Quantum algorithms and the finite element method

TL;DR

This paper evaluates the potential of quantum algorithms to accelerate the finite element method for solving boundary value problems, finding a polynomial speedup that depends on the problem's dimension.

Contribution

It provides a detailed comparison of quantum and classical algorithms for solving linear systems in finite element methods, including runtime analysis and limitations.

Findings

Quantum algorithms offer polynomial speedup over classical methods.

Speedup increases with the dimension of the PDE.

No super-polynomial speedup is possible under certain conditions.

Abstract

The finite element method is used to approximately solve boundary value problems for differential equations. The method discretises the parameter space and finds an approximate solution by solving a large system of linear equations. Here we investigate the extent to which the finite element method can be accelerated using an efficient quantum algorithm for solving linear equations. We consider the representative general question of approximately computing a linear functional of the solution to a boundary value problem, and compare the quantum algorithm's theoretical performance with that of a standard classical algorithm -- the conjugate gradient method. Prior work had claimed that the quantum algorithm could be exponentially faster, but did not determine the overall classical and quantum runtimes required to achieve a predetermined solution accuracy. Taking this into account, we find that the quantum algorithm can achieve a polynomial speedup, the extent of which grows with the dimension of the partial differential equation. In addition, we give evidence that no improvement of the quantum algorithm could lead to a super-polynomial speedup when the dimension is fixed and the solution satisfies certain smoothness properties.