## Physics & Astronomy ETDs

## Publication Date

8-27-2012

## 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.

## Degree Name

Physics

## Level of Degree

Doctoral

## Department Name

Physics & Astronomy

## First Committee Member (Chair)

Caves, Carlton

## Second Committee Member

Deutsch, Ivan

## Third Committee Member

Landahl, Andrew

## Project Sponsors

National Science Foundation, Army Research Office

## Language

English

## Keywords

Quantum computers, Computer algorithms, Symmetry (Mathematics), Fourier transformations, Representations of groups.

## Document Type

Dissertation

## Recommended Citation

Denney, Aaron Jafar. "Quantum algorithms, symmetry, and Fourier analysis." (2012). https://digitalrepository.unm.edu/phyc_etds/14