Exploring quantum walks: weighted paths and quotient graphs unveiled

• Ano de defesa: 2023
• Autor(a) principal: Frederico Cançado Pereira
• Orientador(a): Gabriel de Morais Coutinho
• Banca de defesa: Aldo Procacci, Balchandra Digambar Thatte
• Tipo de documento: Dissertação
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Universidade Federal de Minas Gerais
• Programa de Pós-Graduação: Programa de Pós-Graduação em Matemática
• Departamento: ICX - DEPARTAMENTO DE MATEMÁTICA
• País: Brasil
• Palavras-chave em Português: Spectral graph theory; Perfect state transfer; Equitable partition; Orthogonal polynomials
• Link de acesso: http://hdl.handle.net/1843/60233

This thesis explores the problem of Perfect State Transfer (PST) in graphs, which has significant implications in quantum computing. The goal is to determine which graphs allow for perfect transfer of the state of one qubit (or vertex) to another qubit within a certain time frame. The text provides an introduction to the topic using techniques from linear algebra, discussing necessary and sufficient conditions to achieve PST, and emphasizing long-distance transfer between qubits. The optimization objective is to minimize the number of quantum components required to achieve perfect state transfer. An important class of graphs that admit PST is weighted paths. For PST between vertices at the endpoints, the problem has been completely solved by exploring the connection of these graphs with orthogonal polynomials. However, the problem becomes considerably more complex for vertices in other positions, leading to new results and connections explored in this document. Among these results, we can mention a formula that uniquely relates the extreme polynomial to another arbitrary polynomial in a sequence of orthogonal polynomials, how to create a sequence of orthogonal polynomials containing two given polynomials, and how PST in weighted paths relates to the Prouhet-Tarry-Escott problem, an open problem in number theory. Finally, the document presents an approach to constructing graphs with PST, exploring weighted paths and equitable partitions. New theorems in this are also presented, which have general relevance to graph theory. These theorems include a criterion for two graphs to have a common symmetrized quotient, how equitable partition matrices relate to projective matrices, and how the set of equitable partitions transforms when the original graph is quotiented.