From Ad-Hoc to Systematic: A Strategy for Imposing General Boundary Conditions in Discretized PDEs in variational quantum algorithm

TL;DR

This paper introduces a quantum algorithm that systematically imposes arbitrary boundary conditions in PDE solutions on NISQ devices, enhancing scalability and practical applicability for real-world engineering problems.

Contribution

It presents a novel variational quantum algorithm capable of handling general boundary conditions in PDEs, overcoming limitations of previous methods constrained by simple boundary assumptions.

Findings

Successfully applied to Euler-Bernoulli beam PDE with four boundary conditions

Demonstrates expectation evaluation independent of problem size

Shows potential for scalable quantum PDE solutions in engineering

Abstract

We proposed a general quantum-computing-based algorithm that harnesses the exponential power of noisy intermediate-scale quantum (NISQ) devices in solving partial differential equations (PDE). This variational quantum eigensolver (VQE)-inspired approach transcends previous idealized model demonstrations constrained by strict and simplistic boundary conditions. It enables the imposition of arbitrary boundary conditions, significantly expanding its potential and adaptability for real-world applications, achieving this "from ad-hoc to systematic" concept. We have implemented this method using the fourth-order PDE (the Euler-Bernoulli beam) as example and showcased its effectiveness with four different boundary conditions. This framework enables expectation evaluations independent of problem size, harnessing the exponentially growing state space inherent in quantum computing, resulting in exceptional scalability. This method paves the way for applying quantum computing to practical engineering applications.