Realistic read-out and control for Si:P based quantum computers by Matthew James Testolin BSc(Hons) Submitted in total fulfilment of the requirements for the degree of Doctor of Philosophy School of Physics The University of Melbourne Australia August, 2008 Abstract This thesis identifies problems with the current operation proposals for Si:P based solid-state quantum computing architectures and outlines realistic alter- natives as an effective fix. The focus is qubit read-out and robust two-qubit control of the exchange interaction in the presence of both systematic and environmental errors. We develop a theoretical model of the doubly occupied D− read-out state found in Si:P based nuclear spin architectures. We test our theory by using it to determine the binding energy of the D− state, comparing to known re- sults. Our model can be used in detailed calculations of the adiabatic read-out protocol proposed for these devices. Regarding this protocol, preliminary cal- culations suggest the small binding energy of the doubly occupied read-out state will result in a state dwell-time less than that required for measurement using a single electron transistor (SET). We propose and analyse an alterna- tive approach to single-spin read-out using optically induced spin to charge transduction, showing that the top gate biases required for qubit selection are significantly less than those demanded by the adiabatic scheme, thereby in- + creasing the D D− lifetime. Implications for singlet-triplet discrimination for electron spin qubits are also discussed. On the topic of robust two-qubit control, we demonstrate how by using two-qubit composite rotations a high fidelity controlled-NOT (CNOT) gate can be constructed, even when the strength of the interaction between qubits is not accurately known. We focus on the exchange interaction oscillation in silicon based solid-state architectures with a Heisenberg Hamiltonian. This method easily applies to a general two-qubit Hamiltonian. We show how the robust CNOT gate can achieve a very high fidelity when a single application of the composite rotations is combined with a modest level of Hamiltonian characterisation. Operating the robust CNOT gate in a suitably characterised system means concatenation of the composite pulse is unnecessary, hence re- ducing operation time, ensuring the gate operates below the threshold required i for fault-tolerant quantum computation. We finish by outlining how the effects of charge noise on a pair of spins cou- pled via the exchange interaction can be calculated by modelling the charge fluctuations as a random telegraph noise (RTN) process using probability den- sity functions. We develop analytic expressions for the time-dependent super- operator of a pair of spins as a function of fluctuation amplitude and rate. We show that the theory can be extended to include multiple fluctuators, in particular, spectral distributions of fluctuators. These superoperators can be included in time-dependent analyses of the state of spin systems designed for spintronics or quantum information processing to determine the decohering effects of exchange fluctuations. We discuss the implications of the charge noise produced from a single fluctuator on the operation fidelity of our robust CNOT gate, comparing the performance to an uncorrected CNOT gate. ii Declaration This is to certify that (i) the thesis comprises only my original work towards the PhD except where indicated in the Introduction, (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 tables, maps, bibliographies, appendices and footnotes. I authorise the Head of the School of Physics to make or have made a copy of this thesis to any person judged to have an acceptable reason for access to the information, i.e., for research, study or instruction. Signature Date iii Acknowledgements This thesis and the work it entailed would not have been possible without the support of a number of people who I now wish to acknowledge. Firstly, I would like to thank Lloyd Hollenberg for his dedicated supervision. His insight, guid- ance and patience have been invaluable throughout my PhD candidature and honours year and his door always open. Most of the early work in this thesis would not have been possible without the assistance of Cameron Wellard. I particularly value Cam’s practical approach and much of his advice has re- mained with me throughout this PhD. Both he and Andrew Greentree never seemed to tire of my numerous questions and concerns and I thank them both for that. I would also like to acknowledge Charles Hill and Jared Cole for their collaboration on parts of this research. Working closely with peers has been one of the most enjoyable of all the PhD experiences. There are many others from the School of Physics that have had either direct or indirect influence on both myself and this PhD thesis, however I would like to highlight the contributions of a few. The members of the DMAP group as well as my roommates (from 402 and 606) in particular Gajendran Kandasamy, Jared Cole, Jason Doukas, John McIntosh, Joo Chew Ang and Vince Conrad. I’m also thankful for the constant source of useful distraction provided by those on the sixth floor. I reserve a very special thanks for Kristian McDonald. Without close friends outside the School of Physics, completing this thesis would have been an onerous task. I’d like to thank all those who helped me to escape from the pressures of a PhD, especially Cameron Grech, Matt Lynch, Anthony Lewis, James Copes, Andrew Allan, Adrian Polizzi, James Hartigan, Gerard McMahon, Wojtek Ruszkowski, Jane Leigh, Sally Malone and Nadia Polizzi. Thanks for your friendship, advice, the taunts... Finally, I would like to thank my family for their loving support. I’m not sure how you have put up with me and continued to encourage me after all these years of study but I am extremely grateful for it. Thank you mum, dad and Sarah. v List of Publications Throughout the course of this project a number of the key results presented in this thesis have appeared in publication. These publications are listed here for reference, along with all conference presentations, which have been given based on the work in this thesis. REFEREED PUBLICATIONS M. J. Testolin, A. D. Greentree, C. J. Wellard and L. C. L. Hollenberg, • “Optically induced spin-to-charge transduction in donor-spin readout”, Physical Review B 72, 195325 (2005). M. J. Testolin, C. D. Hill, C. J. Wellard and L. C. L. Hollenberg, “Ro- • bust controlled-NOT gate in the presence of large fabrication-induced variations of the exchange interaction strength”, Physical Review A 76, 012302 (2007). REFEREED CONFERENCE PROCEEDINGS M. J. Testolin, L. C. L. Hollenberg, A. D. Greentree and C. J. Wellard, • “Single-spin detection and read-out for the solid-state quantum computer via resonant techniques”, Proceedings of SPIE International Society of Optical Engineers 5650, 516-526 (2005). CONFERENCE ABSTRACTS M. J. Testolin, L. C. L. Hollenberg, C. J. Wellard and A. D. Greentree, • “Single-spin optical readout scheme for solid-state quantum computers”, SPIE International Symposium. Smart Materials, Nano-, and Micro- Smart Systems, Sydney, Australia, December 12-15, 2004. vii M. J. Testolin, L. C. L. Hollenberg, C. J. Wellard and A. D. Green- • tree, “Optical readout of single-spins for solid-state quantum comput- ing”, Australian Institute of Physics - 16th National Congress, Canberra, Australia, January 30 - February 4, 2005. M. J. Testolin, C. D. Hill, C. J. Wellard, A. D. Greentree and L. C. L. Hol- • lenberg, “Robust CNOT gates to correct for variations in donor based ex- change coupling”, Sir Mark Oliphant International Frontiers of Science and Technology Conference on Quantum Nanoscience, Noosa, Queens- land, January 22-26, 2006. M. J. Testolin, C. D. Hill, C. J. Wellard, and L. C. L. Hollenberg, “Imple- • menting a robust CNOT gate to correct for fabrication induced variations in donor based exchange coupling”, Australian Institute of Physics - 17th National Congress, Brisbane, Australia, December 3-8, 2006. viii Contents Abstract.................................. i Declaration ................................ iii Acknowledgements ............................ v ListofPublications. vii TableofContents............................. ix ListofFigures............................... xi ListofTables ...............................xix 1 Introduction 1 1.1 Structureofthethesis . .. .. 2 2 Background 5 2.1 Anewparadigmforcomputation . 5 2.2 Thequantumcircuitmodel . 7 2.3 Devicearchitectures . 12 2.3.1 TheKanearchitecture . 12 2.3.2 The global electron spin architecture . 14 3 The read-out state for Si:P based architectures 17 3.1 Kaneadiabaticread-outprotocol . 18 3.2 Modelling the Si:P qubit . 24 3.2.1 Thesiliconlattice. 25 3.2.2 Impurity donors and effective mass theory . 28 3.2.3 Binding energy of a phosphorus donor in silicon . 33 3.3 Determining the binding energy of the D− read-outstate . 35 0 3.4 Improving the modelling of the D and D− states . 38 ix 3.5 Summary .............................. 39 4 Optically induced charge transfer for donor spin read-out 41 4.1 Gatedresonantspintransfer . 43 4.2 ResonantFIRlasertransfer . 49 4.3 Chargetransferfidelity . 53 4.4 Singlet-triplet read-out for electron spin qubits . ...... 55 4.5 Summary .............................. 57 5 A CNOT gate immune to large fabrication induced variations 59 5.1 Constructing robust gates using composite rotations . ..... 61 5.2 Correcting for an unknown exchange interaction strength.... 66 5.2.1 Gatecount ......................... 69 5.2.2 Gatetime .......................... 70 5.3 The role of two-qubit Hamiltonian characterisation . .... 71 5.4 Summary .............................. 77 6 Modelling effects of charge noise on the exchange interaction 79 6.1 Thenoisemodel........................... 80 6.2 Calculating the probability density function . ... 84 6.2.1 Approximating Ω (ξ, T )................... 85 6.3 Using the PDF to determine Q (t)................. 87 6.4 Multiplefluctuators. 90 6.5 Usingthesuperoperators.