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Decoherence, Control, and Symmetry in Quantum Computers Decoherence, Control, and Symmetry in Quantum Computers by Dave Morris Bacon Double B.S. with Honors, (California Institute of Technology) 1997 A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the GRADUATE DIVISION of the UNIVERSITY of CALIFORNIA at BERKELEY arXiv:quant-ph/0305025v1 5 May 2003 Committee in charge: Professor K. Birgitta Whaley, Chair Professor Raymond Chiao Professor Dan Stamper-Kurn Professor Umesh Vazirani 2001 The dissertation of Dave Morris Bacon is approved: Chair Date Date Date Date University of California at Berkeley 2001 Decoherence, Control, and Symmetry in Quantum Computers Copyright 2001 by Dave Morris Bacon 1 Abstract Decoherence, Control, and Symmetry in Quantum Computers by Dave Morris Bacon Doctor of Philosophy in Physics University of California at Berkeley Professor K. Birgitta Whaley, Chair Computers built on the physical principles of quantum theory offer the possibility of tremendous computational advantages over conventional computers. To actually realize such quantum computers will require technologies far beyond present day capabilities. One problem which particularly plagues quantum computers is the cou- pling of the quantum computer to an environment and the subsequent destruction of the quantum information in the computer through the process known as deco- herence. In this thesis we describe methods for avoiding the detrimental effects of decoherence while at the same time still allowing for computation of the quantum information. The philosophy of our method is to use a symmetry of the decoherence mechanism to find robust encodings of the quantum information. The theory of such decoherence-free systems is developed in this thesis with a particular emphasis on the manipulation of the decoherence-free information. Stability, control, and methods for using decoherence-free information in a quantum computer are presented. Spe- cific emphasis is put on decoherence due to a collective coupling between the system and its environment. Universal quantum computation on such collective decoherence decoherence-free encodings is demonstrated. Along the way, rigorous definitions of control and the use of encoded universality in the physical implementations of quan- tum computers are addressed. Explicit gate constructions for encoded universality on ion trap and exchange based quantum computers are given. The second part of the thesis is devoted to methods of reducing the decoherence problem which rely on more classically motivated reasoning for the robust storage of information. We ex- amine quantum systems that can store information in their ground state such that decoherence processes are prohibited via considerations of energetics. We present the theory of supercoherent systems whose ground states are quantum error detecting codes and give examples of supercoherent systems which allow universal quantum computation. We also give examples of a spin ladder whose ground state has both the error detecting properties of supercoherence as well as error correcting properties. We present the first example of a quantum error correcting ground state which is a natural error correcting code under reasonable physical assumptions. We conclude 2 by discussing the radical possibility of a naturally fault-tolerant quantum computer. Professor K. Birgitta Whaley Dissertation Committee Chair iii Dedicated to my parents Nancy and Larry Bacon “My home...I think I would be lost without it. I spend so much time in my head, and what I do there is so different...so bizarre to most people. If I didn’t have a center, a place to return to, I’d get lost.” –Patricia Vasquez, Eon[16] and to the memory of my grandfathers Lee Morris (1912-1996) Glen Bacon (1908-1994) iv Contents List of Figures x List of Tables xii I Quantum Computation: Decoherence and Control 1 1 Philosonomicon 2 1.1 Prologue.................................. 2 1.2 Argument via the inevitability of technology . ..... 3 1.3 Theriseofthequantumalgorithm . 5 1.4 Controlandthequantumcomputer . 7 1.5 Thedecoherenceroadblock. 8 1.6 Quantum Gemini: decoherence and control . .. 10 1.7 Thesisoutline ............................... 11 2 The Pain of Isolating Quantum Information: Decoherence 13 2.1 Thedegradationofquantuminformation . .. 13 2.2 Quantumoperations ........................... 15 2.2.1 Derivation............................. 15 2.2.2 FixedbasisOSR ......................... 17 2.2.3 ExampleOSR........................... 18 2.2.4 OSRandunitaryevolution. 19 2.2.5 LimitsoftheOSR ........................ 20 2.3 Masterequations ............................. 21 2.3.1 Discrete coarse graining derivation of the SME . ... 22 2.3.2 Resolving a mystery in decoherence rates . 25 2.3.3 DiagonalformoftheSME . 28 2.4 Decoherence................................ 29 3 Quantum Control 30 3.1 Controlandmeasurement . 30 v 3.2 Conditionsforcontrol .......................... 31 3.2.1 Orthogonal pure state stationary control . .. 31 3.2.2 Commuting mixed state stationary control . 33 3.2.3 Non-stationary control and throwing the switch . ... 34 3.3 Controlexamples ............................. 35 3.4 Controlwithcoherentstates . 36 3.5 Theunitarycontrolquestion. 38 3.5.1 Denselyfilledgroup. 38 3.5.2 Hamiltoniancontrol . 39 3.5.3 LiestructureofHamiltoniancontrol . 39 3.6 Controlandapproximation. 41 3.6.1 Approximate unitary evolution . 41 3.6.2 ApproximateOSRevolution . 43 3.7 Control .................................. 44 4 Universal Quantum Computation 45 4.1 Quantumsubsystems ........................... 46 4.2 Thequantumcircuitmodel . 48 4.3 Universality ................................ 50 4.3.1 The inductive subsystem lemma . 52 4.3.2 Universalgatesetexample . 52 4.3.3 TheKitaev-Solovaytheorem . 53 4.3.4 Discrete versus Hamiltonian control . 54 4.3.5 Example use of Lie algebraic structure . 54 4.4 Encodeduniversality ........................... 55 4.4.1 The fungible nature of quantum information . 55 4.4.2 Encoded universality constructions . 56 4.4.3 Few subsystems encoded universality example . .. 58 4.5 What to do with the strange irreducible representations ........ 59 4.5.1 WhattodowithLieAlgebraX? . 60 4.5.2 Whatisaqubit? ......................... 60 4.6 Subsystem growth of a Lie algebra and quantum computation .... 62 4.7 Universalquantumcomputation . 63 II Decoherence-Free Quantum Computation 66 5 Decoherence-Free Conditions 67 5.1 Protecting information by encoding . .. 67 5.2 TheOSRAlgebra............................. 68 5.3 Decoherence-free subspaces . 71 5.3.1 Decoherence-free subspace condition . 72 vi 5.3.2 Example decoherence-free subspace . 73 5.3.3 SystemHamiltonianandtheDFS . 74 5.3.4 Decoherence-free subspaces and quantum error correction. 75 5.4 Decoherence-free subsystems . 75 5.4.1 Representation theory for the OSR algebra . 77 5.4.2 Decoherence-free subsystem condition . .. 79 5.4.3 ThecommutantoftheOSRalgebraandDFSs . 80 5.4.4 Example decoherence-free subsystem . 80 5.5 Master equation decoherence-free conditions . ...... 82 5.6 Inducing decoherence-free conditions . .... 83 5.7 A brief history of decoherence-free conditions . ....... 85 5.8 Decoherence-freeconditions . 86 6 Stability of Decoherence-Free Systems 87 6.1 Stability example for a decoherence-free subspace . ....... 87 6.2 Stability under the operator-sum representation . ....... 88 6.3 Stability under the semigroup master equation . ..... 92 6.4 Dynamicalstability............................ 94 6.5 Stability .................................. 94 7 Decoherence-Free Subsystems and the Quantum Computer 96 7.1 Quantum computation and decoherence-free subspaces . ...... 96 7.2 DFSsforquantumcomputation . 97 7.3 The commutant and universal quantum computation . ... 100 7.3.1 Example of the commutant condition . 100 7.3.2 Why the commutant condition is sufficient but not necessary . 102 7.3.3 Representation theory and the commutant . 102 7.3.4 Existential universality on a DFS . 103 7.4 Measurement on DFSs for quantum computation . 103 7.5 Makingleakageintonoise . 104 7.6 Decoherence-free subsystems as components of a quantum computer . 105 8 Collective Decoherence 106 8.1 Collective coupling to an environment . ... 106 8.2 Collectivedephasing ........................... 107 8.2.1 Master equation collective dephasing . 108 8.3 Collective amplitude damping . 112 8.3.1 Master equation collective amplitude damping . ... 112 8.4 Collectivedecoherence . 113 8.5 Weak collective decoherence DFSs . 114 8.5.1 TheweakDFSbasis ....................... 115 8.6 Strong collective decoherence DFSs . 117 vii 8.6.1 ThestrongDFSbasis. 120 8.7 Collective amplitude damping DF subspaces . ... 121 8.8 Collectivedecoherence . 123 9 Universality on Collective Decoherence Decoherence-Free Subsys- tems 124 9.1 Nontrivial one and two qubit interactions on the collectiveDFSs . 124 9.1.1 Weak collective decoherence DFS commutant operations ... 125 9.1.2 Strong collective decoherence DFS commutant operations. 125 9.2 Weak collective decoherence DFS universality . ..... 126 9.2.1 Conjoining weak collective decoherence DFSs and universality 127 9.3 Strong collective decoherence DFS universality . ....... 128 9.3.1 Conjoining strong collective decoherence DFSs for universality 129 9.4 Weak collective decoherence DFS preparation and measurement . 130 9.5 Strong collective decoherence DFS preparation and measurement . 132 9.6 Fault-tolerant quantum computation and collective decoherence DFSs 133 9.7 Collective decoherence and quantum computation . ..... 134 10 The
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