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 />
