In this work we tackle the challenge of designing quantum unitary operators which represent solutions to optimization problems. We start with a novel method which combines an evolutionary algorithm known as an Evolution Strategy (ES) with a method to randomly generate unitary operators. With this new method, a quantum operator is represented for the first time using real-valued vectors and can be "evolved" or designed to meet certain target criteria. This criteria could be the solution to an optimization problem. With the ability to evolve quantum operators, we attempt to evolve various known single and multi-qubit quantum gates as well as quantum oracles. We evolve quantum operators which solve instance problems of a known NP-Hard problem and even attempt to evolve a generalized solution operator. We evolve multiple operators with varying size and investigate their properties through eigenanalysis methods as well as by synthesizing them into quantum logic gates using the quantum compiler Qubiter. We also present a new quantum logic algebra which offers a new way to represent quantum circuits and demonstrate its immediate uses in quantum computing.

Thesis (Ph.D.)--University of Missouri-Columbia, 2006. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file viewed on (May 2, 2007) Vita. Includes bibliographical references.

Thesis (Ph. D.)--University of North Texas, May, 2008. / Title from title page display. Includes bibliographical references.

Thesis (Ph. D.)--University of Alabama at Birmingham, 2007. / Description based on contents viewed Feb. 12, 2009; title from PDF t.p. Includes bibliographical references (p. 75-77).

Let H be a separable infinite dimensional complex Hilbert space, let U(H) be the Polish topological group of unitary operators on H, let G be a Polish topological group and φ:G→U(H) an algebraic isomorphism. Then φ is a topological isomorphism. The same theorem holds for the projective unitary group, for the group of *-automorphisms of L(H) and for the complex isometry group. If H is a separable real Hilbert space with dim(H)≥3, the theorem is also true for the orthogonal group O(H), for the projective orthogonal group and for the real isometry group. The theorem fails for U(H) if H is finite dimensional complex Hilbert space.

Submitted by Bruna Rodrigues (bruna92rodrigues@yahoo.com.br) on 2016-10-10T13:51:29Z No. of bitstreams: 1 DissWM.pdf: 576272 bytes, checksum: 2aed9c1a9025d2818155f1b5a16ffc54 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-21T13:33:51Z (GMT) No. of bitstreams: 1 DissWM.pdf: 576272 bytes, checksum: 2aed9c1a9025d2818155f1b5a16ffc54 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-21T13:34:02Z (GMT) No. of bitstreams: 1 DissWM.pdf: 576272 bytes, checksum: 2aed9c1a9025d2818155f1b5a16ffc54 (MD5) / Made available in DSpace on 2016-10-21T13:34:12Z (GMT). No. of bitstreams: 1 DissWM.pdf: 576272 bytes, checksum: 2aed9c1a9025d2818155f1b5a16ffc54 (MD5) Previous issue date: 2016-02-17 / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / We present a study of rank-one perturbations of self-adjoint and unitary operators, in particular related to the so-called Kotani’s trick. In Chapter 2 we recall relevant definitions and elementary facts. In Chapter 3 we present recent results by Marx, which involve self-adjoint operators and Hausdorff measures. A generalization of such results to the set of unitary operators is described in Chapter 4. / Apresentaremos um estudo realizados sobre pertubações de posto 1 de operadores auto-adjuntos e unitários, particularmente relacionados ao chamado truque de Kotani. O Capítulo 2 consiste na exposição de definições elementares e fatos básicos que serão utilizados nos outros capítulos. O Capítulo 3 consiste na exposição de um estudo feito de um artigo recente de Marx sobre o tema, envolvendo operadores autoadjuntos e medidas de Hausdorff. Uma generalização dos resultados obtidos no artigo já mencionado para o contexto de operadores unitários é discutida no Capítulo 4.