Quantum algorithm for matrix logarithm by integral formula

In scientific computing, one can find a wide application of the matrix-vector product f ( A ) b . Recently, a quantum algorithm that computes the state | f ⟩ corresponding to f ( A ) b has been proposed in Takahira et al. (Quantum Inf Comput 20(1/2):14–36, 2020). However, this important algorithm can not be directly applied to the matrix logarithm, which is one of the significant matrix functions. In this paper, we propose an original quantum algorithm to compute the state | f ⟩ = log ( A ) | b ⟩ / ‖ log ( A ) | b ⟩ ‖ , via the integral representation of log ( A ) and the Gauss–Legendre quadrature rule, using LCU method and block-encoding technique as subroutines.