High-order quantum algorithm for solving linear differential equations

TL;DR

This paper introduces a high-order quantum algorithm for efficiently solving general inhomogeneous sparse linear differential equations, extending quantum simulation capabilities to broader classical systems.

Contribution

It develops a high-order quantum algorithm that improves efficiency in solving linear differential equations beyond previous methods.

Findings

Achieves near quadratic scaling in evolution time Δt

Extends quantum simulation to inhomogeneous sparse systems

Enables extraction of global features from quantum solutions

Abstract

Linear differential equations are ubiquitous in science and engineering. Quantum computers can simulate quantum systems, which are described by a restricted type of linear differential equations. Here we extend quantum simulation algorithms to general inhomogeneous sparse linear differential equations, which describe many classical physical systems. We examine the use of high-order methods to improve the efficiency. These provide scaling close to $\Delta t^2$ in the evolution time $\Delta t$. As with other algorithms of this type, the solution is encoded in amplitudes of the quantum state, and it is possible to extract global features of the solution.