Exact results on matrix inversion polynomials

The quantum singular value transformation (QSVT) allows for the implementation of certain polynomial functions of matrices on quantum computers. Of particular importance are matrix inversion polynomials: uniform polynomial approximants of the function 1/x on appropriate domains, whose application via the QSVT achieves numerical inversion of matrices. We present a number of exact and explicit results for matrix inversion polynomials, including representations in terms of classical orthogonal polynomials, uniform error bounds in both non-asymptotic and asymptotic regimes. Our results are supplemented by numerical studies of matrix inversion polynomials obtained via the Remez algorithm