Quantum Algorithms for Solving Ordinary Differential Equations

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. Applied to two-dimensional toy model differential equations, we devise and simulate quantum circuits for realizing (i), and implement and run a 6^th order Gauss-Legendre collocation method on a D-Wave 2000Q system, showing good agreement with the reference solution. As promising future scenario, the digital arithmetic method could be employed as ``oracle'' within quantum search algorithms for inverse problems. The quantum annealing approach exhibits the largest potential for high-order implicit integration methods.