2020/12/31 by Benjamin Zanger, Christian B. Mendl, Martin Schulz +1
Computer Science · Mathematics · Physics and Astronomy · #Algorithm #Applied mathematics #Computer science #Differential equation #Mathematical analysis #Mathematics #Numerical Methods and Algorithms #Ordinary differential equation #Physics #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum algorithm #Quantum annealing #Quantum computer #Quantum mechanics #quant-ph
paper · pdf · doi:10.22331/q-2021-07-13-502
published as Quantum 5, 502 (2021)
arxiv created 2021/07/12 · openalex publication_date 2021/07/13 · arxiv updated 2021/07/14 · openalex created_date 2021/07/19 · openalex updated_date 2026/08/05
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<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi/><mml:mrow class="MJX-TeXAtom-ORD"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">t</mml:mi><mml:mi mathvariant="normal">h</mml:mi></mml:mrow></mml:mrow></mml:msup></mml:math> 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.