Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2005 Symmetry and Control in Spin-Based Quantum Computing Dimitrije Stepanenko Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] THE FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES SYMMETRY AND CONTROL IN SPIN-BASED QUANTUM COMPUTING By DIMITRIJE STEPANENKO A Dissertation submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy Degree Awarded: Summer Semester, 2005 The members of the Committee approve the dissertation of Dimitrije Stepanenko defended on June 10, 2005. Nicholas E. Bonesteel Professor Directing Thesis Washington Mio Outside Committee Member Vladimir Dobrosavljevi´c Committee Member Stephan von Molnar Committee Member Mark Riley Committee Member The Office of Graduate Studies has verified and approved the above named committee members. ii To Andrija iii ACKNOWLEDGEMENTS It was unique privilege to spend my graduate school years working with my adviser and mentor, Professor Nick Bonesteel. The most important lessons he had taught me are that intuitive models and insights based on them are at least as important as the technical skills and that clarity and simplicity are necessary ingredients of every successful research. People who contributed to research presented in this thesis, both directly and through the inspiring discussions are Nicholas Bonesteel and Layla Hormozi from the Florida State University, Kerwin Foster from the Dillard University, David DiVincenzo from the IBM T. J. Watson Research Center and Guido Burkard and Daniel Loss from the University of Basel. I enjoyed the conversations about condensed matter physics with Professors Vladimir Dobrosavljevi´c, Pedro Schlottmann and Stephan von Molnar that kept me aware that there are many interesting faces of every problem. Darko, George, Kenny, Curt, Layla and John helped create an environment that made the long hours at the desk enjoyable. Sherry Tointigh and late Kate Caudill have shown a great deal of patience while dealing with my chaotic manner of handling paperwork, from the first day at FSU till the last signature on the last form. I would like to thank Professor LJubiˇsa Adamovi´c, Vlatko, Snjeˇzana, Marko, Vlada, Dragana, Olan, Joe, Donnato, Cheryl, Eddy, Lucas, Ann, Myongshik, Kan, Balˇsa, Ana, Ivana, Cira,´ Aleksa, Ilka, Vincent and Patricia, Marjan, Bo´ca and Dejan for their friendship and beautiful time spent together. I am especially grateful to my mother Slavica and mother in law Vanja for being with my family and me when we needed help. Without their devotion and assistance, my time in graduate school would have been much longer. My wife Jelena and son Andrija filled the years in graduate school with love and joy. I will always be thankful for their patience and understanding. iv TABLE OF CONTENTS List of Tables ................................................... .... vii List of Figures ................................................... viii Abstract ................................................... ......... x 1. INTRODUCTION ............................................... 1 1.1 Overview ........................................ ............. 2 1.2 Background on Quantum Computing .................... ........... 3 1.2.1 ClassicalComputers ............................ ............ 3 1.2.2 QuantumComputers.............................. .......... 7 1.2.3 Church-Turing Thesis ........................... ............ 11 1.3 Physical Realizations ............................ ................ 14 1.3.1 The Quantum Dot Quantum Computer .................. ....... 17 1.4 Spin-OrbitCoupling .............................. .............. 21 1.5 SymmetryandControl .............................. ............ 22 1.5.1 Reduction of the Effects of Spin-Orbit Coupling in Exchange Gates.... 24 1.5.2 Using Spin-Orbit Coupling Effects for Spin Control . ............. 26 1.6 SummaryofResults................................ ............. 28 2. SPIN-ORBIT COUPLING AS A PROBLEM: REDUCING THE GATE ANISOTROPY .................................................. 29 2.1 Reduction of Control Complexity .................... .............. 29 2.2 IsotropicExchange............................... ............... 31 2.2.1 SimpleControl................................. ............ 34 2.2.2 Universal Gate Set Using Isotropic Exchange . ............. 35 2.3 Anisotropy...................................... .............. 37 2.4 Time Dependence, Interaction and the Gate . .............. 38 2.4.1 Control through Pulse Shaping .................... ............ 39 2.4.2 Voltage Control Program......................... ............ 40 2.5 Simplifications from Symmetry ...................... .............. 42 2.6 Removal of First Order Anisotropy ................... .............. 44 2.6.1 CalculationoftheGate .......................... ............ 44 2.6.1.1 Parity and Time Reversal ....................... ......... 45 2.6.1.2 Time Reversal Symmetric Interactions and Time Symmetric Pulses 46 2.6.2 Perturbative Evaluation of the Gate . .. .. .. ............. 48 2.6.3 LocalSymmetry ................................. .......... 52 2.7 SummaryoftheResults............................. ............. 60 v 3. MICROSCOPIC DERIVATION OF SPIN INTERACTION .......... 62 3.1 The Model of Quantum Dots Quantum Computer . ......... 64 3.1.1 Confinement to Two Dimensions . .......... 65 3.1.2 Confinement in Plane, from Two to Zero Dimensions . .......... 69 3.1.3 ControlMechanisms ............................. ........... 71 3.2 Hund-Mulliken Approximation . .............. 74 3.2.1 Spin Orbit Coupling Hamiltonian . ............ 79 3.3 Axial Symmetry of the Effective Spin Interaction . ............... 84 3.4 Pulses of Hund-Mulliken Hamiltonian . ............... 86 3.4.1 Slightly asymmetric pulses . .............. 89 3.5 RobustCNOT...................................... ........... 90 4. SPIN-ORBIT COUPLING AS A RESOURCE: USING THE GATE ANISOTROPY .................................................. 95 4.1 Universal Gate Set Construction .................... ............... 96 4.1.1 SingleQubitGates .............................. ........... 100 4.1.2 TwoQubitGates................................. .......... 103 4.2 ControlRange .................................... ............. 107 4.2.1 Control at Fabrication Level ..................... ............. 108 4.2.2 Control through Time Dependence .................. ........... 116 4.3 Summary......................................... ............ 122 APPENDIX A: Axially Symmetric Gate for CNOT Construction ......... 124 APPENDIX B: Matrix Elements of Hund-Mulliken Hamiltonian ......... 127 REFERENCES ................................................... 129 BIOGRAPHICAL SKETCH .......................................... 134 vi LIST OF TABLES 2.1 Behavior of gate parameters under parity (P ) and time reversal (T ). Quantities that are even(odd) under parity are of even(odd) order in spin interaction anisotropy. Gate parameters odd under time reversal vanish from the gates implemented by time-symmetric pulses. .............. 45 3.1 Symmetry properties of the pulse parameters r and s, and gate parameters λ, α, β and γ under parity P and time reversal T . ........................ 90 vii LIST OF FIGURES 1.1 TuringMachine ................................... .............. 5 1.2 Universal Classical Logic Gates .................... ................. 6 1.3 Classical Circuit Model ........................... ................ 7 1.4 Elements of Quantum Circuit Model . .............. 9 1.5 Universal Set of Quantum Gates ...................... .............. 10 1.6 Quantum Circuit Complexity . ............... 11 1.7 Comparing Classical and Quantum Complexity . ............... 13 1.8 Quantum Dot Quantum Computer ....................... ........... 18 1.9 Universal Quantum Gates Using Isotropic Exchange . ................ 20 2.1 Controlled Not Using Isotropic Exchange . ................. 36 2.2 LocalReferenceFrames ............................ ............... 55 2.3 Constant Anisotropy Pulses ........................ ................ 57 2.4 Loop and Local Reference Frames ..................... .............. 58 2.5 TreeTopology .................................... .............. 59 3.1 The Quantum Dot Quantum Computer.................... ........... 64 3.2 BandBending ..................................... ............. 66 3.3 Voltagecontrolled gate........................... ................. 72 3.4 Isotropic Spin Interaction Strength . .. .. .. .................. 73 3.5 The Model Potential of Coupled Quantum Dots . ............. 75 3.6 Orbital Excitations and Failure Modes of a Spin Qubit . ................ 77 3.7 Zinc-Blende Crystal Structure ..................... ................. 81 3.8 Precession andHopping ............................ ............... 83 3.9 Anisotropy in Spin Interaction ..................... ................. 86 3.10 Time Dependent Hund-Mulliken Hamiltonian . ................ 89 3.11 GateHamiltonian................................ ................ 91 3.12 RobustCNOT ..................................... ............. 93 4.1 Range of Anisotropic Control....................... ................ 100 viii 4.2 Sequence for pseudospin x rotation .................................. 101 4.3 Two-SpinQubits .................................. .............. 104 4.4 CNOTcircuit..................................... .............. 106 4.5 Angle