Table of contents for Mathematics of quantum computation and quantum technology / editors, Goong Chen, Louis Kauffmann, Samuel J. Lomonaco.

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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 . . . . . . . . . . . . . . . . 41
2 A Realization Scheme for Quantum Multi-Object
Search 47
Zijian Diao, Goong Chen, and Peter Shiue
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2 Circuit design for the multi-object sign-flipping Operator . . . 50
xi
xii
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 complexity
. . . . . . . . . . . . . . . . . . . . . . . . . 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 Qp: proving the main
theorem . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.3.2 Detecting square-freeness: proving corollary 3.1 . . . 83
4 Quantum Algorithms for Hamiltonian Simulation 89
DominicW. 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
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
5.6 Electrons in electric and magnetic fields:
the quantum Hall effect . . . . . . . . . . . . . . . . . . . . . 134
5.7 The semiclassical method . . . . . . . . . . . . . . . . . . . . 138
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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
8.6 Cluster-state approach and gate fidelity . . . . . . . . . . . . . 245
8.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
Quantum Technology 256
xiv
Quantum Information 256
9 Exploiting Entanglement in Quantum Cryptographic
Probes 259
Howard E. Brandt
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
9.2 Probe designs based on U(1) . . . . . . . . . . . . . . . . . . 263
9.3 Probe designs based on U(2) . . . . . . . . . . . . . . . . . . 271
9.4 Probe designs based on U(3) . . . . . . . . . . . . . . . . . . 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 Error bases . . . . . . . . . . . . . . . . . . . . . . . 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 Fq . . . . . . . . . . . . . . . . . . . . . 293
10.3.2 Codes over Fq2 . . . . . . . . . . . . . . . . . . . . . 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
11.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 310
11.2.1 States and measurements . . . . . . . . . . . . . . . . 310
11.2.2 Entropy and information . . . . . . . . . . . . . . . . 313
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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: Z3 symmetry in two dimensions . . . . . . . . 329
11.6.3 Six-states protocol: symmetric group S3 . . . . . . . . 331
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
12.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
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
13.5 Two further applications . . . . . . . . . . . . . . . . . . . . 397
13.5.1 Toeplitz states . . . . . . . . . . . . . . . . . . . . . . 397
13.5.2 Constraints on relaxation rates . . . . . . . . . . . . . 400
13.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
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Quantum Topology, Categorical Algebra and Logic 406
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 . . . . . . . . 420
14.3.1 What is a quantum computer? . . . . . . . . . . . . . 422
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 . . . 431
14.6 The Aravind hypothesis . . . . . . . . . . . . . . . . . . . . . 433
14.7 SU(2) representations of the Artin braid group . . . . . . . . . 435
14.8 The bracket polynomial and the Jones polynomial . . . . . . . 440
14.8.1 Quantum computation of the Jones polynomial . . . . 446
14.9 Quantum topology, cobordism categories, Temperley¿Lieb algebra
and topological quantum field theory . . . . . . . . . . 451
14.10Braiding and topological quantum field theory . . . . . . . . . 462
14.11Spin networks and Temperley¿Lieb recoupling theory . . . . . 474
14.11.1 Evaluations . . . . . . . . . . . . . . . . . . . . . . . 478
14.11.2 Symmetry and unitarity . . . . . . . . . . . . . . . . 482
14.12Fibonacci particles . . . . . . . . . . . . . . . . . . . . . . . 485
14.13The Fibonacci recoupling model . . . . . . . . . . . . . . . . 492
14.14Quantum computation of colored Jones polynomials and the
Witten¿Reshetikhin¿Turaev invariant . . . . . . . . . . . . . 507
15 Temperley¿Lieb Algebra: From Knot Theory to Logic and Computation
via Quantum Mechanics 519
Samson Abramsky
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 520
15.1.1 Knot theory . . . . . . . . . . . . . . . . . . . . . . . 520
15.1.2 Categorical quantum mechanics . . . . . . . . . . . . 520
15.1.3 Logic and computation . . . . . . . . . . . . . . . . . 521
15.1.4 Outline of the paper . . . . . . . . . . . . . . . . . . 521
15.2 The Temperley¿Lieb algebra . . . . . . . . . . . . . . . . . . 522
15.2.1 Temperley¿Lieb algebra: generators and relations . . . 522
15.2.2 Diagram monoids . . . . . . . . . . . . . . . . . . . . 523
15.2.3 Expressiveness of the generators . . . . . . . . . . . . 524
15.2.4 The trace . . . . . . . . . . . . . . . . . . . . . . . . 525
15.2.5 The connection to knots . . . . . . . . . . . . . . . . 526
15.3 The Temperley¿Lieb category . . . . . . . . . . . . . . . . . 527
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15.3.1 Pivotal categories . . . . . . . . . . . . . . . . . . . . 528
15.3.2 Pivotal dagger categories . . . . . . . . . . . . . . . . 531
15.4 Factorization and idempotents . . . . . . . . . . . . . . . . . 532
15.5 Categorical quantum mechanics . . . . . . . . . . . . . . . . 534
15.5.1 Outline of the approach . . . . . . . . . . . . . . . . . 535
15.5.2 Quantum non-logic vs. quantum hyper-logic . . . . . 537
15.5.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . 540
15.6 Planar geometry of interaction and the Temperley¿Lieb algebra 541
15.6.1 Some preliminary notions . . . . . . . . . . . . . . . 542
15.6.2 Formalizing diagrams . . . . . . . . . . . . . . . . . 543
15.6.3 Characterizing planarity . . . . . . . . . . . . . . . . 544
15.6.4 The Temperley¿Lieb category . . . . . . . . . . . . . 548
15.7 Planar l-calculus . . . . . . . . . . . . . . . . . . . . . . . . 552
15.7.1 The l-calculus . . . . . . . . . . . . . . . . . . . . . 553
15.7.2 Types . . . . . . . . . . . . . . . . . . . . . . . . . . 553
15.7.3 Interpretation in pivotal categories . . . . . . . . . . . 555
15.7.4 An example . . . . . . . . . . . . . . . . . . . . . . . 556
15.7.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . 557
15.8 Further directions . . . . . . . . . . . . . . . . . . . . . . . . 557
16 Quantum measurements without sums 563
Bob Coecke and Dusko Pavlovic
16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
16.2 Categorical semantics . . . . . . . . . . . . . . . . . . . . . . 566
16.2.1 ¿-compact categories . . . . . . . . . . . . . . . . . . 566
16.2.2 Graphical calculus . . . . . . . . . . . . . . . . . . . 568
16.2.3 Scalars, trace, and partial transpose . . . . . . . . . . 570
16.3 Sums and bases in Hilbert spaces . . . . . . . . . . . . . . . . 572
16.3.1 Sums in quantum mechanics . . . . . . . . . . . . . . 572
16.3.2 No-cloning and existence of a natural diagonal . . . . 573
16.3.3 Measurement and bases . . . . . . . . . . . . . . . . 574
16.3.4 Vanishing of non-diagonal elements and deletion . . . 575
16.3.5 Canonical bases . . . . . . . . . . . . . . . . . . . . . 576
16.4 Classical objects . . . . . . . . . . . . . . . . . . . . . . . . . 576
16.4.1 Special ¿-compact Frobenius algebras . . . . . . . . . 577
16.4.2 Self-adjointness relative to a classical object . . . . . . 580
16.4.3 GHZ-states as classical objects . . . . . . . . . . . . . 582
16.4.4 Extracting the classical world . . . . . . . . . . . . . 583
16.5 Quantum spectra . . . . . . . . . . . . . . . . . . . . . . . . 584
16.5.1 Coalgebraic characterization of spectra . . . . . . . . 585
16.5.2 Characterization of X-concepts . . . . . . . . . . . . . 586
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16.6 Quantum measurements . . . . . . . . . . . . . . . . . . . . . 586
16.6.1 The CPM-construction . . . . . . . . . . . . . . . . . 588
16.6.2 Formal decoherence . . . . . . . . . . . . . . . . . . 589
16.6.3 Demolition measurements . . . . . . . . . . . . . . . 590
16.7 Quantum teleportation . . . . . . . . . . . . . . . . . . . . . 591
16.8 Dense coding . . . . . . . . . . . . . . . . . . . . . . . . . . 596
Appendix Panel Report on the Forward Looking Discussion 601

Library of Congress Subject Headings for this publication:

Quantum computers -- Mathematics.
Quantum theory -- Mathematics.