Variational Quantum Simulation of Partial Differential Equations: Applications in Colloidal Transport

TL;DR

This paper explores variational quantum algorithms for solving partial differential equations, demonstrating their effectiveness in simulating colloidal transport phenomena with potential applications on near-term quantum hardware.

Contribution

It introduces a graphical mapping technique for encoding impulse functions and analyzes the impact of ansatz design and over-parameterization on solution fidelity.

Findings

Real-amplitude ansaetze with full entangling layers yield higher fidelity solutions.

Graphical mapping efficiently encodes impulse functions with minimal gate operations.

Over-parameterization and higher-order time-stepping improve boundary condition satisfaction and reduce errors.

Abstract

We assess the use of variational quantum imaginary time evolution for solving partial differential equations. Our results demonstrate that real-amplitude ansaetze with full circular entangling layers lead to higher-fidelity solutions compared to those with partial or linear entangling layers. To efficiently encode impulse functions, we propose a graphical mapping technique for quantum states that often requires only a single bit-flip of a parametric gate. As a proof of concept, we simulate colloidal deposition on a planar wall by solving the Smoluchowski equation including the Derjaguin-Landau-Verwey-Overbeek (DLVO) potential energy. We find that over-parameterization is necessary to satisfy certain boundary conditions and that higher-order time-stepping can effectively reduce norm errors. Together, our work highlights the potential of variational quantum simulation for solving partial differential equations using near-term quantum devices.