Solving Fractional Differential Equations on a Quantum Computer: A Variational Approach

TL;DR

This paper presents a novel variational hybrid quantum-classical algorithm for efficiently solving Caputo time-fractional PDEs, demonstrating lower resource costs and robustness to noise in engineering applications.

Contribution

It introduces a new quantum algorithm for fractional PDEs that is more efficient in time and memory than classical methods, with practical validation on noisy hardware.

Findings

Solution fidelity is insensitive to fractional index.

Gradient evaluation cost scales efficiently with time steps.

Algorithm performs well on noisy quantum hardware.

Abstract

We introduce an efficient variational hybrid quantum-classical algorithm designed for solving Caputo time-fractional partial differential equations. Our method employs an iterable cost function incorporating a linear combination of overlap history states. The proposed algorithm is not only efficient in time complexity, but has lower memory costs compared to classical methods. Our results indicate that solution fidelity is insensitive to the fractional index and that gradient evaluation cost scales economically with the number of time steps. As a proof of concept, we apply our algorithm to solve a range of fractional partial differential equations commonly encountered in engineering applications, such as the sub-diffusion equation, the non-linear Burgers' equation and a coupled diffusive epidemic model. We assess quantum hardware performance under realistic noise conditions, further validating the practical utility of our algorithm.