Pretty good state transfer

• Ano de defesa: 2021
• Autor(a) principal: Pedro Vinícius Ferreira Baptista
• Orientador(a): Gabriel de Morais Coutinho
• Banca de defesa: Mario Sérgio Ferreira Alvim Júnior, Carlos Felipe Lardizabal Rodrigues, Raphael Campos Drumond
• 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 Ciência da Computação
• Departamento: ICX - DEPARTAMENTO DE CIÊNCIA DA COMPUTAÇÃO
• País: Brasil
• Palavras-chave em Português: Quantum walks; State transfer; Quantum computing; Algebraic graph theory
• Link de acesso: http://hdl.handle.net/1843/43127

Continuous-time quantum walks is a recent area of research, both in quantum computing and algebraic graph theory. It has applications in quantum search algorithms, state transfer and, more recently, in creating a set of universal quantum gates. In terms of state transfer, its main motivation is to research in which cases there can be transfer of states in a network comprised of vertices of a graph and how much time must we wait for a specific probability. One of the main topics of research in state transfer in quantum walks is about two types of transfer: perfect state transfer and pretty good state transfer. In so far, the former has been thoroughly researched with characterizations for its occurrence on the Adjacency and Laplacian matrices. Furthermore, it was also shown that we can verify its occurrence in a graph using a classical algorithm in polynomial time. As for the pretty good state transfer, some results are known in terms of its occurrence for some classes of graphs and also for a characterization of its occurrence. The main problem is that the characterizations we know demand some work to verify it for a graph, and no exact algorithm was known to do it. Furthermore, for some common classes of graphs, it was shown that perfect state transfer is rare. This is mostly due to the restrictions it imposes on the eigenspaces of the graph. Therefore, since we cannot have state transfer with 1 probability, it is natural to check for state transfer with probability close to 1 and at what time cost it demands. In this master’s thesis, we present the first exact algorithm for verifying pretty good state transfer in graphs. Another path of research was to try to replicate some known results of state transfer, perfect and pretty good state transfers, for the adjacency and the Laplacian matrix in the Normalized Laplacian. The motivation for that arises in the connection of the Normalized Laplacian with the Classical Random Walk.