Analysis of variational quantum algorithms for differential equations in the presence of quantum noise : application to the stationary Gross-Pitaevskii equation ; Analyse d'algorithmes quantiques variationnels pour la résolution d'équations différentielles en présence de bruit quantique : application à l'équation de Gross-Pitaevskii stationnaire
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| Názov: | Analysis of variational quantum algorithms for differential equations in the presence of quantum noise : application to the stationary Gross-Pitaevskii equation ; Analyse d'algorithmes quantiques variationnels pour la résolution d'équations différentielles en présence de bruit quantique : application à l'équation de Gross-Pitaevskii stationnaire |
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| Autori: | Serret, Michel Fabrice |
| Prispievatelia: | Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Sorbonne Université, Laurent Boudin, Yvon Maday, Thomas Ayral |
| Zdroj: | https://theses.hal.science/tel-04833499 ; Numerical Analysis [math.NA]. Sorbonne Université, 2024. English. ⟨NNT : 2024SORUS298⟩. |
| Informácie o vydavateľovi: | CCSD |
| Rok vydania: | 2024 |
| Predmety: | Quantum computing, Numerical analysis, Differential equations, Numerical resolution, Quantum, Informatique quantique, Analyse numérique, Équations différentielles, Résolution numérique, Quantique, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] |
| Popis: | Variational quantum algorithms (VQAs) have been proposed for solvingpartial differential equations on quantum computers. This thesis focuses on analyzingVQAs for the stationary Gross-Pitaevskii Equation (GPE) both under ideal (noiseless) conditionsand in the presence of quantum noise, providing error bounds, convergence properties, and estimatesfor the number of samples required.A central concept, make use of a relationship between the representation of functions and functional operators on dyadic rationals, through the Walsh basis, and the encoding offunctions and operators for N-qubit quantum systems through the Pauli operators and their eigenstates.In chapter 1, we link Pauli operators of N-qubit quantum systems with the Walsh basis on N-bit dyadic rationals, presenting new error bounds for the convergence of the N-bit Walsh series for functions in H^1(0,1) and presenting some results on the representation of Fourier basis functions in the Walsh basis.In chapter 2 we analyse VQAs for the GPE without noise, detailing the mathematicalsetting, discretization, and a-priori analysis. We introduce new energyestimators, either based on the Walsh decomposition of operators or obtained through inductive methods, and compare them to directsampling, in the diagonal basis of the operators, and the Hadamard-test method. Our results show that in the absence of noise the most promising methods for energyestimation is direct sampling in the diagonal basis, yielding the lowest variance and sample requirements.In chapter 3, we further examine the impact of quantum noise on energy estimation.Depolarizing noise introduces bias and shifts the variance of estimators. We show that the Pauli estimators proves least affected bynoise, due to their lower circuit size requirement,outperforming others both without mitigation, due to a lower bias, and with mitigation, as its sample efficiency is less affected.In the last chapter, devoted to work in progress, we present some preliminary results on decomposing differential operators in the ... |
| Druh dokumentu: | doctoral or postdoctoral thesis |
| Jazyk: | English |
| Relation: | NNT: 2024SORUS298 |
| Dostupnosť: | https://theses.hal.science/tel-04833499 https://theses.hal.science/tel-04833499v1/document https://theses.hal.science/tel-04833499v1/file/143095_SERRET_2024_archivage.pdf |
| Rights: | info:eu-repo/semantics/OpenAccess |
| Prístupové číslo: | edsbas.CE8C930B |
| Databáza: | BASE |
Nájsť tento článok vo Web of Science | Abstrakt: | Variational quantum algorithms (VQAs) have been proposed for solvingpartial differential equations on quantum computers. This thesis focuses on analyzingVQAs for the stationary Gross-Pitaevskii Equation (GPE) both under ideal (noiseless) conditionsand in the presence of quantum noise, providing error bounds, convergence properties, and estimatesfor the number of samples required.A central concept, make use of a relationship between the representation of functions and functional operators on dyadic rationals, through the Walsh basis, and the encoding offunctions and operators for N-qubit quantum systems through the Pauli operators and their eigenstates.In chapter 1, we link Pauli operators of N-qubit quantum systems with the Walsh basis on N-bit dyadic rationals, presenting new error bounds for the convergence of the N-bit Walsh series for functions in H^1(0,1) and presenting some results on the representation of Fourier basis functions in the Walsh basis.In chapter 2 we analyse VQAs for the GPE without noise, detailing the mathematicalsetting, discretization, and a-priori analysis. We introduce new energyestimators, either based on the Walsh decomposition of operators or obtained through inductive methods, and compare them to directsampling, in the diagonal basis of the operators, and the Hadamard-test method. Our results show that in the absence of noise the most promising methods for energyestimation is direct sampling in the diagonal basis, yielding the lowest variance and sample requirements.In chapter 3, we further examine the impact of quantum noise on energy estimation.Depolarizing noise introduces bias and shifts the variance of estimators. We show that the Pauli estimators proves least affected bynoise, due to their lower circuit size requirement,outperforming others both without mitigation, due to a lower bias, and with mitigation, as its sample efficiency is less affected.In the last chapter, devoted to work in progress, we present some preliminary results on decomposing differential operators in the ... |
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