Quantum-Enhanced Spectral Solution of the Poisson Equation

TL;DR

This paper introduces a hybrid quantum-classical method utilizing the Quantum Fourier Transform to efficiently solve the Poisson equation, significantly reducing computational complexity and improving accuracy compared to classical approaches.

Contribution

It presents a novel quantum-enhanced numerical method for solving PDEs that directly estimates solution coefficients, bypassing traditional integration-heavy calculations.

Findings

Significant reduction in time complexity.

Improved solution accuracy.

Validated effectiveness through numerical experiments.

Abstract

We present a hybrid numerical-quantum method for solving the Poisson equation under homogeneous Dirichlet boundary conditions, leveraging the Quantum Fourier Transform (QFT) to enhance computational efficiency and reduce time and space complexity. This approach bypasses the integration-heavy calculations of classical methods, which have to deal with high computational costs for large number of points. The proposed method estimates the coefficients of the series expansion of the solution directly within the quantum framework. Numerical experiments validate its effectiveness and reveal significant improvements in terms of time and space complexity and solution accuracy, demonstrating the capability of quantum-assisted techniques to contribute in solving partial differential equations (PDEs). Despite the inherent challenges of quantum implementation, the present work serves as a starting point for future researches aimed at refining and expanding quantum numerical methods.