Quantum Algorithms for Solving Ordinary Differential Equations via Classical Integration Methods

Benjamin Zanger; Christian B. Mendl; Martin Schulz; Martin Schreiber

doi:10.22331/q-2021-07-13-502

Quantum Algorithms for Solving Ordinary Differential Equations via Classical Integration Methods

Benjamin Zanger¹, Christian B. Mendl^1,2, Martin Schulz^1,3, 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

Published:	2021-07-13, volume 5, page 502
Eprint:	arXiv:2012.09469v2
Doi:	https://doi.org/10.22331/q-2021-07-13-502
Citation:	Quantum 5, 502 (2021).

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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.

Featured image: Explicit Euler circuit for a 2D ODE.

► BibTeX data

@article{Zanger2021quantumalgorithms,
  doi = {10.22331/q-2021-07-13-502},
  url = {https://doi.org/10.22331/q-2021-07-13-502},
  title = {Quantum {A}lgorithms for {S}olving {O}rdinary {D}ifferential {E}quations via {C}lassical {I}ntegration {M}ethods},
  author = {Zanger, Benjamin and Mendl, Christian B. and Schulz, Martin and Schreiber, Martin},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {502},
  month = jul,
  year = {2021}
}

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