Quantum algorithms for powering stable Hermitian matrices

TL;DR

This paper explores quantum algorithms to accelerate matrix powering of sparse stable Hermitian matrices, demonstrating potential speedups and establishing limitations for non-Hermitian cases.

Contribution

It introduces two quantum algorithms for faster matrix powering of Hermitian matrices and provides no-go theorems for non-Hermitian matrices.

Findings

Quantum algorithms achieve speedup for Hermitian matrices.

No-go theorems limit quantum speedups for non-Hermitian matrices.

Applications include solving differential equations and Markov chain analysis.

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) 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.