Quantum Algorithm for Matrix Logarithm by Integral Formula

Abstract

The matrix logarithm is one of the important matrix functions. Recently, a quantum algorithm that computes the state |f 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 = log(A)|b / log(A)|b corresponding to log(A)b via the integral representation of log(A) and the Gauss-Legendre quadrature rule.