Quantum algorithms based on the block-encoding framework for matrix functions by contour integrals (pp965-979)
          Souichi Takahira, Asuka Ohashi, Tomohiro Sogabe,
          and Tsuyoshi S. Usuda
          doi: https://doi.org/10.26421/QIC22.11-12-4
Abstracts: 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 propose a quantum algorithm for matrix functions based on a procedure to implement the linear combination of the inverses on quantum computers. Compared with the previous study [S. Takahira, A. Ohashi, T. Sogabe, and T.S. Usuda, Quant. Inf. Comput., \textbf{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. This is an advantage compared with quantum singular value transformation.
Key words: Quantum algorithm, Block-encoding, Matrix functions, Cauchy's integral formula