Quantum evolution: The case of weak localization for a 3D alloy-type Anderson model and application to Hamiltonian based quantum computation

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dc.contributor.author Cao, Zhenwei en_US
dc.date.accessioned 2013-02-19T22:35:52Z
dc.date.available 2013-02-19T22:35:52Z
dc.date.issued 2012-12-11 en_US
dc.identifier.other vt_gsexam:98 en_US
dc.identifier.uri http://hdl.handle.net/10919/19205
dc.description.abstract Over the years, people have found Quantum Mechanics to be extremely useful in <br />explaining various physical phenomena from a microscopic point of view.<br />Anderson localization, named after physicist P. W. Anderson, states that<br />disorder in a crystal can cause non-spreading of wave packets, which is one possible mechanism (at single electron level) to explain metal-insulator transitions. The theory of quantum computation promises to bring greater computational power over classical computers by making use of<br />some special features of Quantum Mechanics.<br />The first part of this dissertation considers a 3D alloy-type<br />model, where the Hamiltonian is the sum of the finite difference Laplacian corresponding to free motion of an electron and a random potential generated by a sign-indefinite single-site potential.<br />The result shows that localization occurs in the weak disorder regime,<br />{\\it i.e.}, when the coupling parameter $\\lambda$ is very small, for energies<br />$E \\le -C\\lambda^2$.<br />The second part of this dissertation considers adiabatic quantum computing (AQC) algorithms for the unstructured search problem to the case when the number of marked items is unknown.<br />In an ideal situation,<br />an explicit quantum algorithm together with a counting subroutine are given that achieve the optimal Grover<br />speedup over classical algorithms, {\\it i.e.}, roughly speaking, reduce $O(2^n)$ to $O(2^{n/2})$, where $n$ is the size of the problem. <br />However, if one considers more realistic settings, the result shows this quantum speedup is achievable <br />only under a very rigid control precision requirement ({\\it e.g.}, exponentially small control error).<br /> en_US
dc.format.medium ETD en_US
dc.publisher Virginia Tech en_US
dc.rights The authors of the theses and dissertations are the copyright owners. Virginia Tech's Digital Library and Archives has their permission to store and provide access to these works. en_US
dc.subject Quantum Mechanics en_US
dc.subject Random Schr en_US
dc.title Quantum evolution: The case of weak localization for a 3D alloy-type Anderson model and application to Hamiltonian based quantum computation en_US
dc.type Other - Dissertation en_US
dc.contributor.department Mathematics en_US
dc.description.degree PHD en_US
thesis.degree.name PHD en_US
thesis.degree.level doctoral en_US
thesis.degree.grantor Virginia Polytechnic Institute and State University en_US
thesis.degree.discipline Mathematics en_US
dc.contributor.committeechair Elgart, Alexander en_US
dc.contributor.committeemember Tauber, Uwe C en_US
dc.contributor.committeemember Hagedorn, George A en_US
dc.contributor.committeemember Stolz, Gunter en_US

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