Solving Helmholtz problems with finite elements on a quantum annealer

TL;DR

This paper explores the potential of quantum annealers to solve Helmholtz problems formulated via finite element methods, analyzing their effectiveness and limitations compared to classical approaches.

Contribution

It introduces a novel approach to map Helmholtz finite element problems into QUBOs for quantum annealing and evaluates key hardware parameters affecting performance.

Findings

Large system condition numbers require finer discretization for convergence.

AQAE's tolerance to hardware errors depends on the specific gEVP.

Lower bounds on annealing time suggest limited quantum advantage.

Abstract

Solving Helmholtz problems using finite elements leads to the resolution of a linear system which is challenging to solve for classical computers. In this paper, we investigate how quantum annealers could address this challenge. We first express the linear system arising from the Helmholtz problem as a generalized eigenvalue problem (gEVP). The obtained gEVP is mapped into quadratic unconstrained binary optimization problems (QUBOs) which we solve using an adaptive quantum annealing eigensolver (AQAE) and its classical equivalent. We identify two key parameters in the success of AQAE for solving Helmholtz problems: the system condition number and the integrated control errors (ICE) in the quantum hardware. Our results show that a large system condition number implies a finer discretization grid for AQAE to converge, leading to a variable overhead, and that AQAE is either tolerant or not with respect to ICE depending on the gEVP. Finally, we establish lower bounds on the annealing time, narrowing the possibility of a quantum advantage for solving Helmholtz problems.