASE OF MULTIPLE SOLUTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . 252 5.6 CAN GROVER'S ALGORITHM BE BEATEN? . . . . . . . . . . . . . . . . 254 5.7 SOME APPLICATIONS OF QUANTUM SEARCH . . . . . . . . . . . . . . 255 5.7.1 SPEEDING UP RANDOMIZED ALGORITHMS . . . . . . . . . . 255 5.7.2 SYNTHESIZING ARBITRARY SUPERPOSITIONS . . . . . . . . . . 256 5.8 QUANTUM SEARCHING OF REAL DATABASES . . . . . . . . . . . . . . . 260 5.9 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 5.10 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 6 CODE BREAKING WITH A QUANTUM COMPUTER . . . . . . . . . . . . . . 263 6.1 CODE-MAKING AND CODE-BREAKING . . . . . . . . . . . . . . . . . 264 6.1.1 CODE-BREAKING: THE ENIGMA CODE AND ALAN TURING . . . 265 6.2 PUBLIC KEY CRYPTOSYSTEMS . . . . . . . . . . . . . . . . . . . . . 267 6.2.1 THE RSA PUBLIC-KEY CRYPTOSYSTEM . . . . . . . . . . . . 267 6.2.2 EXAMPLE OF THE RSA CRYPTOSYSTEM . . . . . . . . . . . . 271 6.3 SHOR'S FACTORING ALGORITHM FOR BREAKING RSA QUANTUMLY . . . . 272 6.3.1 THE CONTINUED FRACTION TRICK AT THE END OF SHOR'S ALGORITHM . . . . . . . . . . . . . . . . . . . . . . . . . 276 6.3.2 EXAMPLE TRACE OF SHOR'S ALGORITHM . . . . . . . . . . . . 280 6.4 BREAKING ELLIPTIC CURVE CRYPTOSYSTEMS WITH A QUANTUM COMPUTER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 6.5 BREAKING DES WITH A QUANTUM COMPUTER . . . . . . . . . . . . . 287 6.6 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 6.7 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 7 SOLVING NP-COMPLETE PROBLEMS WITH A QUANTUM COMPUTER . . . . . 293 7.1 IMPORTANCE AND UBIQUITY OF NP-COMPLETE PROBLEMS . . . . . . . 295 7.1.1 WORST CASE COMPLEXITY OF SOLVING NP-COMPLETE PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . 296 7.2 PHYSICS-INSPIRED VIEW OF COMPUTATIONAL COMPLEXITY . . . . . . . 297 7.2.1 PHASE TRANSITION PHENOMENA IN PHYSICS . . . . . . . . . 297 7.2.2 PHASE TRANSITION PHENOMENA IN MATHEMATICS . . . . . . . 299 7.2.3 COMPUTATIONAL PHASE TRANSITIONS . . . . . . . . . . . . . 299 7.2.4 WHERE ARE THE REALLY HARD PROBLEMS? . . . . . . . . . . 302 7.3 QUANTUM ALGORITHMS FOR NP-COMPLETE PROBLEMS . . . . . . . . . 302 IMAGE 6 XVI CONTENTS 7.3.1 QUANTUM SOLUTION USING GROVER'S ALGORITHM . . . . . . . 303 7.3.2 STRUCTURED SEARCH SPACES: TREES AND LATTICES . . . . . . . 304 7.4 QUANTUM SOLUTION USING NESTED GROVER'S ALGORITHM . . . . . . . 308 7.4.1 THE CORE QUANTUM ALGORITHM . . . . . . . . . . . . . . . 308 7.4.2 ANALYSIS OF QUANTUM STRUCTURED SEARCH . . . . . . . . . 309 7.4.3 QUANTUM CIRCUIT FOR QUANTUM STRUCTURED SEARCH . . . . . 312 7.4.4 QUANTUM AVERAGE-CASE COMPLEXITY . . . . . . . . . . . 312 7.5 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 7.6 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 8 QUANTUM SIMULATION WITH A QUANTUM COMPUTER . . . . . . . . . . . 319 8.1 CLASSICAL COMPUTER SIMULATIONS OF QUANTUM PHYSICS . . . . . . . 320 8.1.1 EXACT SIMULATION AND THE PROBLEM OF MEMORY . . . . . . 321 8.1.2 EXACT SIMULATION AND THE PROBLEM OF ENTANGLEMENT . . . 321 8.1.3 APPROXIMATE SIMULATION AND THE PROBLEM OF FIDELITY . . 322 8.2 QUANTUM COMPUTER SIMULATIONS OF QUANTUM PHYSICS . . . . . . . 325 8.2.1 FEYNMAN CONCEIVES OF A UNIVERSAL QUANTUM SIMULATOR . 326 8.2.2 QUANTUM SYSTEMS WITH LOCAL INTERACTIONS . . . . . . . . 326 8.2.3 LLOYD-ZALKA-WIESNER QUANTUM SIMULATION ALGORITHM . . 327 8.3 EXTRACTING RESULTS FROM QUANTUM SIMULATIONS EFFICIENTLY . . . . . 328 8.3.1 SINGLE ANCILLA-ASSISTED READOUT . . . . . . . . . . . . . 328 8.3.2 MULTI-ANCILLA-ASSISTED READOUT . . . . . . . . . . . . . . 330 8.3.3 TOMOGRAPHY VERSUS SPECTROSCOPY . . . . . . . . . . . . . 332 8.3.4 EVALUATING CORRELATION FUNCTIONS . . . . . . . . . . . . . 333 8.4 FERMIONIC SIMULATIONS ON QUANTUM COMPUTERS . . . . . . . . . . 334 8.4.1 INDISTINGUISHABILITY AND IMPLICATIONS FOR PARTICLE STATISTICS . . . . . . . . . . . . . . . . . . . . . . . . . . 334 8.4.2 SYMMETRIC VERSUS ANTI-SYMMETRIC STATE VECTORS . . . . . 335 8.4.3 BOSONS AND FERMIONS . . . . . . . . . . . . . . . . . . . 336 8.4.4 BOSE-EINSTEIN STATISTICS . . . . . . . . . . . . . . . . . . 337 8.4.5 PAULI EXCLUSION PRINCIPLE AND FERMI-DIRAC STATISTICS . . . 337 8.4.6 FERMIONIC SIMULATIONS VIA THE JORDAN-WIGNER TRANSFORMATION . . . . . . . . . . . . . . . . . . . . . . 339 8.4.7 FERMIONIC SIMULATIONS VIA TRANSFORMATION TO NON-INTERACTING HAMILTONIANS . . . . . . . . . . . . . 341 8.5 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 8.6 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 9 QUANTUM CHEMISTRY WITH A QUANTUM COMPUTER . . . . . . . . . . . 349 9.1 CLASSICAL COMPUTING APPROACH TO QUANTUM CHEMISTRY . . . . . . 349 9.1.1 CLASSICAL EIGENVALUE ESTIMATION VIA THE LANCZOS ALGORITHM . . . . . . . . . . . . . . . . . . . . . . . . . 351 9.2 QUANTUM EIGENVALUE ESTIMATION VIA PHASE ESTIMATION . . . . . . 352 9.2.1 THE "PHASE" STATE . . . . . . . . . . . . . . . . . . . . . 352 9.2.2 BINARY FRACTION REPRESENTATION OF THE PHASE FACTOR . . . 353 9.3 QUANTUM PHASE ESTIMATION . . . . . . . . . . . . . . . . . . . . . 354 IMAGE 7 CONTENTS XVII 9.4 EIGENVALUE KICK-BACK FOR SYNTHESIZING THE PHASE STATE . . . . . . 357 9.5 QUANTUM EIGENVALUE ESTIMATION ALGORITHMS . . . . . . . . . . . . 361 9.5.1 ABRAMS-LLOYD EIGENVALUE ESTIMATION ALGORITHM . . . . . 361 9.5.2 KITAEV EIGENVALUE ESTIMATION ALGORITHM . . . . . . . . . 361 9.6 QUANTUM CHEMISTRY BEYOND EIGENVALUE ESTIMATION . . . . . . . . 364 9.7 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 9.8 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 10 MATHEMATICS ON A QUANTUM COMPUTER . . . . . . . . . . . . . . . . . 369 10.1 QUANTUM FUNCTIONAL ANALYSIS . . . . . . . . . . . . . . . . . . . 369 10.1.1 QUANTUM MEAN ESTIMATION . . . . . . . . . . . . . . . . 370 10.1.2 QUANTUM COUNTING . . . . . . . . . . . . . . . . . . . . 371 10.2 QUANTUM ALGEBRAIC NUMBER THEORY . . . . . . . . . . . . . . . . 375 10.2.1 THE CATTLE PROBLEM OF ARCHIMEDES AND PELL'S EQUATION . 375 10.2.2 WHY SOLVING PELL'S EQUATION IS HARD . . . . . . . . . . . 376 10.2.3 SOLUTION BY FINDING THE "REGULATOR" . . . . . . . . . . . 377 10.2.4 THE REGULATOR AND PERIOD FINDING . . . . . . . . . . . . 378 10.2.5 QUANTUM CORE OF HALLGREN'S ALGORITHM . . . . . . . . . . 378 10.2.6 HALLGREN'S QUANTUM ALGORITHM FOR SOLVING PELL'S EQUATION . . . . . . . . . . . . . . . . . . . . . . . . . . 378 10.2.7 WHAT IS THE SIGNIFICANCE OF PELL'S EQUATION? . . . . . . . 381 10.3 QUANTUM SIGNAL, IMAGE, AND DATA PROCESSING . . . . . . . . . . . 382 10.3.1 CLASSICAL-TO-QUANTUM ENCODING . . . . . . . . . . . . . . 382 10.3.2 QUANTUM IMAGE PROCESSING: 2D QUANTUM TRANSFORMS . . 384 10.3.3 QUANTUM-TO-CLASSICAL READOUT . . . . . . . . . . . . . . 385 10.4 QUANTUM WALKS . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 10.4.1 ONE-DIMENSIONAL QUANTUM WALKS . . . . . . . . . . . . 387 10.4.2 EXAMPLE: BIASED INITIAL COIN STATE & HADAMARD COIN . . 389 10.4.3 EXAMPLE: SYMMETRIC INITIAL COIN STATE & HADAMARD COIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 10.4.4 EXAMPLE: CHIRAL INITIAL COIN STATE & HADAMARD COIN . . 392 10.4.5 EXAMPLE: SYMMETRIC INITIAL COIN STATE & NON-HADAMARD COIN . . . . . . . . . . . . . . . . . . . 393 10.4.6 QUANTUM WALKS CAN SPREAD FASTER THAN CLASSICAL WALKS . 395 10.5 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 10.6 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 PART III WHAT CAN YOU DO WITH QUANTUM INFORMATION? 11 QUANTUM INFORMATION . . . . . . . . . . . . . . . . . . . . . . . . . . 403 11.1 WHAT IS CLASSICAL INFORMATION? . . . . . . . . . . . . . . . . . . . 404 11.1.1 CLASSICAL SOURCES: THE SHANNON ENTROPY . . . . . . . . . 405 11.1.2 MAXIMAL COMPRESSION (SOURCE CODING THEOREM) . . . . 407 11.1.3 RELIABLE TRANSMISSION (CHANNEL CODING THEOREM) . . . . 408 11.1.4 UNSTATED ASSUMPTIONS REGARDING CLASSICAL INFORMATION . 410 IMAGE 8 XVIII CONTENTS 11.2 WHAT IS QUANTUM INFORMATION? . . . . . . . . . . . . . . . . . . . 411 11.2.1 PURE STATES CF. MIXED STATES . . . . . . . . . . . . . . . . 411 11.2.2 MIXED STATES FROM PARTIAL KNOWLEDGE: THE DENSITY OPERATOR . . . . . . . . . . . . . . . . . . . . . . . . . . 411 11.2.3 MIXED STATES FROM PARTIAL IGNORANCE: THE PARTIAL TRACE . 417 11.2.4 MIXED STATES AS PARTS OF LARGER PURE STATES: "PURIFICATIONS" . . . . . . . . . . . . . . . . . . . . . . . 419 11.2.5 QUANTIFYING MIXEDNESS . . . . . . . . . . . . . . . . . . 420 11.3 ENTANGLEMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 11.3.1 SEPARABLE STATES VERSUS ENTANGLED STATES . . . . . . . . . 422 11.3.2 SIGNALLING ENTANGLEMENT VIA ENTANGLEMENT WITNESSES . . 423 11.3.3 SIGNALLING ENTANGLEMENT VIA THE PERES-HORODECKI CRITERION . . . . . . . . . . . . . . . . . . . . . . . . . . 425 11.3.4 QUANTIFYING ENTANGLEMENT . . . . . . . . . . . . . . . . . 429 11.3.5 MAXIMALLY ENTANGLED PURE STATES . . . . . . . . . . . . . 431 11.3.6 MAXIMALLY ENTANGLED MIXED STATES . . . . . . . . . . . . 432 11.3.7 THE SCHMIDT DECOMPOSITION OF A PURE ENTANGLED STATE . 433 11.3.8 ENTANGLEMENT DISTILLATION . . . . . . . . . . . . . . . . . 436 11.3.9 ENTANGLEMENT SWAPPING . . . . . . . . . . . . . . . . . . 441 11.3.10 ENTANGLEMENT IN "WARM" BULK MATTER . . . . . . . . . . 443 11.4 COMPRESSING QUANTUM INFORMATION . . . . . . . . . . . . . . . . 444 11.4.1 QUANTUM SOURCES: THE VON NEUMANN ENTROPY . . . . . . 445 11.4.2 SCHUMACHER-JOZSA QUANTUM DATA COMPRESSION . . . . . 445 11.4.3 "DISCARD-ON-FAIL" QUANTUM DATA COMPRESSION PROTOCOL . 447 11.4.4 "AUGMENT-ON-FAIL" QUANTUM DATA COMPRESSION PROTOCOL . . . . . . . . . . . . . . . . . . . . . . . . . . 449 11.4.5 QUANTUM CIRCUIT FOR SCHUMACHER-JOZSA COMPRESSOR . . . 450 11.4.6 IS EXPONENTIAL COMPRESSION POSSIBLE? . . . . . . . . . . 452 11.5 SUPERDENSE CODING . . . . . . . . . . . . . . . . . . . . . . . . . 453 11.5.1 BELL STATES . . . . . . . . . . . . . . . . . . . . . . . . . 454 11.5.2 INTERCONVERSION BETWEEN BELL STATES BY LOCAL ACTIONS . . 455 11.5.3 SUPERDENSE CODING PROTOCOL . . . . . . . . . . . . . . . 455 11.6 CLONING QUANTUM INFORMATION . . . . . . . . . . . . . . . . . . . 457 11.6.1 HISTORICAL ROOTS AND IMPORTANCE OF QUANTUM CLONING . . 457 11.6.2 IMPOSSIBILITY OF EXACT DETERMINISTIC QUANTUM CLONING . 458 11.6.3 UNIVERSAL APPROXIMATE QUANTUM CLONING . . . . . . . . 460 11.6.4 CIRCUIT FOR QUANTUM CLONING . . . . . . . . . . . . . . . 463 11.6.5 USABILITY OF THE QUANTUM CLONES . . . . . . . . . . . . . 464 11.6.6 UNIVERSAL PROBABILISTIC QUANTUM CLONING . . . . . . . . . 468 11.6.7 BROADCASTING QUANTUM INFORMATION . . . . . . . . . . . . 470 11.7 NEGATING QUANTUM INFORMATION . . . . . . . . . . . . . . . . . . 470 11.7.1 UNIVERSAL QUANTUM NEGATION CIRCUIT . . . . . . . . . . . 471 11.7.2 EXPECTATION VALUE OF AN OBSERVABLE BASED ON THE NEGATED STATE . . . . . . . . . . . . . . . . . . . . . . . 472 11.8 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 11.9 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 IMAGE 9 CONTENTS XIX 12 QUANTUM TELEPORTATION . . . . . . . . . . . . . . . . . . . . . . . . . 483 12.1 UNCERTAINTY PRINCIPLE AND "IMPOSSIBILITY" OF TELEPORTATION . . . . 483 12.1.1 HEISENBERG UNCERTAINTY PRINCIPLE . . . . . . . . . . . . . 484 12.2 PRINCIPLES OF TRUE TELEPORTATION . . . . . . . . . . . . . . . . . . 486 12.2.1 LOCAL VERSUS NON-LOCAL INTERACTIONS . . . . . . . . . . . 486 12.2.2 NON-LOCALITY: EINSTEIN'S "SPOOKY ACTION AT A DISTANCE" . 488 12.2.3 BELL'S INEQUALITY . . . . . . . . . . . . . . . . . . . . . . 489 12.3 EXPERIMENTAL TESTS OF BELL'S INEQUALITY . . . . . . . . . . . . . . 492 12.3.1 SPEED OF NON-LOCAL INFLUENCES . . . . . . . . . . . . . . . 494 12.4 QUANTUM TELEPORTATION PROTOCOL . . . . . . . . . . . . . . . . . . 496 12.4.1 TELEPORTATION DOES NOT IMPLY SUPERLUMINAL COMMUNICATION . . . . . . . . . . . . . . . . . . . . . . 499 12.5 WORKING PROTOTYPES . . . . . . . . . . . . . . . . . . . . . . . . . 500 12.6 TELEPORTING LARGER OBJECTS . . . . . . . . . . . . . . . . . . . . . 501 12.7 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 12.8 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 13 QUANTUM CRYPTOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . 507 13.1 NEED FOR STRONGER CRYPTOGRAPHY . . . . . . . . . . . . . . . . . . 508 13.1.1 SATELLITE COMMUNICATIONS CAN BE TAPPED . . . . . . . . 508 13.1.2 FIBER-OPTIC COMMUNICATIONS CAN BE TAPPED . . . . . . . 510 13.1.3 GROWING REGULATORY PRESSURES FOR HEIGHTENED SECURITY . 512 13.1.4 ARCHIVED ENCRYPTED MESSAGES RETROACTIVELY VULNERABLE . 512 13.2 AN UNBREAKABLE CRYPTOSYSTEM: THE ONE TIME PAD . . . . . . . . 515 13.2.1 SECURITY OF OTP: LOOPHOLES IF USED IMPROPERLY . . . . . 518 13.2.2 PRACTICALITY OF OTP: PROBLEM OF KEY DISTRIBUTION . . . . 519 13.3 QUANTUM KEY DISTRIBUTION . . . . . . . . . . . . . . . . . . . . . 520 13.3.1 CONCEPT OF QKD . . . . . . . . . . . . . . . . . . . . . . 520 13.3.2 SECURITY FOUNDATIONS OF QKD . . . . . . . . . . . . . . . 520 13.3.3 OTP MADE PRACTICAL BY QKD . . . . . . . . . . . . . . . 521 13.3.4 VARIETIES OF QKD . . . . . . . . . . . . . . . . . . . . . 521 13.4 PHYSICS BEHIND QUANTUM KEY DISTRIBUTION . . . . . . . . . . . . 522 13.4.1 PHOTON POLARIZATION . . . . . . . . . . . . . . . . . . . . 522 13.4.2 SINGLE PHOTON SOURCES . . . . . . . . . . . . . . . . . . . 523 13.4.3 ENTANGLED PHOTON SOURCES . . . . . . . . . . . . . . . . . 524 13.4.4 CREATING TRULY RANDOM BITS . . . . . . . . . . . . . . . . 525 13.4.5 ENCODING KEYS IN POLARIZED PHOTONS . . . . . . . . . . . 526 13.4.6 MEASURING PHOTON POLARIZATION WITH A BIREFRINGENT CRYSTAL . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 13.4.7 MEASURING PHOTON POLARIZATION WITH A POLARIZING FILTER . 529 13.5 BENNETT AND BRASSARD'S BB84 QKD SCHEME . . . . . . . . . . . . 529 13.5.1 THE BB84 QKD PROTOCOL . . . . . . . . . . . . . . . . . 531 13.5.2 EXAMPLE: BB84 QKD IN THE ABSENCE OF EAVESDROPPING . 534 13.5.3 EXAMPLE: BB84 QKD IN THE PRESENCE OF EAVESDROPPING . 536 13.5.4 SPEDALIERI'S ORBITAL ANGULAR MOMENTUM SCHEME FOR BB84 . . . . . . . . . . . . . . . . . . . . . . . . . . 537 IMAGE 10 XX CONTENTS 13.5.5 GENERALIZATION OF BB84: BRUSS' 6-STATE PROTOCOL . . . . . 538 13.6 BENNETT'S 2-STATE PROTOCOL (B92) . . . . . . . . . . . . . . . . . . 539 13.6.1 THE B92 QKD PROTOCOL . . . . . . . . . . . . . . . . . . 539 13.6.2 THREAT OF "DISCARD-ON-FAIL" UNAMBIGUOUS STATE DISCRIMINATION . . . . . . . . . . . . . . . . . . . . . . . 540 13.7 EKERT'S ENTANGLEMENT-BASED PROTOCOL . . . . . . . . . . . . . . . 541 13.7.1 THE E91 PROTOCOL . . . . . . . . . . . . . . . . . . . . . 541 13.8 ERROR RECONCILIATION AND PRIVACY AMPLIFICATION . . . . . . . . . . 542 13.8.1 ERROR RECONCILIATION . . . . . . . . . . . . . . . . . . . . 543 13.8.2 PRIVACY AMPLIF ICATION . . . . . . . . . . . . . . . . . . . 544 13.9 IMPLEMENTATIONS OF QUANTUM CRYPTOGRAPHY . . . . . . . . . . . . 545 13.9.1 FIBER-OPTIC IMPLEMENTATIONS OF QUANTUM CRYPTOGRAPHY . 545 13.9.2 EXTENDING THE RANGE OF QKD WITH QUANTUM REPEATERS . 547 13.9.3 EARTH-TO-SPACE QUANTUM CRYPTOGRAPHY . . . . . . . . . . 548 13.9.4 HIJACKING SATELLITES . . . . . . . . . . . . . . . . . . . . 550 13.9.5 COMMERCIAL QUANTUM CRYPTOGRAPHY SYSTEMS . . . . . . 554 13.10 BARRIERS TO WIDESPREAD ADOPTION OF QUANTUM CRYPTOGRAPHY . . . 555 13.10.1 WILL PEOPLE PERCEIVE A NEED FOR STRONGER CRYPTOGRAPHY? . . . . . . . . . . . . . . . . . . . . . . . 555 13.10.2 WILL PEOPLE BELIEVE THE FOUNDATIONS OF QKD ARE SOLID? . . . . . . . . . . . . . . . . . . . . . . . . . . . 556 13.10.3 WILL PEOPLE TRUST THE WARRANTIES OF CERTIFICATION AGENCIES? . . . . . . . . . . . . . . . . . . . . . . . . . 556 13.10.4 WILL WIDE AREA QUANTUM CRYPTOGRAPHY NETWORKS BE PRACTICAL? . . . . . . . . . . . . . . . . . . . . . . . . . 557 13.10.5 WILL KEY GENERATION RATE BE HIGH ENOUGH TO SUPPORT OTP? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 13.10.6 WILL SECURITY BE THE DOMINANT CONCERN? . . . . . . . . . 558 13.11 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 13.12 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560 PART IV TOWARDS PRACTICAL QUANTUM COMPUTERS 14 QUANTUM ERROR CORRECTION . . . . . . . . . . . . . . . . . . . . . . . 567 14.1 HOW ERRORS ARISE IN QUANTUM COMPUTING . . . . . . . . . . . . . 568 14.1.1 DISSIPATION-INDUCED BIT FLIP ERRORS . . . . . . . . . . . . 568 14.1.2 DECOHERENCE-INDUCED PHASE SHIFT ERRORS . . . . . . . . . 569 14.1.3 NATURAL DECOHERENCE TIMES OF PHYSICAL SYSTEMS . . . . . 570 14.1.4 WHAT MAKES QUANTUM ERROR CORRECTION SO HARD? . . . . 571 14.2 QUANTUM ERROR REDUCTION BY SYMMETRIZATION . . . . . . . . . . . 573 14.2.1 THE SYMMETRIZATION TRICK . . . . . . . . . . . . . . . . . 574 14.2.2 QUANTUM CIRCUIT FOR SYMMETRIZATION . . . . . . . . . . . 576 14.2.3 EXAMPLE: QUANTUM ERROR REDUCTION VIA SYMMETRIZATION . 577 14.3 PRINCIPLES OF QUANTUM ERROR CORRECTING CODES (QECCS) . . . . . 579 14.3.1 CLASSICAL ERROR CORRECTING CODES . . . . . . . . . . . . . 579 IMAGE 11 CONTENTS XXI 14.3.2 ISSUES UNIQUE TO QUANTUM ERROR CORRECTING CODES . . . . 580 14.3.3 MODELING ERRORS IN TERMS OF ERROR OPERATORS . . . . . . . 581 14.3.4 PROTECTING QUANTUM INFORMATION VIA ENCODING . . . . . . 583 14.3.5 DIGITIZING AND DIAGNOSING ERRORS BY MEASURING ERROR SYNDROMES . . . . . . . . . . . . . . . . . . . . . . . . . 585 14.3.6 REVERSING ERRORS VIA INVERSE ERROR OPERATORS . . . . . . . 585 14.3.7 ABSTRACT VIEW OF QUANTUM ERROR CORRECTING CODES . . . 585 14.4 OPTIMAL QUANTUM ERROR CORRECTING CODE . . . . . . . . . . . . . 588 14.4.1 LAFLAMME-MIQUEL-PAZ-ZUREK'S 5-QUBIT CODE . . . . . . . 588 14.4.2 ERROR OPERATORS FOR THE 5-QUBIT CODE . . . . . . . . . . . 588 14.4.3 ENCODING SCHEME FOR THE 5-QUBIT CODE . . . . . . . . . . 589 14.4.4 ERROR SYNDROMES & CORRECTIVE ACTIONS FOR THE 5-QUBIT CODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 14.4.5 EXAMPLE: CORRECTING A BIT-FLIP . . . . . . . . . . . . . . 592 14.5 OTHER ADDITIVE QUANTUM ERROR CORRECTING CODES . . . . . . . . . 593 14.5.1 SHOR'S 9-QUBIT CODE . . . . . . . . . . . . . . . . . . . . 593 14.5.2 STEANE'S 7-QUBIT CODE . . . . . . . . . . . . . . . . . . . 594 14.6 STABILIZER FORMALISM FOR QUANTUM ERROR CORRECTING CODES . . . . 594 14.6.1 GROUP THEORY FOR STABILIZER CODES . . . . . . . . . . . . 595 14.6.2 THE STABILIZER . . . . . . . . . . . . . . . . . . . . . . . 595 14.6.3 EXAMPLE: A STABILIZER FOR THE 5-QUBIT CODE . . . . . . . 596 14.6.4 USING A STABILIZER TO FIND THE CODEWORDS IT STABILIZES . . 597 14.6.5 HOW THE STABILIZER IS RELATED TO THE ERROR OPERATORS . . . 599 14.6.6 EXAMPLE: STABILIZERS AND ERROR OPERATORS FOR THE 5-QUBIT CODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 14.6.7 STABILIZER-BASED ERROR CORRECTION: THE ENCODING STEP . . 603 14.6.8 STABILIZER-BASED ERROR CORRECTION: INTRODUCTION OF THE ERROR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603 14.6.9 STABILIZER-BASED ERROR CORRECTION: ERROR DIAGNOSIS & RECOVERY . . . . . . . . . . . . . . . . . . . . . . . . . 603 14.6.10 STABILIZERS FOR OTHER CODES . . . . . . . . . . . . . . . . 604 14.7 BOUNDS ON QUANTUM ERROR CORRECTING CODES . . . . . . . . . . . 605 14.7.1 QUANTUM HAMMING BOUND . . . . . . . . . . . . . . . . 606 14.7.2 QUANTUM SINGLETON BOUND . . . . . . . . . . . . . . . . 606 14.7.3 QUANTUM GILBERT-VARSHAMOV BOUND . . . . . . . . . . . 607 14.7.4 PREDICTING UPPER AND LOWER BOUNDS ON ADDITIVE CODES . 607 14.7.5 TIGHTEST PROVEN UPPER AND LOWER BOUNDS ON ADDITIVE CODES . . . . . . . . . . . . . . . . . . . . . . . . . . . 611 14.8 NON-ADDITIVE (NON-STABILIZER) QUANTUM CODES . . . . . . . . . . . 611 14.9 FAULT-TOLERANT QUANTUM ERROR CORRECTING CODES . . . . . . . . . . 611 14.9.1 CONCATENATED CODES AND THE THRESHOLD THEOREM . . . . . 617 14.10 ERRORS AS ALLIES: NOISE-ASSISTED QUANTUM COMPUTING . . . . . . . 620 14.11 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 14.12 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 IMAGE 12 XXII CONTENTS 15 ALTERNATIVE MODELS OF QUANTUM COMPUTATION . . . . . . . . . . . . . 627 15.1 DESIGN PRINCIPLES FOR A QUANTUM COMPUTER . . . . . . . . . . . . 627 15.2 DISTRIBUTED QUANTUM COMPUTER . . . . . . . . . . . . . . . . . . 628 15.3 QUANTUM CELLULAR AUTOMATA MODEL . . . . . . . . . . . . . . . . 630 15.4 MEASUREMENT I: TELEPORTATION-BASED QUANTUM COMPUTER . . . . . 633 15.5 MEASUREMENT II: ONE-WAY QUANTUM COMPUTER . . . . . . . . . . 640 15.6 TOPOLOGICAL QUANTUM COMPUTER . . . . . . . . . . . . . . . . . . 641 15.6.1 TOPOLOGICAL QUANTUM EFFECTS . . . . . . . . . . . . . . . 642 15.6.2 BEYOND FERMIONS AND BOSONS-ANYONS . . . . . . . . . 643 15.6.3 ABELIAN VERSUS NON-ABELIAN ANYONS . . . . . . . . . . . 644 15.6.4 QUANTUM GATES BY BRAIDING NON-ABELIAN ANYONS . . . . 644 15.6.5 DO NON-ABELIAN ANYONS EXIST? . . . . . . . . . . . . . . 649 15.7 ADIABATIC QUANTUM COMPUTING . . . . . . . . . . . . . . . . . . 649 15.8 ENCODED UNIVERSALITY USING ONLY SPIN-SPIN EXCHANGE INTERACTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 15.8.1 THE EXCHANGE INTERACTION . . . . . . . . . . . . . . . . . 653 15.8.2 SWAP * VIA THE EXCHANGE INTERACTION . . . . . . . . . . . 654 15.8.3 PROBLEM: ALTHOUGH SWAP * IS EASY 1-QUBITS GATES ARE HARD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 15.8.4 SOLUTION: USE AN ENCODED BASIS . . . . . . . . . . . . . . 655 15.8.5 U 1 , 2 L , U 2 , 3 L , AND U 1 , 3 L . . . . . . . . . . . . . . . . . . . . 656 15.8.6 R Z GATES IN ENCODED BASIS . . . . . . . . . . . . . . . . 657 15.8.7 R X GATES IN ENCODED BASIS . . . . . . . . . . . . . . . . 657 15.8.8 R Y GATES IN ENCODED BASIS . . . . . . . . . . . . . . . . 658 15.8.9 CNOT IN ENCODED BASIS . . . . . . . . . . . . . . . . . . 658 15.9 EQUIVALENCES BETWEEN ALTERNATIVE MODELS OF QUANTUM COMPUTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 15.10 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660 15.11 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689
any_adam_object	1
author	Williams, Colin P.
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spelling	Williams, Colin P. Verfasser (DE-588)118111493 aut Explorations in quantum computing Colin P. Williams second edition London Springer 2011 XXII, 717 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Texts in computer science Quantenphysik (DE-588)4266670-3 gnd rswk-swf Quantencomputer (DE-588)4533372-5 gnd rswk-swf Computerphysik (DE-588)4273564-6 gnd rswk-swf Quantenphysik (DE-588)4266670-3 s Computerphysik (DE-588)4273564-6 s DE-604 Quantencomputer (DE-588)4533372-5 s Erscheint auch als Online-Ausgabe 978-1-84628-887-6 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2932196&prov=M&dok_var=1&dok_ext=htm Inhaltstext SWB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020345660&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Williams, Colin P. Explorations in quantum computing Quantenphysik (DE-588)4266670-3 gnd Quantencomputer (DE-588)4533372-5 gnd Computerphysik (DE-588)4273564-6 gnd
subject_GND	(DE-588)4266670-3 (DE-588)4533372-5 (DE-588)4273564-6
title	Explorations in quantum computing
title_auth	Explorations in quantum computing
title_exact_search	Explorations in quantum computing
title_full	Explorations in quantum computing Colin P. Williams
title_fullStr	Explorations in quantum computing Colin P. Williams
title_full_unstemmed	Explorations in quantum computing Colin P. Williams
title_short	Explorations in quantum computing
title_sort	explorations in quantum computing
topic	Quantenphysik (DE-588)4266670-3 gnd Quantencomputer (DE-588)4533372-5 gnd Computerphysik (DE-588)4273564-6 gnd
topic_facet	Quantenphysik Quantencomputer Computerphysik
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work_keys_str_mv	AT williamscolinp explorationsinquantumcomputing

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