## Quantum algorithms, symmetry, and Fourier analysis

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

Title

Quantum algorithms, symmetry, and Fourier analysis

Author(s)

Denney, Aaron Jafar

Advisor(s)

Moore, Cristopher

Committee Member(s)

Caves, Carlton

Deutsch, Ivan

Landahl, Andrew

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

hidden subgroup problem

quantum algorithms

quantum computation

geometry

optics

LC Subject(s)

Quantum computers.

Computer algorithms.

Symmetry (Mathematics)

Fourier transformations.

Representations of groups.

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.

Date

May 2012

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##### Collections

- Physics [55]
- Physics & Astronomy Dissertations [41]