Quantum Information Engineering: Concepts to Quantum Technologies. Simon John Devitt, B.Sc (Honours) Melbourne. Submitted in partial fulfilment of the requirements of the degree of Doctor of Philosophy. November, 2007. Center For Quantum Computing Technology. School of Physics, University of Melbourne. Victoria, Australia. Acid Free Paper. ii Quantum Mechanics: It’s all fun and games until somebody loses an i. iii The University of Melbourne. ABSTRACT Quantum Information Engineering: Concepts to Quantum Technologies. by Simon John Devitt. Academic Supervisor: Lloyd C. L. Hollenberg. This thesis investigates several broad areas related to the effective implementation of quantum information processing, from large scale quantum algorithms and error correction, through to system identification and characterization techniques, efficient designs for quantum computing architectures and the design of small devices which utilize quantum effects. The discussion begins with the introduction of a quantum circuit appropri- ate for implementing Shor’s factoring algorithm on Linear Nearest Neighbor qubit arrays such as the Kane phosphorus in silicon system. Detailed numerical sim- ulations are then presented, demonstrating the sensitivity of the circuit under coherent quantum errors. The concepts of Quantum Error Correction and Fault-tolerant computation are reviewed with original work carried out to show the relative robustness and practicality of Fault-tolerant computation for logical state preparation. Methods of intrinsic system identification and characterization are proposed. Protocols for characterizing both the confinement of a multi-level system to the qubit subspace and the Hamiltonian dynamics governing two-qubit interactions are presented as well as a brief review of characterization techniques already developed for single qubit dynamics. A quantum bus protocol is introduced which can be applied to several sys- tems, allowing for a highly distributed architecture which is invulnerable to in- formation loss through the transport bus. Several preliminary architecture de- signs are presented, including solid state and atom/cavity quantum processors. A more detailed, distributed system in ion traps utilizing this protocol is introduced which demonstrates both large scale quantum computation and a smaller proof of principle experiment which will be possible in the near future. Finally, I present a design for a small scale quantum device which can be used to deterministically prepare a large class of highly entangled quantum states using photonic qubits. This device, which is dubbed the “photnic module”, would be extremely versatile with a variety of uses, from communication to computation. iv Declaration This is to certify that: (i) the thesis comprises only my original work towards the PhD except where indicated, (ii) due acknowledgement has been made in the text to all other material used, (iii) the thesis is less than 100,000 words in length, exclusive of table, maps, bibliographies, appendices and footnotes. Simon John Devitt, July 2007. v vi Publications During the course of this project, a number of refereed journal and conference articles have been published based on the work presented in this thesis. They are listed here for reference. Journal Articles A.G. Fowler, S.J. Devitt and L.C.L. Hollenberg, Implementation of • Shor’s Algorithm on a Linear Nearest Neighbour Qubit Array, Quant. Inf. Comp. 4, 237-251 (2004). S.J. Devitt, A.G. Fowler and L.C.L. Hollenberg, Robustness of Shor’s • algorithm, Quant. Inf. Comp. 6, 616-629 (2006). A.D. Greentree, S.J. Devitt and L.C.L. Hollenberg, Quantum infor- • mation transport to multiple receivers, Phys. Rev. A. 73, 032319 (2006). S.J. Devitt, J.H. Cole and L.C.L. Hollenberg, Scheme for direct mea- • surement of a general two-qubit Hamiltonian, Phys. Rev. A 73, 052317 (2006). D.K.L. Oi, S.J. Devitt and L.C.L. Hollenberg, Scalable Error Cor- • rection in Distributed Ion Trap Computers, Phys. Rev. A. 74, 052313 (2006). J.H. Cole, S.J. Devitt and L.C.L. Hollenberg, Precision character- • isation of two-qubit Hamiltonians via entanglement mapping, J. Phys. A: Math.Gen, 39 (2006), 14649-14658. S.J. Devitt, A.D.Greentree and L.C.L. Hollenberg, Information free • quantum bus for generating stabilized states, Quant. Inf. Proc. 6, 229 (2007). S.J. Devitt, S.G. Schirmer, D.K.L. Oi, J.H. Cole and L.C.L. Hol- • lenberg, Subspace Confinement: How good is your qubit? New J. Phys. 9 (2007) 384. vii S.J. Devitt, A.D. Greentree, R. Ionicioiu, J.L. O’Brien, W.J. • Munro and L.C.L. Hollenberg, The Photonic Module: and on-demand source for photonic entanglement. Phys. Rev. A. 76, 052312 (2007). Conference Articles S.J. Devitt, A.G. Fowler and L.C.L. Hollenberg, Investigating the • practical implementation of Shor’s algorithm, Proc. SPIE 5650, 483 (2005). A.G. Fowler, S.J. Devitt and L.C.L. Hollenberg, Constructing Steane • code fault-tolerant gates, Proc. SPIE 5650, 557 (2005). S.J. Devitt, A.G. Fowler and L.C.L. Hollenberg, Practicality of Fault- • Tolerant Quantum Computation, Proc. Australian Institute of Physics 16th National Congress (2005). L.C.L. Hollenberg, A.D. Greentree, C.J. Wellard, A.G. Fowler, • S.J. Devitt and J.H. Cole, Qubit Transport and Fault-tolerant Architec- tures in Silicon, Proc. 2006 International Conference On Nanoscience and Nanotechnology. (2006). S.J. Devitt, S.G. Schirmer, D.K.L. Oi, J.H. Cole and L.C.L. Hol- • lenberg, Subspace confinement of qubit systems, Proc. Australian Institute of Physics 17th National Congress (2006). A.D. Greentree, S.J. Devitt and L.C.L. Hollenberg, Adiabatic proto- • cols for operator measurement based entanglement and quantum computing, Proc. Australian Institute of Physics 17th National Congress (2006). L. C. L. Hollenberg, A. D. Greentree, C. J. Wellard, A. G. Fowler, • S. J. Devitt, J. H. Cole, and A. Stephens, Qubit Transport and fault- tolerant architectures for Si:P quantum computing, Proc. Australian Insti- tute of Physics 17th National Congress (2006). Un-refereed Articles A.M. Stephens, S.J.Devitt, A.G. Fowler, J.C. Ang, L.C.L. Hollen- • berg, Gate-Level Simulation of logical state preparation, quant-ph/0608112, (2006). viii Patents S.J. Devitt, Implementation of Shor’s Algorithm on a Linear Nearest • Neighbour Qubit Array, Australian Provisional Patent, QUCOR Pty. Ltd. Sydney. Patent number:2004900951. ix x Acknowledgements This thesis would never have left the ground if it were not for a number of peo- ple. First I would like to thank my supervisor, Lloyd Hollenberg for exceptional support and guidance and giving me the freedom to pursue aspects of quantum computing which I found interesting and compelling. Secondly to Jared Cole, Andrew Greentree and Austin Fowler, three people whom I have collaborated closely with over the last three years and who have not only brought out the best in my own research, but who have always been up for a few drinks wherever we have been in the world. During my Ph.D I have visited several institutions, including an extended 12 month visiting student-ship in 2006 to Cambridge which was generously funded by the Rae & Edith Bennett Foundation. I wish to acknowledge the tremendous help and entertainment provided by Sonia Schirmer and Daniel Oi of Cambridge University who have been close collaborators and graciously hosted me during my brief visit in 2005 and my extended student-ship in 2006, along with the extraordinary administrative skill of Kaija Hampson who helped me settle into a new environment. Thanks also go to Frank Wilhelm and Jan von Delft at LMU in M¨unchen for hosting me for a month in 2005. Personally I wish to thank the great people at the School of Physics, both the 1st and 2nd generation of inhabitance of room 408 and 412, the guys and girls on the 6th floor, Sean Crosby for continued access to my favorite tool of procrastination, the DMP group at large and scores of others who made my time extremely enjoyable. Finally I wish to acknowledge the support from both my family and non- physics friends (In Australia, the U.K. and Hungary). Their continued tolerance with my moods and tendency to disappear for weeks at a time gave me a much needed break from the world of physics. xi xii Contents 1 Introduction 1 2 Background 5 2.1 Emergence of the Quantum from the Classical. 5 2.2 Quantum Information Processing: Computational Models..... 8 2.2.1 The Quantum Circuit Model. 8 2.2.2 ClusterStateModel . 12 2.3 Conclusion............................... 18 3 Shor’s Factoring Algorithm 19 3.1 Introduction.............................. 20 3.1.1 FromFactoringtoPeriodFinding. 21 3.1.2 The Quantum Period Finding subroutine . 22 3.1.3 Classicalpost-processing . 24 3.2 Linear Nearest Neighbor quantum circuit. 27 3.2.1 Decomposing the QPF subroutine . 28 3.2.2 Conclusion........................... 40 3.3 Stability of Shor’s Algorithm under errors . .. 42 3.3.1 Errormodelsandanalysis . 44 3.3.2 Stability under a fixed number of errors . 47 3.3.3 Conclusion........................... 55 4 Quantum Error Correction and Fault-tolerance. 57 4.1 Introduction.............................. 59 4.2 QuantumErrorCorrection . 61 4.2.1 The3-qubitcode ....................... 62 xiii CONTENTS 4.2.2 StabilizerFormalism . 64 4.2.3 QECwithstabilizercodes . 66 4.3 DigitizationofQuantumErrors . 72 4.3.1 Systematicgateerrors . 72 4.3.2 Environmental decoherence . 73 4.4 Fault-tolerant Quantum Error Correction and the threshold theorem. 76 4.4.1 Fault-tolerance . .. .. 76 4.4.2 ThresholdTheorem. 77 4.5 Fault-tolerant operations on encoded data . .. 80 4.5.1 SingleQubitOperations . 80 4.5.2 Two-qubitgate......................... 82 4.6 Fault-tolerant circuit design for logical state preparation ..... 86 4.7 Simulations ofLogical StatePreparation . .. 90 4.7.1 LNN circuit for logical encoding . 90 4.7.2 Results............................. 93 4.7.3 Conclusions .......................... 94 5 Intrinsic Characterization of qubit systems 97 5.1 Introduction.............................