Methods for Improved Analysis and Implementation of Algorithms for Local Hamiltonians

Morales Soler, Mauricio Enrique

Methods for Improved Analysis and Implementation of Algorithms for Local Hamiltonians

Morales Soler, Mauricio Enrique

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Publication Type:: Thesis
Issue Date:: 2023

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Field	Value	Language
dc.contributor.author	Morales Soler, Mauricio Enrique
dc.date.accessioned	2024-03-25T03:39:54Z
dc.date.available	2024-03-25T03:39:54Z
dc.date.issued	2023
dc.identifier.uri	http://hdl.handle.net/10453/177083
dc.description	University of Technology Sydney. Faculty of Engineering and Information Technology.	en_US.UTF-8
dc.description.abstract	One of the main applications of quantum computers is expected to be computing properties of physical systems. Of particular interest is that of computing ground states and of performing evolution of a system with a given Hamiltonian. Related to these problems are the questions of when can we expect a quantum advantage and how to best implement a quantum algorithm when such advantage is expected. In this thesis we seek to give partial answers to these questions for certain versions of the local Hamiltonian problem and the Hamiltonian evolution problem. In the first part of this thesis we study a parameterized version of the local Hamiltonian problem where the ground-state of interest can be expressed as a superposition of computational basis states with a fixed Hamming weight. We find that this problem is unlikely to be tractable for quantum computers and moreover find connections with a quantum version of the exponential time hypothesis. The second part of this thesis provides a quantum sampling scheme based on Fermions which is inspired on the Boson Sampling scheme. The hardness guarantees in our scheme are comparable to those of Random Circuit Sampling, moreover we prove an anticoncentration property for this scheme in an improvement over what is known for Boson Sampling. In the final part we present results on product formulas for Hamiltonian simulation. This work improves over previous methods to implement higher-order product formulae by finding new product formulae which greatly reduce the error and thus require fewer gate counts to implement. Moreover, we compare the performance of these product formulae in practice and find that our product formulae could be a better choice for quantum chemistry.	en_US.UTF-8
dc.format	Thesis (PhD)
dc.language.iso	en_US	en_US.UTF-8
dc.relation	https://opus.lib.uts.edu.au/bitstream/10453/177083/1/thesis.pdf
dc.rights	info:eu-repo/semantics/openAccess
dc.rights	The author owns the copyright in this thesis including all reproduction and reuse rights for the work. The work may not be altered without the permission of the copyright owner. Attribution is essential when quoting or paraphrasing from this thesis.
dc.rights	© 2023 Mauricio Enrique Morales Soler
dc.rights	au.edu.uts.lib/cph
dc.title	Methods for Improved Analysis and Implementation of Algorithms for Local Hamiltonians	en_US.UTF-8
dc.type	Thesis
utslib.copyright.status	open_access	*

Abstract:

One of the main applications of quantum computers is expected to be computing properties of physical systems. Of particular interest is that of computing ground states and of performing evolution of a system with a given Hamiltonian. Related to these problems are the questions of when can we expect a quantum advantage and how to best implement a quantum algorithm when such advantage is expected. In this thesis we seek to give partial answers to these questions for certain versions of the local Hamiltonian problem and the Hamiltonian evolution problem. In the first part of this thesis we study a parameterized version of the local Hamiltonian problem where the ground-state of interest can be expressed as a superposition of computational basis states with a fixed Hamming weight. We find that this problem is unlikely to be tractable for quantum computers and moreover find connections with a quantum version of the exponential time hypothesis. The second part of this thesis provides a quantum sampling scheme based on Fermions which is inspired on the Boson Sampling scheme. The hardness guarantees in our scheme are comparable to those of Random Circuit Sampling, moreover we prove an anticoncentration property for this scheme in an improvement over what is known for Boson Sampling. In the final part we present results on product formulas for Hamiltonian simulation. This work improves over previous methods to implement higher-order product formulae by finding new product formulae which greatly reduce the error and thus require fewer gate counts to implement. Moreover, we compare the performance of these product formulae in practice and find that our product formulae could be a better choice for quantum chemistry.

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http://hdl.handle.net/10453/177083