Quantum Algorithm for Matrix Logarithm by Integral Formula

TL;DR

This paper introduces a quantum algorithm for computing the matrix logarithm applied to a vector, utilizing integral representation, LCU, and block-encoding techniques, advancing quantum matrix function computations.

Contribution

It presents the first quantum algorithm for the matrix logarithm based on integral formulas, extending previous methods limited to other matrix functions.

Findings

Uses integral representation and quadrature for approximation

Employs LCU and block-encoding as subroutines

Enables quantum computation of matrix logarithm applied to vectors

Abstract

The matrix logarithm is one of the important matrix functions. Recently, a quantum algorithm that computes the state $|f\rangle$ corresponding to matrix-vector product $f(A)b$ is proposed in [Takahira, et al. Quantum algorithm for matrix functions by Cauchy's integral formula, QIC, Vol.20, No.1\&2, pp.14-36, 2020]. However, it can not be applied to matrix logarithm. In this paper, we propose a quantum algorithm, which uses LCU method and block-encoding technique as subroutines, to compute the state $|f\rangle = \log(A)|b\rangle / \|\log(A)|b\rangle\|$ corresponding to $\log(A)b$ via the integral representation of $\log(A)$ and the Gauss-Legendre quadrature rule.