Quantum Algorithms for Solving Ordinary Differential Equations via Classical Integration Methods

TL;DR

This paper investigates quantum algorithms for solving ordinary differential equations, comparing digital quantum and quantum annealing approaches, and demonstrates their potential through simulations and experiments.

Contribution

It introduces two quantum methods for differential equations, including digital quantum circuits and quantum annealing, highlighting their advantages and potential applications.

Findings

Quantum annealing shows promise for high-order implicit methods.

Digital quantum circuits can simulate differential equations accurately.

Quantum approaches could enhance inverse problem solving.

Abstract

Identifying computational tasks suitable for (future) quantum computers is an active field of research. Here we explore utilizing quantum computers for the purpose of solving differential equations. We consider two approaches: (i) basis encoding and fixed-point arithmetic on a digital quantum computer, and (ii) representing and solving high-order Runge-Kutta methods as optimization problems on quantum annealers. As realizations applied to two-dimensional linear ordinary differential equations, we devise and simulate corresponding digital quantum circuits, and implement and run a 6$^{\mathrm{th}}$ order Gauss-Legendre collocation method on a D-Wave 2000Q system, showing good agreement with the reference solution. We find that the quantum annealing approach exhibits the largest potential for high-order implicit integration methods. As promising future scenario, the digital arithmetic method could be employed as an "oracle" within quantum search algorithms for inverse problems.