  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.
