Quantum Algorithms for Solving Ordinary Differential Equations via Classical Integration Methods

Benjamin Zanger1, Christian B. Mendl1,2, Martin Schulz1,3, and Martin Schreiber1

1Technical University of Munich, Department of Informatics, Boltzmannstraße 3, 85748 Garching, Germany
2TUM Institute for Advanced Study, Lichtenbergstraße 2a, 85748 Garching, Germany
3Leibniz Supercomputing Centre, Boltzmannstraße 1, 85748 Garching, Germany

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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.

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