H-DES: a Quantum-Classical Hybrid Differential Equation Solver

TL;DR

This paper presents H-DES, a hybrid quantum-classical algorithm that uses variational quantum techniques to solve differential equations by transforming them into optimization problems, demonstrated on emulators and hardware.

Contribution

The paper introduces a novel hybrid quantum-classical algorithm for differential equations using spectral decomposition and variational circuits, with detailed pseudo-code and complexity analysis.

Findings

Applicable to diverse differential equations

Demonstrated on emulators and real hardware

Shows potential advantages of quantum-classical hybrid methods

Abstract

In this article, we introduce an original hybrid quantum-classical algorithm based on a variational quantum algorithm for solving systems of differential equations. The algorithm relies on a spectral decomposition of the trial functions that are encoded directly in the quantum states generated by different parametrized circuits, and transforms the task of solving the differential equations into an optimization problem. We first describe the principle of the algorithm from a theoretical point of view. We provide a detailed pseudo-code of the algorithm, on which we elaborate preliminary elements for a complexity analysis to highlight some of its scaling properties. We apply our algorithm to a set of examples, running on emulators and real hardware showcasing its applicability across diverse sets of differential equations. We discuss the advantages of our method and potential avenues for further exploration and refinement.