  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)
an adaption of quantum-walk based fast forwarding algorithm (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.
