Mathematics of Quantum Computation and Quantum Technology

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CHAPMAN & HALL/CRC APPLIED MATHEMATICS. AND NONLINEAR ... Chapman & Hall/CRC is an imprint of the ... 4.2 Simulation method of Lloyd. 91.
CHAPMAN & HALL/CRC APPLIED MATHEMATICS AND NONLINEAR SCIENCE SERIES

Mathematics of Quantum Computation and Quantum Technology Edited by

Goong Chen Louis Kauffman Samuel J. Lomonaco

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Chapman & HaII/CRC Taylor &Francis Group

Boca Raton London New York

Chapman & Hall/CRC is an imprint of the Taylor & Francis Group, an informa business

Contents

Preface Quantum Computation

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1 Quantum Hidden Subgroup Algorithms: An Algorithmic Toolkit Samuel J. Lomonaco and Louis H. Kauffman 1.1 Introduction 1.2 An example of Shor's quantum factoring algorithm 1.3 Definition of quantum hidden subgroup (QHS) algorithms 1.4 The generic QHS algorithm 1.5 Pushing and Lifting hidden subgroup problems (HSPs) 1.6 Shor's algorithm revisited 1.7 Wandering Shor algorithms, a.k.a. vintage Shor algorithms 1.8 Continuous (variable) Shor algorithms 1.9 The quantum circle and the dual Shor algorithms 1.10 A QHS algorithm for Feynman integrals 1.11 QHS algorithms an free groups 1.12 Generalizing Shor's algorithm to free groups 1.13 Is Grover's algorithm a QHS algorithm? 1.14 Beyond QHS algorithms: A suggestion of a meta-scheme for creating new quantum algorithms 2 A Realization Scheme for Quantum Multi-Object Search Zijian Diao, Goong Chen, and Peter Shiue 2.1 Introduction

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2.2 Circuit design for the multi-object sign-flipping Operator . . . 50 57 2.3 Additional discussion 58 2.4 Complexity issues 3 On Interpolating between Quantum and Classical 67 Complexity Classes J. Maurice Rojas 67 3.1 Introduction and main results 3.1.1 Open questions and the relevance of ultrametric com72 plexity 73 3.2 Background and ancillary results 77 3.2.1 Review of Riemann hypotheses 80 3.3 The proofs of our main results 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 Dominic W. Berry, Graeme Ahokas, Richard Cleve, and Barry C. Sanders 4.1 Introduction 4.2 Simulation method of Lloyd 4.3 Simulation method of ATS 4.4 Higher order integrators 4.5 Linear limit an simulation time 4.6 Efficient decomposition of Hamiltonian 4.7 Conclusions

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Quantum Technology

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5 New Mathematical Tools for Quantum Technology C. Bracher, M. Kleber, and T. Kramer 5.1 Physics in small dimensions 5.2 Propagators and Green functions 5.3 Quantum sources 5.3.1 Photoelectrons emitted from a quantum source 5.3.2 Currents generated by quantum sources 5.3.3 Recovering Fermi's golden rule 5.3.4 Photodetachment and Wigner's threshold laws 5.4 Spatially extended sources: the atom laser 5.5 Ballistic tunneling: STM

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5.6 Electrons in electric and magnetic fields: the quantum Hall effect 5.7 The semiclassical method 6 The Probabilistic Nature of Quantum Mechanics

134 138 149

Leon Cohen

6.1 Introduction 6.2 Are there wave functions in standard probability theory? 6.2.1 The Khinchin theorem 6.3 Two variables 6.3.1 Generalized characteristic function 6.4 Visualization of quantum wave functions 6.5 Local kinetic energy 6.6 Conclusion 7 Superconducting Quantum Computing Devices

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Zhigang Zhang and Goong Chen

7.1 Introduction 7.2 Superconductivity 7.3 More an Cooper pairs and Josephson junctions 7.4 Superconducting circuits: classical 7.4.1 Current-biased JJ 7.4.2 Single Cooper-pair box (SCB) 7.4.3 rf- or ac-SQUID 7.4.4 dc-SQUID 7.5 Superconducting circuits: quantum 7.6 Quantum gates 7.6.1 Some basic facts about SU(2) and SO(3) 7.6.2 One qubit operations (I): charge-qubit 7.6.3 One qubit operations (II): flux-qubit 7.6.4 Charge-flux qubit and Phase qubit 7.6.5 Two qubit operations: charge and flux qubits 7.6.6 Measurement of charge qubit

171 172 176 178 179 182 183 184 186 188 191 192 197 205 206 215

8 Nondeterministic Logic Gates in Optical Quantum Computing 223 Federico M. Spedalieri, Jonathan P. Dowling, and Hwang Lee

8.1 Introduction 8.2 Photon as a qubit 8.3 Linear optical quantum computing 8.4 Nondeterministic two-qubit gate 8.5 Ancilla-state preparation

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8.6 Cluster-state approach and gate fidelity 8.7 Appendices Quantum Information

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9 Exploiting Entanglement in Quantum Cryptographic 259 Probes Howard E. Brandt 260 9.1 Introduction 263 9.2 Probe designs based on U (1) 271 9.3 Probe designs based on U (. 2 ) 273 9.4 Probe designs based on 1.1 (3) 276 9.5 Conclusion 277 Appendix A Renyi information gain 281 Appendix B Maximum Renyi information gain 10 Nonbinary Stabilizer Codes Pradeep Kiran Sarvepalli, Salah A. Aly, and Andreas Klappenecker 10.1 Introduction 10.2 Stabilizer codes 10.2.1 Error bases 10.2.2 Stabilizer codes 10.2.3 Stabilizer and error correction 10.2.4 Minimum distance 10.2.5 Pure and irrpure codes 10.2.6 Encoding quantum codes 10.3 Quantum codes and classical codes 10.3.1 Codes over Fq 10.3.2 Codes over F 2 10.4 Bounds on quantum codes 10.5 Families of quantum codes 10.5.1 Quantum m-adic residue codes 10.5.2 Quantum projective Reed–Muller codes 10.5.3 Puncturing quantum codes 10.6 Conclusion 11 Accessible information about quantum states: An open optimization problem Jun Suzuki, Syed M. Assad, and Berthold-Georg Englert 11.1 Introduction

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11.2 Preliminaries 11.2.1 States and measurements 11.2.2 Entropy and information 11.3 The optimization problem 11.4 Theorems 11.4.1 Concavity and convexity 11.4.2 Necessary condition 11.4.3 Some basic theorems 11.4.4 Group-covariant case 11.5 Numerical search 11.6 Examples 11.6.1 Two quantum states in two dimensions 11.6.2 Trine: Z3 symmetry in two dimensions 11.6.3 Six-states protocol: symmetric group S3 11.6.4 Four-group in four dimensions 11.7 Summary and outlook

12 Quantum Entanglement: Concepts and Criteria Fu-li Li and M. Suhail Zubairy 12.1 Introduction 12.2 EPR correlations and quantum entanglement 12.3 Entanglement of pure states 12.4 Criteria an entanglement of mixed states 12.4.1 Peres–Horodecki criterion 12.4.2 Simon criterion 12.4.3 Duan–Giedke–Cirac–Zoller criterion 12.4.4 Hillery–Zubairy criterion 12.4.5 Shchukin–Vogel criterion 12.5 Coherence-induced entanglement 12.6 Correlated spontaneous emission laser as an entanglement amplifier 12.6 Remarks

13 Parametrizations of Positive Matrices With Applications M. Tseng, H. Zhou, and V Ramakrishna 13.1 Introduction 13.2 Sources of positive matrices in quantum theory 13.3 Characterizations of positive matrices 13.4 A different parametrization of positive matrices

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13.5 Two further applications 13.5.1 Toeplitz states 13.5.2 Constraints an relaxation rates 13.6 Conclusions

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Quantum Topology, Categorical Algebra, and Logic

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14 Quantum Computing and Quantum Topology Louis H. Kauffman and Samuel J. Lomonaco 14.1 Introduction 14.2 Knots and braids 14.3 Quantum mechanics and quantum computation 14.3.1 What is a quantum computer? 14.4 Braiding operators and universal quantum gates 14.4.1 Universal gates 14.5 A remark about EPR, entanglement and Bell's inequality 14.6 The Aravind hypothesis 14.7 SU(2) representations of the Artin braid group 14.8 The bracket polynomial and the Jones polynomial 14.8.1 Quantum computation of the Jones polynomial 14.9 Quantum topology, cobordism categories, Temperley—Lieb algebra, and topological quantum field theory 14.10Braiding and topological quantum field theory 14.11 Spin networks and Temperley—Lieb recoupling theory 14.11.1 Evaluations 14.11.2 Symmetry and unitarity 14.12Fibonacci particles 14.13The Fibonacci recoupling model 14.14Quantum computation of colored Jones polynomials and the Witten—Reshetikhin—Turaev invariant

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15 Temperley—Lieb Algebra: From Knot Theory to Logic and Computation via Quantum Mechanics Samson Abramsky 15.1 Introduction 15.1.1 Knot theory 15.1.2 Categorical quantum mechanics 15.1.3 Logic and computation 15.1.4 Outline of the paper

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15.2 The Temperley–Lieb algebra 15.2.1 Temperley–Lieb algebra: generators and relations 15.2.2 Diagram monoids 15.2.3 Expressiveness of the generators 15.2.4 The trace 15.2.5 The connection to knots 15.3 The Temperley–Lieb category 15.3.1 Pivotal categories 15.3.2 Pivotal dagger categories 15.4 Factorization and idempotents 15.5 Categorical quantum mechanics 15.5.1 Outline of the approach 15.5.2 Quantum non-logic vs. quantum hyper-logic 15.5.3 Remarks 15.6 Planar geometry of interaction and the Temperley–Lieb algebra 15.6.1 Some preliminary notions 15.6.2 Formalizing diagrams 15.6.3 Characterizing planarity 15.6.4 The Temperley–Lieb category 15.7 Planar 2u-calculus 15.7.1 The Ä,-calculus 15.7.2 Types 15.7.3 Interpretation in pivotal categories 15.7.4 An example 15.7.5 Discussion 15.8 Further directions 16 Quantum measurements without sums Bob Coecke and Dusko Pavlovic 16.1 Introduction 16.2 Categorical semantics 16.2.1 t-compact categories 16.2.2 Graphical calculus 16.2.3 Scalars, trace, and partial transpose 16.3 Sums and bases in Hilbert spaces 16.3.1 Sums in quantum mechanics 16.3.2 No-cloning and existence of a natural diagonal 16.3.3 Measurement and bases 16.3.4 Vanishing of non-diagonal elements and deletion 16.3.5 Canonical bases

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16.4 Classical objects 16.4.1 Special t-compact Frobenius algebras 16.4.2 Self-adjointness relative to a classical object 16.4.3 GHZ states as classical objects 16.4.4 Extracting the classical world 16.5 Quantum spectra 16.5.1 Coalgebraic characterization of spectra 16.5.2 Characterization of X-concepts 16.6 Quantum measurements 16.6.1 The CPM-construction 16.6.2 Formal decoherence 16.6.3 Demolition measurements 16.7 Quantum teleportation 16.8 Dense coding

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Appendix Panel Report an the Forward Looking Discussion

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Index

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