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Quantum algorithms, symmetry, and Fourier analysis

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Please use this identifier to cite or link to this item: http://hdl.handle.net/1928/21025

Quantum algorithms, symmetry, and Fourier analysis

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Title: Quantum algorithms, symmetry, and Fourier analysis
Author: Denney, Aaron Jafar
Advisor(s): Moore, Cristopher
Committee Member(s): Caves, Carlton
Deutsch, Ivan
Landahl, Andrew
Department: University of New Mexico. Dept. of Physics & Astronomy
Subject(s): Fourier
hidden subgroup problem
quantum algorithms
quantum computation
geometry
optics
LC Subject(s): Quantum computers.
Computer algorithms.
Symmetry (Mathematics)
Fourier transformations.
Representations of groups.
Degree Level: Doctoral
Abstract: I describe the role of symmetry in two quantum algorithms, with a focus on how that symmetry is made manifest by the Fourier transform. The Fourier transform can be considered in a wider context than the familiar one of functions on Rn or Z/nZ; instead it can be defined for an arbitrary group where it is known as representation theory. The first quantum algorithm solves an instance of the hidden subgroup problem--distinguishing conjugates of the Borel subgroup from each other in groups related to PSL(2; q). I use the symmetry of the subgroups under consideration to reduce the problem to a mild extension of a previously solved problem. This generalizes a result of Moore, Rockmore, Russel and Schulman[33] by switching to a more natural measurement that also applies to prime powers. In contrast to the first algorithm, the second quantum algorithm is an attempt to use naturally continuous spaces. Quantum walks have proved to be a useful tool for designing quantum algorithms. The natural equivalent to continuous time quantum walks is evolution with the Schrodinger equation, under the kinetic energy Hamiltonian for a massive particle. I take advantage of quantum interference to find the center of spherical shells in high dimensions. Any implementation would be likely to take place on a discrete grid, using the ability of a digital quantum computer to simulate the evolution of a quantum system. In addition, I use ideas from the second algorithm on a different set of starting states, and find that quantum evolution can be used to sample from the evolute of a plane curve. The method of stationary phase is used to determine scaling exponents characterizing the precision and probability of success for this procedure.
Graduation Date: May 2012
URI: http://hdl.handle.net/1928/21025

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