Practicality of Quantum Information Processing by Hoi-Kwan Lau A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Physics University of Toronto c Copyright 2014 by Hoi-Kwan Lau Abstract Practicality of Quantum Information Processing Hoi-Kwan Lau Doctor of Philosophy Graduate Department of Physics University of Toronto 2014 Quantum Information Processing (QIP) is expected to bring revolutionary enhancement to various technological areas. However, today’s QIP applications are far from being practical. The problem involves both hardware issues, i.e., quantum devices are imper- fect, and software issues, i.e., the functionality of some QIP applications is not fully understood. Aiming to improve the practicality of QIP, in my PhD research I have studied various topics in quantum cryptography and ion trap quantum computation. In quantum cryp- tography, I first studied the security of position-based quantum cryptography (PBQC). I discovered a wrong assumption in the previous literature that the cheaters are not al- lowed to share entangled resources. I proposed entanglement attacks that could cheat all known PBQC protocols. I also studied the practicality of continuous-variable (CV) quantum secret sharing (QSS). While the security of CV QSS was considered by the literature only in the limit of infinite squeezing, I found that finitely squeezed CV resources could also provide finite secret sharing rate. Our work relaxes the stringent resources requirement of implementing QSS. In ion trap quantum computation, I studied the phase error of quantum information induced by dc Stark effect during ion transportation. I found an optimized ion trajectory for which the phase error is the minimum. I also defined a threshold speed, above which ion transportation would induce significant error. ii In addition, I proposed a new application for ion trap systems as universal bosonic simulators (UBS). I introduced two architectures, and discussed their respective strength and weakness. I illustrated the implementations of bosonic state initialization, transfor- mation, and measurement by applying radiation fields or by varying the trap potential. When comparing with conducting optical experiments, the ion trap UBS is advantageous in higher state initialization efficiency and higher measurement accuracy. Finally, I proposed a new method to re-cool ion qubits during quantum computation. The idea is to transfer the motional excitation of a qubit to another ion that is prepared in the motional ground state. I showed that my method could be ten times faster than current laser cooling techniques, and thus could improve the speed of ion trap quantum computation. iii Acknowledgements First of all, may I express my sincere gratitude to my supervisors: Prof. Daniel James and Prof. Hoi-Kwong Lo. I really appreciate that they agreed to co-supervise my PhD thesis; this was an adventurous decision as they work in different directions. I am very fortunate to have had numerous inspiring discussions with my supervisors, through which I have learnt a lot in both physics knowledge and the ways to tackle problems. I am obliged for their patience and tolerance, particularly they relentlessly gave me friendly advices even when I held a strong opposing opinion. Besides, I would like to thank for their effort and consideration in helping me in job-hunting and in training me to be a better physicist and person. In summary, my PhD period would not be so fruitful and enjoyable without the kindness of my supervisors. I would like to thank my supervisory committee members, Joseph Thywissen and John Sipe, and my examiners, Jonathan Dowling and John Wei, for their valuable comments and beneficial advices. I also want to thank my colleagues, including Eric Chitambar, Serge Fehr, John Gaebler, Barry Sanders, Marcos Villagra, Christian Weedbrook, and Andrew White, for their illuminating discussions and useful helps. In addition, I want to thank Mr. David Tang and Ms. Kristy Kwok for their hospitality during my stay in Toronto. It is my honour to acknowledge the support from the Kwok Sau Po Scholarship, the E. F. Burton Fellowship, the Lachlan Gilchrist Fellowship Fund, and the Queen Elizabeth II Graduate Scholarship in Science and Technology. Finally, I would like to thank my beloved wife, Dan Sun, for agreeing to accompanying me in the rest of my life. iv Contents Glossaries and Acronyms xi 1 Introduction 1 2 Position-Based Quantum Cryptography 6 2.1 Introduction ................................. 6 2.2 Classical Position-Based Cryptrography . 8 2.2.1 Individually Cheating . 10 2.2.2 Collaborative Cheating . 10 2.3 PBQCProtocols ............................... 10 2.3.1 ProtocolA............................... 11 2.3.2 ProtocolB............................... 12 2.4 Cheating in N =2Case ........................... 12 2.4.1 CheatingagainstProtocolA . 12 2.4.2 CheatingagainstProtocolB . 15 2.5 Cheating in N > 2Case ........................... 17 2.5.1 CheatingagainstProtocolA. 17 2.5.2 CheatingagainstProtocolB . 20 2.6 Principle of the cheating schemes . 22 2.6.1 ProtocolA............................... 22 2.6.2 ProtocolB............................... 23 2.7 ModifiedPBQCProtocol .......................... 24 2.8 Summary ................................... 26 3 Quantum Secret Sharing with Continuous-Variable Cluster States 28 3.1 Introduction.................................. 28 3.2 Background .................................. 30 3.2.1 QuantumSecretSharing ...................... 30 v 3.2.2 Continuous-Variable Cluster States . 32 3.2.2.1 Nullifier representation . 32 3.2.2.2 Wigner function representation . 33 3.2.2.3 Correlations of measurement . 34 3.2.2.4 Cluster-class state . 35 3.3 CCQuantumsecretsharing . .. .. 35 3.4 CQQuantumsecretsharing . .. .. 38 3.4.1 Equivalence of CQ Quantum Secret Sharing and QKD . 40 3.4.2 SecretSharingRate ......................... 41 3.4.3 Simplification . 45 3.4.3.1 MixedStateApproach . 45 3.4.3.2 Classical Memory . 46 3.4.3.3 LocalMeasurement. 48 3.4.3.4 Simplified CQ Protocol . 50 3.5 QQQuantumsecretsharing . .. .. 51 3.6 Conclusion .................................. 53 4 Motional States of Trapped Ions 56 4.1 TrapandIonMotion ............................. 57 4.2 InternalStructureandLaserOperation . .. 59 4.2.1 Transition............................... 59 4.2.2 Measurement ............................. 61 4.3 IonMotioninTrapPotential ........................ 61 4.3.1 Generalized Harmonic Oscillator . 63 4.3.2 InteractionPicture .......................... 67 5 Decoherence Induced by dc Electric field During Ion Transport 69 5.1 Speedofiontrapquantumcomputer . 69 5.2 Motionofion ................................. 71 5.3 PhaseshiftduetodcStarkeffect . 72 5.4 Minimum possible phase shift . 73 5.5 Threshold speed of transporting ion qubits . 76 5.6 Non-encodingStateExcitation . 77 5.7 SummaryandDiscussion........................... 80 6 Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap 82 6.1 Introduction ................................. 82 vi 6.2 Layoutofthesystem ............................. 84 6.3 Universal Bosonic Simulation . 85 6.4 Laser Implementation of Basic Operations . 87 6.4.1 DisplacementOperator. .. .. 89 6.4.2 Phase-ShiftOperator . .. .. 90 6.4.3 SqueezingOperator.......................... 90 6.4.4 NonlinearOperator.......................... 90 6.4.5 BeamSplitter............................. 91 6.5 Readout .................................... 95 6.5.1 Adiabaticpassage........................... 95 6.5.2 ResonantPulses............................ 99 6.5.2.1 Post-selection Method . 100 6.5.2.2 Multiple Electronic State Method . 103 6.6 Initialization . 104 7 Ion Trap Bosonic Simulator 2: Ions in Separate Trap 107 7.1 Introduction.................................. 107 7.2 Model ..................................... 108 7.3 SingleModeOperations ........................... 109 7.3.1 DisplacementOperator . 110 7.3.2 SqueezingOperator ......................... 111 7.3.3 Phase-ShiftOperator . 112 7.3.4 NonlinearOperator ......................... 113 7.4 Two-modeOperation ............................ 114 7.4.1 IonTransportandPick-up . 119 7.4.2 Accuracyofbeamsplitter . 120 7.5 Initialization and readout . 122 7.6 Conclusion .................................. 123 8 Rapid ion re-cooling by swapping beam splitter 126 8.1 Introduction ................................. 126 8.2 Model ..................................... 127 8.3 Cooling .................................... 130 8.4 SwappingBeamSplitter ........................... 131 8.4.1 Ansatz................................. 133 8.5 Groundstatequbitpair ........................... 133 8.6 Transportbetweentraps ........................... 136 vii 8.7 Implementationofpotential . 137 8.8 Anharmonicity ................................ 138 8.9 Fluctuation .................................. 139 8.10Conclusion................................... 140 9 Summary 142 9.1 Prospects ................................... 145 A Appendix 148 A.1 SecurityofModifiedProtocol . 148 A.1.1 Security Against Attacks with One Entangled Qubit . 148 A.1.2 Security Against Attacks with One Entangled Qutrit . 151 A.2 ExampleofCCQSS ............................. 154 A.2.1 Example1:(2,3)-CCprotocol . 154 A.2.1.1 Parties 1,2 collaboration. 155 { } A.2.1.2 Parties 2,3 collaboration. 156 { } A.2.2 Example2:(3,5)-CCprotocol . 157 A.2.2.1 Parties 1,2,3 collaboration. 158 { } A.2.2.2 Parties 1,3,4 collaboration. 159 { } A.3 ExamplesofCQQSS ............................ 161 A.3.1