On the geometry of Variational Quantum Algorithms: Hopf fibrations, entanglement and the natural gradient descent

Detalhes bibliográficos
Autor(a) principal: Habermann, Gustavo Schranck
Data de Publicação: 2024
Tipo de documento: Dissertação
Idioma: eng
Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
Texto Completo: https://www.teses.usp.br/teses/disponiveis/76/76134/tde-28082024-160650/
Resumo: Variational quantum algorithms were proposed as a strategy to obtain quantum advantage using the current state of the art quantum devices, the so called noisy intermediate scale quantum (NISQ) devices, by combining low-depth parameterized quantum circuits and classical optimization methods, such as the natural gradient descent. In this work we investigate geometrical aspects of variational quantum algorithms and the natural gradient descent in one and two qubits settings using tools of Riemannian geometry and information geometry, exploring the geometry of quantum states and the associated Hopf fibration structure as well as the behaviour of the natural gradient descent on statistical models contained in Riemannian manifolds. In addittion, recent results connecting entanglement, scalar curvature and two qubits parameterized quantum circuit performance are discussed and compared to our own calculations.
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spelling On the geometry of Variational Quantum Algorithms: Hopf fibrations, entanglement and the natural gradient descentSobre a geometria de Algoritmos Variacionais Quânticos: fibrações de Hopf, emaranhamento e a descida de gradiente naturalAlgoritmos quânticos variacionaisCircuitos quânticos parametrizadosEmaranhamentoEntanglementFibração de HopfFubini-StudyFubini-StudyHopf FibrationParameterized quantum circuitsVariational quantum algorithmsVariational quantum algorithms were proposed as a strategy to obtain quantum advantage using the current state of the art quantum devices, the so called noisy intermediate scale quantum (NISQ) devices, by combining low-depth parameterized quantum circuits and classical optimization methods, such as the natural gradient descent. In this work we investigate geometrical aspects of variational quantum algorithms and the natural gradient descent in one and two qubits settings using tools of Riemannian geometry and information geometry, exploring the geometry of quantum states and the associated Hopf fibration structure as well as the behaviour of the natural gradient descent on statistical models contained in Riemannian manifolds. In addittion, recent results connecting entanglement, scalar curvature and two qubits parameterized quantum circuit performance are discussed and compared to our own calculations.Algoritmos variacionais quânticos foram propostos como uma estratégia para obter vantagem quantica utilizando os dipositivos quânticos do altual estado da arte, chamados de dispositivos noisy intermediate scale quantum (NISQ), através da combinação de circuitos quânticos de baixa profundidade e métodos classicos de otimização, como a descida por gradiente natural. Neste trabalho investigamos aspectos geométricos de algoritmos variacionais quânticos usando ferramentas da geometria Riemanniana e da geometria da informação, explorando geometria de estados quânticos e a estrutura e sua estrutura de fibrados de Hopf bem como o comportamento da descida de gradiente natural em modelos estatísticos contidos em variedades Riemannianas. Além disso, discutimos resultados recentes conectando emaranhamento, curvatura escalar e a performance de circuitos quânticos parametrizados de dois qubits, comparando-os com nossos próprios cálculos.Biblioteca Digitais de Teses e Dissertações da USPPinto, Diogo de Oliveira SoaresHabermann, Gustavo Schranck2024-07-11info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/masterThesisapplication/pdfhttps://www.teses.usp.br/teses/disponiveis/76/76134/tde-28082024-160650/reponame:Biblioteca Digital de Teses e Dissertações da USPinstname:Universidade de São Paulo (USP)instacron:USPLiberar o conteúdo para acesso público.info:eu-repo/semantics/openAccesseng2024-08-29T19:47:03Zoai:teses.usp.br:tde-28082024-160650Biblioteca Digital de Teses e Dissertaçõeshttp://www.teses.usp.br/PUBhttp://www.teses.usp.br/cgi-bin/mtd2br.plvirginia@if.usp.br|| atendimento@aguia.usp.br||virginia@if.usp.bropendoar:27212024-08-29T19:47:03Biblioteca Digital de Teses e Dissertações da USP - Universidade de São Paulo (USP)false
dc.title.none.fl_str_mv On the geometry of Variational Quantum Algorithms: Hopf fibrations, entanglement and the natural gradient descent
Sobre a geometria de Algoritmos Variacionais Quânticos: fibrações de Hopf, emaranhamento e a descida de gradiente natural
title On the geometry of Variational Quantum Algorithms: Hopf fibrations, entanglement and the natural gradient descent
spellingShingle On the geometry of Variational Quantum Algorithms: Hopf fibrations, entanglement and the natural gradient descent
Habermann, Gustavo Schranck
Algoritmos quânticos variacionais
Circuitos quânticos parametrizados
Emaranhamento
Entanglement
Fibração de Hopf
Fubini-Study
Fubini-Study
Hopf Fibration
Parameterized quantum circuits
Variational quantum algorithms
title_short On the geometry of Variational Quantum Algorithms: Hopf fibrations, entanglement and the natural gradient descent
title_full On the geometry of Variational Quantum Algorithms: Hopf fibrations, entanglement and the natural gradient descent
title_fullStr On the geometry of Variational Quantum Algorithms: Hopf fibrations, entanglement and the natural gradient descent
title_full_unstemmed On the geometry of Variational Quantum Algorithms: Hopf fibrations, entanglement and the natural gradient descent
title_sort On the geometry of Variational Quantum Algorithms: Hopf fibrations, entanglement and the natural gradient descent
author Habermann, Gustavo Schranck
author_facet Habermann, Gustavo Schranck
author_role author
dc.contributor.none.fl_str_mv Pinto, Diogo de Oliveira Soares
dc.contributor.author.fl_str_mv Habermann, Gustavo Schranck
dc.subject.por.fl_str_mv Algoritmos quânticos variacionais
Circuitos quânticos parametrizados
Emaranhamento
Entanglement
Fibração de Hopf
Fubini-Study
Fubini-Study
Hopf Fibration
Parameterized quantum circuits
Variational quantum algorithms
topic Algoritmos quânticos variacionais
Circuitos quânticos parametrizados
Emaranhamento
Entanglement
Fibração de Hopf
Fubini-Study
Fubini-Study
Hopf Fibration
Parameterized quantum circuits
Variational quantum algorithms
description Variational quantum algorithms were proposed as a strategy to obtain quantum advantage using the current state of the art quantum devices, the so called noisy intermediate scale quantum (NISQ) devices, by combining low-depth parameterized quantum circuits and classical optimization methods, such as the natural gradient descent. In this work we investigate geometrical aspects of variational quantum algorithms and the natural gradient descent in one and two qubits settings using tools of Riemannian geometry and information geometry, exploring the geometry of quantum states and the associated Hopf fibration structure as well as the behaviour of the natural gradient descent on statistical models contained in Riemannian manifolds. In addittion, recent results connecting entanglement, scalar curvature and two qubits parameterized quantum circuit performance are discussed and compared to our own calculations.
publishDate 2024
dc.date.none.fl_str_mv 2024-07-11
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/masterThesis
format masterThesis
status_str publishedVersion
dc.identifier.uri.fl_str_mv https://www.teses.usp.br/teses/disponiveis/76/76134/tde-28082024-160650/
url https://www.teses.usp.br/teses/disponiveis/76/76134/tde-28082024-160650/
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv
dc.rights.driver.fl_str_mv Liberar o conteúdo para acesso público.
info:eu-repo/semantics/openAccess
rights_invalid_str_mv Liberar o conteúdo para acesso público.
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.coverage.none.fl_str_mv
dc.publisher.none.fl_str_mv Biblioteca Digitais de Teses e Dissertações da USP
publisher.none.fl_str_mv Biblioteca Digitais de Teses e Dissertações da USP
dc.source.none.fl_str_mv
reponame:Biblioteca Digital de Teses e Dissertações da USP
instname:Universidade de São Paulo (USP)
instacron:USP
instname_str Universidade de São Paulo (USP)
instacron_str USP
institution USP
reponame_str Biblioteca Digital de Teses e Dissertações da USP
collection Biblioteca Digital de Teses e Dissertações da USP
repository.name.fl_str_mv Biblioteca Digital de Teses e Dissertações da USP - Universidade de São Paulo (USP)
repository.mail.fl_str_mv virginia@if.usp.br|| atendimento@aguia.usp.br||virginia@if.usp.br
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