Variational quantum algorithm for solving Helmholtz problems with high order finite elements

TL;DR

This paper explores the use of variational quantum algorithms to efficiently solve high-order finite element discretizations of Helmholtz problems, demonstrating potential quantum advantages for large-scale linear systems.

Contribution

It introduces a quantum algorithm with block encoding for Helmholtz operators from high-order finite elements, showing how to implement it for regular meshes.

Findings

Quantum circuit depth scales as $ ext{O}(p^3 ext{poly}\log(Np))$

Algorithm applied successfully to 1D Helmholtz problems with various boundary conditions

Potential for quantum speedup in solving large Helmholtz discretizations

Abstract

Discretizing Helmholtz problems via finite elements yields linear systems whose efficient solution remains a major challenge for classical computation. In this paper, we investigate how variational quantum algorithms could address this challenge. We first show that, for regular meshes, a block encoding of the operators $A$ and $A^\dagger A$ arising from the high-order finite element discretisation of Helmholtz problems can be designed, resulting in a quantum circuit of depth $\mathcal{O}(p^3\mathrm{poly}\log(Np))$ with $N$ the number of elements and $p$ the order of the finite elements. Then we apply our algorithm to a one-dimensional Helmholtz problem with Dirichlet and Neumann boundary conditions for various wavenumbers.