Problema do subgrupo oculto em grupos nilpotentes

• Ano de defesa: 2008
• Autor(a) principal: Fernandes, Tharso Dominisini
• Orientador(a): Portugal, Renato
• Banca de defesa: Garcia, Eduardo Lúcio Mendes , Kritz, Mauricio Vieira , Lavor, Carlile Campos , Leal, Guilherme Augusto de La Roque
• Tipo de documento: Dissertação
• 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: BR
• Palavras-chave em Português: Computadores quânticos; Problema do Subgrupo Oculto; Algoritmos Quânticos
• Palavras-chave em Inglês: Quantum computers; Hidden subgroup problem; Quantum algorithms
• 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/83

Quantum computers may solve certain problems asymptotically faster than the classical computers. Quantum algorithms, such as Shor s algorithm, may be considered as a particular case of the Hidden Subgroup Problem (HSP). The HSP consists in finding a subgroup H of a group G by evaluating a function f, which is constant in cosets of H and distinct for each coset. The HSP for Abelian groups is efficiently solved in a quantum computer, but is quantum computers can solve the HSP in non-Abelian groups efficiently? This question has been regularly discussed by the scientific community due to the importance of some applications, such as the graph isomorphism problem and the short vector in a lattice. In this dissertation we review the Ivanyos et al. (2007a) that address HSP in nilpotent groups of class 2. We make a brief review on Quantum Computing; we address some characteristics of nilpotent groups and solvable groups, with special attention to nilpotent groups of class 2; we discuss the standard method of solution of the HSP in Abelian groups; we present the main characteristics of the polycyclic sequences and important reductions of the HSP in classes of nilpotent groups using the properties of polycyclic sequences. Finally, we present an efficient algorithm to solve the HSP in nilpotent groups of class 2.