Explorations in quantum computing:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
London
Springer
2011
|
| Ausgabe: | second edition |
| Schriftenreihe: | Texts in computer science
|
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | XXII, 717 Seiten Diagramme |
| ISBN: | 9781846288869 |
Internformat
MARC
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| 015 | |a 07,N15,0081 |2 dnb | ||
| 020 | |a 9781846288869 |c hbk |9 978-1-84628-886-9 | ||
| 035 | |a (OCoLC)705560561 | ||
| 035 | |a (DE-599)BVBBV036474030 | ||
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| 100 | 1 | |a Williams, Colin P. |e Verfasser |0 (DE-588)118111493 |4 aut | |
| 245 | 1 | 0 | |a Explorations in quantum computing |c Colin P. Williams |
| 250 | |a second edition | ||
| 264 | 1 | |a London |b Springer |c 2011 | |
| 300 | |a XXII, 717 Seiten |b Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Texts in computer science | |
| 650 | 0 | 7 | |a Quantenphysik |0 (DE-588)4266670-3 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Quantencomputer |0 (DE-588)4533372-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Computerphysik |0 (DE-588)4273564-6 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Quantenphysik |0 (DE-588)4266670-3 |D s |
| 689 | 0 | 1 | |a Computerphysik |0 (DE-588)4273564-6 |D s |
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| 689 | 1 | 0 | |a Quantencomputer |0 (DE-588)4533372-5 |D s |
| 689 | 1 | |5 DE-604 | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-1-84628-887-6 |
| 856 | 4 | 2 | |q text/html |u http://deposit.dnb.de/cgi-bin/dokserv?id=2932196&prov=M&dok_var=1&dok_ext=htm |3 Inhaltstext |
| 856 | 4 | 2 | |m SWB Datenaustausch |q application/pdf |u 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 |3 Inhaltsverzeichnis |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-020345660 | |
Datensatz im Suchindex
| _version_ | 1805094095463383040 |
|---|---|
| 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. |
| author_GND | (DE-588)118111493 |
| author_facet | Williams, Colin P. |
| author_role | aut |
| author_sort | Williams, Colin P. |
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| building | Verbundindex |
| bvnumber | BV036474030 |
| classification_rvk | ST 152 |
| ctrlnum | (OCoLC)705560561 (DE-599)BVBBV036474030 |
| discipline | Informatik |
| edition | second edition |
| format | Book |
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| id | DE-604.BV036474030 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-20T10:37:57Z |
| institution | BVB |
| isbn | 9781846288869 |
| language | English |
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| physical | XXII, 717 Seiten Diagramme |
| publishDate | 2011 |
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| publisher | Springer |
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| series2 | Texts in computer science |
| 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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