Quantum algorithms for powering stable Hermitian matrices

Abstract

Matrix powering is a fundamental computational primitive in linear algebra. It has widespread applications in scientific computing and engineering and underlies the solution of time-homogeneous linear ordinary differential equations, simulation of discrete-time Markov chains, or discovering the spectral properties of matrices with iterative methods. In this paper, we investigate the possibility of speeding up matrix powering of sparse stable Hermitian matrices on a quantum computer. We present two quantum algorithms that can achieve speedup over the classical matrix powering algorithms: (i) a fast-forwarding algorithm that builds on construction of Apers and Sarlette [Quantum Inf. Comput. 19, 181 (2019)] and (ii) an algorithm based on Hamiltonian simulation. Furthermore, by mapping the N-bit parity determination problem to a matrix powering problem, we provide no-go theorems that limit the quantum speedups achievable in powering non-Hermitian matrices.