Quantum Algorithms for Solving Ordinary Differential Equations via Classical Integration Methods

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