Block-encoding dense and full-rank kernels using hierarchical matrices: applications in quantum numerical linear algebra

• URL: http://arxiv.org/abs/2201.11329v4
• Date: Wed, 7 Dec 2022 02:05:04 GMT
• Title: Block-encoding dense and full-rank kernels using hierarchical matrices: applications in quantum numerical linear algebra
• Authors: Quynh T. Nguyen, Bobak T. Kiani, Seth Lloyd
• Abstract summary: We propose a block-encoding scheme of the hierarchical matrix structure on a quantum computer. Our method can improve the runtime of solving quantum linear systems of dimension $N$ to $O(kappa operatornamepolylog(fracNvarepsilon))$.
• Score: 6.338178373376447
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $O(\kappa \operatorname{polylog}(\frac{N}{\varepsilon}))$, where $\kappa$ and $\varepsilon$ 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.

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