FEqa: Finite Element Computations on Quantum Annealers

TL;DR

This paper introduces FEqa, a hybrid quantum-classical method for solving finite element problems using quantum annealers, demonstrating advantages in computational time and solution accuracy.

Contribution

FEqa is a novel hybrid approach that formulates finite element problems on classical computers and minimizes residuals with quantum annealers, enabling scalable and precise solutions.

Findings

Quantum annealer reduces computational time compared to simulated annealing.

FEqa enforces boundary conditions a priori, improving solution validity.

The method is scalable and effective on D-Wave quantum hardware.

Abstract

The solution of physical problems discretized using the finite element methods using quantum computers remains relatively unexplored. Here, we present a unified formulation (FEqa) to solve such problems using quantum annealers. FEqa is a hybrid technique in which the finite element problem is formulated on a classical computer, and the residual is minimized using a quantum annealer. The advantages of FEqa include utilizing a single qubit per degree of freedom, enforcing Dirichlet boundary conditions a priori, reaching arbitrary solution precision, and eliminating the possibility of the annealer generating invalid results. FEqa is scalable on the classical portion of the algorithm due to its Single Program Multiple Data (SPMD) nature and does not rely on ground state solutions from the annealer. The exponentially large number of collocation points used in quantum annealing are investigated for their cosine measures, and new iterative techniques are developed to exploit their properties. The quantum annealer has clear advantages in computational time over simulated annealing, for the example problems presented in this paper solved on the D-Wave machine. The presented work provides a pathway to solving physical problems using quantum annealers.