Block-encoding dense and full-rank kernels using hierarchical matrices: applications in quantum numerical linear algebra
Name
q-2022-12-13-876.pdf
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2.15 MB
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Checksum (MD5)
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Author(s)
Nguyen, Quynh T.
Kiani, Bobak T.
Lloyd, Seth
Date Issued
December 13, 2022
Journal
Quantum
Publisher
Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften
Citation
Nguyen, Quynh T., Kiani, Bobak T. and Lloyd, Seth. 2022. "Block-encoding dense and full-rank kernels using hierarchical matrices: applications in quantum numerical linear algebra." Quantum, 6.
Version
Final published version
Abstract
Many quantum algorithms for numerical linear algebra assume black-box access to a block-encoding of the matrix of interest, which is a strong assumption when the matrix is not sparse. Kernel matrices, which arise from discretizing a kernel function k(x,x′), have a variety of applications in mathematics and engineering. They are generally dense and full-rank. Classically, the celebrated fast multipole method performs matrix multiplication on kernel matrices of dimension N in time almost linear in N by using the linear algebraic framework of hierarchical matrices. In light of this success, we propose a block-encoding scheme of the hierarchical matrix structure on a quantum computer. When applied to many physical kernel matrices, our method can improve the runtime of solving quantum linear systems of dimension N to O(κpolylog⁡(Nε)), where κ and ε are the condition number and error bound of the matrix operation. This runtime is near-optimal and, in terms of N, exponentially improves over prior quantum linear systems algorithms in the case of dense and full-rank kernel matrices. We discuss possible applications of our methodology in solving integral equations and accelerating computations in N-body problems.
Subjects
Physics and Astronomy (miscellaneous)
Atomic and Molecular Physics, and Optics
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Department of Physics
Massachusetts Institute of Technology. Research Laboratory of Electronics
Massachusetts Institute of Technology. Department of Mechanical Engineering
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Creative Commons Attribution
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.22331/q-2022-12-13-876
