Theory of measurement-based quantum computing

In the study of quantum computation, data is represented in terms of linear operators which form a generalized model of probability, and computations are most commonly described as products of unitary transformations, which are the transformations which preserve the quality of the data in a precise sense. This naturally leads to "unitary circuit models", which are models of computation in which unitary operators are expressed as a product of "elementary" unitary transformations. However, unitary transformations can also be effected as a composition of operations which are not all unitary themselves: the "one-way measurement model" is one such model of quantum computation. In this thesis, we examine the relationship between representations of unitary operators and decompositions of those operators in the one-way measurement model. In particular, we consider different circumstances under which a procedure in the one-way measurement model can be described as simulating a unitary circuit, by considering the combinatorial structures which are common to unitary circuits and two simple constructions of one-way based procedures. These structures lead to a characterization of the one-way measurement patterns which arise from these constructions, which can then be related to efficiently testable properties of graphs. We also consider how these characterizations provide automatic techniques for obtaining complete measurement-based decompositions, from unitary transformations which are specified by operator expressions bearing a formal resemblance to path integrals. These techniques are presented as a possible means to devise new algorithms in the one-way measurement model, independently of algorithms in the unitary circuit model.Comment: Ph.D. thesis in Combinatorics and Optimization. 199 pages main text, 26 PDF figures. Official electronic version available at http://hdl.handle.net/10012/413

Applying quantum information to fingerprinting schemes and algebraic structures

Bibliography: p. 280-283This research consists of three topics in quantum information science. First is an examination of the advantages that quantum information gives over classical information in very small instances of the fingerprinting problem. The second topic presents the properties of quantum Fourier transforms (QFTs) with respect to finite abelian groups, extends these ideas to finite unitary rings, and demonstrates how these properties characterize QFTs. The final topic applies some of the properties of the QFT to solve an oracle problem defined in terms of scrambled or "hidden" polynomial functions. Understanding quantum information often requires a major change in perspective, and thus has a reputation of being very difficult. Muddled and vague "popular presentations" contribute to the problem. In reaction to these issues, an introduction is presented here which relies on mathematics and supplementary commentary to present the ideas of quantum information, suitable for any reader with basic knowledge of linear algebra