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

TL;DR

This paper introduces a quantum block-encoding scheme for hierarchical matrices, enabling efficient quantum linear algebra computations on dense, full-rank kernel matrices with exponential speedup over previous methods.

Contribution

It proposes a novel quantum block-encoding approach for hierarchical matrices, significantly improving the efficiency of quantum algorithms for dense kernel matrices.

Findings

Achieves near-linear quantum runtime for solving linear systems with kernel matrices.

Provides exponential speedup over prior quantum algorithms for dense, full-rank matrices.

Applicable to integral equations and N-body problem computations.

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.