Quantum Algorithm for Matrix Logarithm by Integral Formula
Songling Zhang, Hua Xiang

TL;DR
This paper introduces a quantum algorithm for computing the matrix logarithm applied to a vector, utilizing integral representation, LCU, and block-encoding techniques, advancing quantum matrix function computations.
Contribution
It presents the first quantum algorithm for the matrix logarithm based on integral formulas, extending previous methods limited to other matrix functions.
Findings
Uses integral representation and quadrature for approximation
Employs LCU and block-encoding as subroutines
Enables quantum computation of matrix logarithm applied to vectors
Abstract
The matrix logarithm is one of the important matrix functions. Recently, a quantum algorithm that computes the state corresponding to matrix-vector product 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 corresponding to via the integral representation of and the Gauss-Legendre quadrature rule.
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Taxonomy
TopicsQuantum Computing Algorithms and Architecture · Matrix Theory and Algorithms · Polynomial and algebraic computation
