Quantum radial basis function method for the Poisson equation

TL;DR

This paper explores how quantum algorithms can accelerate the radial basis function method for solving high-dimensional Poisson equations, demonstrating a polynomial speedup over classical methods.

Contribution

It introduces a quantum algorithm for the RBF method applied to the Poisson problem and compares its performance with classical algorithms.

Findings

Quantum algorithm achieves polynomial speedup

Potential for faster high-dimensional PDE solutions

Comparison with conjugate gradient method

Abstract

The radial basis function (RBF) method is used for the numerical solution of the Poisson problem in high dimension. The approximate solution can be found by solving a large system of linear equations. Here we investigate the extent to which the RBF method can be accelerated using an efficient quantum algorithm for linear equations. We compare the theoretical performance of our quantum algorithm with that of a standard classical algorithm, the conjugate gradient method. We find that the quantum algorithm can achieve a polynomial speedup.