Quantum Algorithm for Matrix Logarithm by Integral Formula
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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.
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