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

• URL: http://arxiv.org/abs/2512.22665v1
• Date: Sat, 27 Dec 2025 17:34:43 GMT
• Title: Variational quantum algorithm for solving Helmholtz problems with high order finite elements
• Authors: Arnaud Rémi, François Damanet, Christophe Geuzaine,
• Abstract summary: Discretizing Helmholtz problems via finite elements yields linear systems whose efficient solution remains a major challenge for classical computation.<n>We first show that, for regular meshes, a block encoding of the operators $A$ and $Adagger A$ arising from the high-order finite element discretisation of Helmholtz problems can be designed.<n>We then apply our algorithm to a one-dimensional Helmholtz problem with Dirichlet and Neumann boundary conditions for various wavenumbers.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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