A Performance Study of Variational Quantum Algorithms for Solving the Poisson Equation on a Quantum Computer

TL;DR

This study evaluates the effectiveness of variational quantum algorithms in solving the Poisson equation on current quantum hardware, revealing significant performance limitations and questioning their near-term practical utility.

Contribution

The paper provides an empirical assessment of VQAs for PDEs on real quantum devices, highlighting their current shortcomings and limitations for practical problem sizes.

Findings

Performance on real quantum devices is poor despite promising noiseless simulations.

Direct amplitude encoding of PDE solutions is not effective on NISQ devices.

VQAs currently face significant challenges for large or nonlinear PDEs.

Abstract

Recent advances in quantum computing and their increased availability has led to a growing interest in possible applications. Among those is the solution of partial differential equations (PDEs) for, e.g., material or flow simulation. Currently, the most promising route to useful deployment of quantum processors in the short to near term are so-called hybrid variational quantum algorithms (VQAs). Thus, variational methods for PDEs have been proposed as a candidate for quantum advantage in the noisy intermediate scale quantum (NISQ) era. In this work, we conduct an extensive study of utilizing VQAs on real quantum devices to solve the simplest prototype of a PDE -- the Poisson equation. Although results on noiseless simulators for small problem sizes may seem deceivingly promising, the performance on quantum computers is very poor. We argue that direct resolution of PDEs via an amplitude encoding of the solution is not a good use case within reach of today's quantum devices -- especially when considering large system sizes and more complicated non-linear PDEs that are required in order to be competitive with classical high-end solvers.