Keywords

Quantum computers, Random walks (Mathematics)

Abstract

In this thesis, I investigate quantum walks in quantum computing from three aspects: the insights, the implementation, and the applications. Quantum walks are the quantum analogue of classical random walks. For the insights of quantum walks, I list and explain the required components for quantizing a classical random walk into a quantum walk. The components are, for instance, Markov chains, quantum phase estimation, and quantum spectrum theorem. I then demonstrate how the product of two reflections in the walk operator provides a quadratic speed-up, in comparison to the classical counterpart. For the implementation of quantum walks, I show the construction of an efficient circuit for realizing one single step of the quantum walk operator. Furthermore, I devise a more succinct circuit to approximately implement quantum phase estimation with constant precision controlled phase shift operators. From an implementation perspective, efficient circuits are always desirable because the realization of a phase shift operator with high precision would be a costly task and a critical obstacle. For the applications of quantum walks, I apply the quantum walk technique along with other fundamental quantum techniques, such as phase estimation, to solve the partition function problem. However, there might be some scenario in which the speed-up of spectral gap is insignificant. In a situation like that that, I provide an amplitude amplification-based iii approach to prepare the thermal Gibbs state. Such an approach is useful when the spectral gap is extremely small. Finally, I further investigate and explore the effect of noise (perturbation) on the performance of quantum walks

Notes

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Graduation Date

2011

Semester

Fall

Advisor

Wocjan, Pawel

Degree

Doctor of Philosophy (Ph.D.)

College

College of Engineering and Computer Science

Department

Electrical Engineering and Computer Science

Degree Program

Computer Science

Format

application/pdf

Identifier

CFE0004094

URL

http://purl.fcla.edu/fcla/etd/CFE0004094

Language

English

Release Date

December 2011

Length of Campus-only Access

None

Access Status

Doctoral Dissertation (Open Access)

Subjects

Dissertations, Academic -- Engineering and Computer Science, Engineering and Computer Science -- Dissertations, Academic

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