Quantum Realization of the Finite Element Method

TL;DR

This paper develops a quantum algorithm for efficiently solving second-order elliptic PDEs discretized by finite elements, demonstrating quantum advantage in two dimensions and feasibility on current quantum hardware.

Contribution

It introduces a quantum algorithm with optimal complexity for finite element PDE solutions, utilizing a BPX preconditioner, and shows practical implementation on existing quantum devices.

Findings

Quantum algorithm achieves order $ ext{tol}^{-1}$ complexity.

Provides quantum advantage in 2D problems.

Demonstrates feasibility on current quantum hardware.

Abstract

This paper presents a quantum algorithm for the solution of prototypical second-order linear elliptic partial differential equations discretized by $d$-linear finite elements on Cartesian grids of a bounded $d$-dimensional domain. An essential step in the construction is a BPX preconditioner, which transforms the linear system into a sufficiently well-conditioned one, making it amenable to quantum computation. We provide a constructive proof demonstrating that, for any fixed dimension, our quantum algorithm can compute suitable functionals of the solution to a given tolerance $\mathtt{tol}$ with an optimal complexity of order $\mathtt{tol}^{-1}$ up to logarithmic terms, significantly improving over existing approaches. Notably, this approach does not rely on regularity of the solution and achieves quantum advantage over classical solvers in two dimensions, whereas prior quantum methods required at least four dimensions for asymptotic benefits. We further detail the design and implementation of a quantum circuit capable of executing our algorithm, present simulator results, and report numerical experiments on current quantum hardware, confirming the feasibility of preconditioned finite element methods for near-term quantum computing.