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.
@article{arxiv.2111.08914,
title = {Quantum Algorithm for Matrix Logarithm by Integral Formula},
author = {Songling Zhang and Hua Xiang},
journal= {arXiv preprint arXiv:2111.08914},
year = {2021}
}