Variational Quantum algorithm for Poisson equation

TL;DR

This paper introduces a variational quantum algorithm tailored for NISQ devices to efficiently solve the Poisson equation by leveraging a tensor product decomposition, reducing quantum resource requirements significantly.

Contribution

The paper presents a novel VQA that transforms the Poisson equation into a linear system with a special structure, enabling resource-efficient quantum computation on NISQ hardware.

Findings

Algorithm effectively solves the Poisson equation numerically.

Reduces quantum measurements to O(log n) due to tensor product structure.

Demonstrates feasibility on NISQ devices with numerical experiments.

Abstract

The Poisson equation has wide applications in many areas of science and engineering. Although there are some quantum algorithms that can efficiently solve the Poisson equation, they generally require a fault-tolerant quantum computer which is beyond the current technology. In this paper, we propose a Variational Quantum Algorithm (VQA) to solve the Poisson equation, which can be executed on Noise Intermediate-Scale Quantum (NISQ) devices. In detail, we first adopt the finite difference method to transform the Poisson equation into a linear system. Then, according to the special structure of the linear system, we find an explicit tensor product decomposition, with only $2\log n+1$ items, of its coefficient matrix under a specific set of simple operators, where $n$ is the dimension of the coefficient matrix. This implies that the proposed VQA only needs $O(\log n)$ measurements, which dramatically reduce quantum resources. Additionally, we perform quantum Bell measurements to efficiently evaluate the expectation values of simple operators. Numerical experiments demonstrate that our algorithm can effectively solve the Poisson equation.