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Applicability of solving the one- and two-dimensional Poisson equations with the quantum Harrow-Hassidim-Lloyd algorithm

Luca Ghafourpour1,2,* and Sylvain Laizet2

  • *Contact author: lghafourpour@ethz.ch
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Phys. Rev. Applied 24, 024032 – Published 13 August, 2025

DOI: https://doi.org/10.1103/ymjf-6n8j

Abstract

This paper assesses the numerical accuracy of the Harrow-Hassidim-Lloyd (HHL) algorithm in solving a finite-difference approximation of the Poisson equation, with the Thomas algorithm used as a comparative benchmark. In the one-dimensional setting with homogeneous boundary conditions, the HHL algorithm exhibits numerical accuracy comparable to the Thomas algorithm; however, the HHL algorithm exhibits increased sensitivity to variations in input parameters for simulations with nonhomogeneous boundary conditions. While the current implementation of the HHL algorithm is limited to solving the one-dimensional Poisson equation, we propose a hybrid framework to extend the applicability of the HHL algorithm to solving the two-dimensional Poisson equation. Our results show that although both algorithms perform similarly for small systems of 8×8 mesh nodes, the HHL algorithm shows increased input-parameter sensitivity for systems of 16×16 mesh nodes, indicating that further development is needed to scale the hybrid approach effectively.

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Phys. Rev. Applied 24, 024032– Published 13 August, 2025
  • Received 8 October 2024
  • Revised 22 May 2025
  • Accepted 1 July 2025
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