A variational quantum algorithm for tackling multi-dimensional Poisson equations with inhomogeneous boundary conditions

TL;DR

This paper introduces a variational quantum algorithm designed to solve multi-dimensional Poisson equations with complex boundary conditions, demonstrating its application in semiconductor electric field calculations and highlighting its potential in computational science.

Contribution

The paper presents the first application of a variational quantum algorithm to multi-dimensional Poisson equations with mixed boundary conditions, including practical circuit design and real-world problem solving.

Findings

Successfully applied to 2D electric field distributions in semiconductors

Demonstrated noise-robust and cost-efficient quantum circuit design

Extended the scope of quantum algorithms to complex boundary value problems

Abstract

We design a variational quantum algorithm to solve multi-dimensional Poisson equations with mixed boundary conditions that are typically required in various fields of computational science. Employing an objective function that is formulated with the concept of the minimal potential energy, we not only present in-depth discussion on the cost-efficient & noise-robust design of quantum circuits that are essential for evaluation of the objective function, but, more remarkably, employ the proposed algorithm to calculate bias-dependent spatial distributions of electric fields in semiconductor systems that are described with a two-dimensional domain and up to 10-qubit circuits. Extending the application scope to multi-dimensional problems with mixed boundary conditions for the first time, fairly solid computational results of this work clearly demonstrate the potential of variational quantum algorithms to tackle Poisson equations derived from physically meaningful problems.