A quantum computer amenable sparse matrix equation solver

dc.contributor.authorPhillips, Christopher
dc.contributor.examiningcommitteeJeffrey, Ian (Electrical and Computer Engineering)en_US
dc.contributor.examiningcommitteeSirker, Jesko (Physics and Astronomy)en_US
dc.contributor.supervisorOkhmatovski, Vladimir
dc.date.accessioned2022-08-12T15:58:10Z
dc.date.available2022-08-12T15:58:10Z
dc.date.copyright2022-08-05
dc.date.issued2022-08-05
dc.date.submitted2022-08-05T17:52:03Zen_US
dc.degree.disciplineElectrical and Computer Engineeringen_US
dc.degree.levelMaster of Science (M.Sc.)en_US
dc.description.abstractQuantum computation offers a promising alternative to classical computing methods in many areas of numerical science, with algorithms that make use of the unique way in which quantum computers store and manipulate data often achieving dramatic improvements in performance over their classical counterparts. The potential efficiency of quantum computers is particularly important for numerical simulations, where the capabilities of classical computing systems are often insufficient for the analysis of real-world problems. In this work, we study problems involving the solution of matrix equations, for which there currently exists no efficient, general quantum procedure. We develop a generalization of the Harrow/Hassidim/Lloyd algorithm by providing an alternative unitary for eigenphase estimation. This unitary, which we have adopted from research in the area of quantum walks, has the advantage of being well defined for any arbitrary matrix equation, thereby allowing the solution procedure to be directly implemented on quantum hardware for any well-conditioned system. The procedure is most useful for sparse matrix equations, as it allows for the inverse of a matrix to be applied with O(N_nz log (N)) complexity, where N is the number of unknowns, and N_nz is the total number of nonzero elements in the system matrix. This efficiency is independent of the matrix structure, and hence the quantum procedure can outperform classical methods for many common system types. We show this using the example of sparse approximate inverse (SPAI) preconditioning, which involves the application of matrix inverses for matrices with N_nz = O(N). While these matrices are indeed sparse, it is often found that their inverses are quite dense, and classical methods can require as much as O(N^3) time to apply an inverse preconditioner.en_US
dc.description.noteOctober 2022en_US
dc.identifier.urihttp://hdl.handle.net/1993/36676
dc.language.isoengen_US
dc.rightsopen accessen_US
dc.subjectQuantum Computingen_US
dc.subjectMatrix Algorithmsen_US
dc.subjectHHLen_US
dc.titleA quantum computer amenable sparse matrix equation solveren_US
dc.typemaster thesisen_US
local.subject.manitobanoen_US
oaire.awardTitleUniversity of Manitoba graduate fellowshipen_US
oaire.awardURIhttps://umanitoba.ca/graduate-studies/funding-awards-and-financial-aid/university-manitoba-graduate-fellowship-umgfen_US
project.funder.identifierhttps://doi.org/10.13039/100010318en_US
project.funder.nameUniversity of Manitobaen_US
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