Quantum Algorithms based on the Block-Encoding Framework for Matrix Functions by Contour Integrals

The matrix functions can be defined by Cauchy's integral formula and can be approximated by the linear combination of inverses of shifted matrices using a quadrature formula. In this paper, we show a concrete construction of a framework to implement the linear combination of the inverses on quantum computers and propose a quantum algorithm for matrix functions based on the framework. Compared with the previous study [S. Takahira, A. Ohashi, T. Sogabe, and T.S. Usuda, Quant. Inf. Comput., 20, 1&2, 14--36, (Feb. 2020)] that proposed a quantum algorithm to compute a quantum state for the matrix function based on the circular contour centered at the origin, the quantum algorithm in the present paper can be applied to a more general contour. Moreover, the algorithm is described by the block-encoding framework. Similarly to the previous study, the algorithm can be applied even if the input matrix is not a Hermitian or normal matrix.