Análise assintótica de passeios quânticos para algoritmos de busca com múltiplos marcados

• Ano de defesa: 2023
• Autor(a) principal: Lugão, Pedro Henrique Gasparetto
• Orientador(a): Portugal, Renato
• Banca de defesa: Portugal, Renato, Giraldi, Gilson Antônio, Oliveira, Marcos Cesar de, Coutinho, Gabriel de Morais
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: por
• Instituição de defesa: Laboratório Nacional de Computação Científica
• Programa de Pós-Graduação: Programa de Pós-Graduação em Modelagem Computacional
• Departamento: Coordenação de Pós-Graduação e Aperfeiçoamento (COPGA)
• País: Brasil
• Palavras-chave em Português: Computação quântica; Passeio aleatório (Matemática); Teoria dos grafos; Análise assintótica
• Área do conhecimento CNPq: CNPQ::CIENCIAS EXATAS E DA TERRA::CIENCIA DA COMPUTACAO::TEORIA DA COMPUTACAO::COMPUTABILIDADE E MODELOS DE COMPUTACAO
• Link de acesso: https://tede.lncc.br/handle/tede/385

Given an arbitrary graph, it is possible to define an operator that represents the evolution of states respecting the graph’s locality. Quantum search, represented by a slight perturbation in certain nodes in this evolution operator to mark them, is based on evolving the quantum state for a certain number of steps until the probability of finding one of the marked nodes is maximized. Finding the number of steps and the final success probability in graphs of interest with an arbitrary number of marked nodes is the main objective of this work. To achieve this, we have developed an analytical method that obtains an asymptotic expression in the number of nodes for the desired quantities, based on the analysis of the two leading eigenvectors of the evolution matrix. The method has been developed for both discrete and continuous walk cases, with examples on the two-dimensional lattice and Johnson graphs, each having two marked elements. For an even more general case, we provide an analysis of continuous walks on t-designs, a mathematical abstraction that generates various bipartite graphs. In this latter case, we derive analytical expressions depending not only on the graph’s parameters but also on the number of marked nodes. The studied examples cover cases where a precise theory regarding the success probability and optimal time of quantum walks is lacking, showcasing the potential of the developed method to address more general problems.