Variational quantum algorithm based on the minimum potential energy for solving the Poisson equation

TL;DR

This paper introduces a variational quantum algorithm that efficiently solves the Poisson equation by minimizing potential energy, demonstrating faster computation than classical methods and previous quantum approaches, suitable for noisy quantum devices.

Contribution

The paper presents a novel variational quantum algorithm that minimizes the potential energy as a Hamiltonian for solving PDEs, with fewer measurements and improved speed.

Findings

Faster computation speed compared to classical methods.

Requires fewer quantum measurements due to Hamiltonian decomposition.

Applicable to noisy intermediate-scale quantum devices.

Abstract

Computer-aided engineering techniques are indispensable in modern engineering developments. In particular, partial differential equations are commonly used to simulate the dynamics of physical phenomena, but very large systems are often intractable within a reasonable computation time, even when using supercomputers. To overcome the inherent limit of classical computing, we present a variational quantum algorithm for solving the Poisson equation that can be implemented in noisy intermediate-scale quantum devices. The proposed method defines the total potential energy of the Poisson equation as a Hamiltonian, which is decomposed into a linear combination of Pauli operators and simple observables. The expectation value of the Hamiltonian is then minimized with respect to a parameterized quantum state. Because the number of decomposed terms is independent of the size of the problem, this method requires relatively few quantum measurements. Numerical experiments demonstrate the faster computing speed of this method compared with classical computing methods and a previous variational quantum approach. We believe that our approach brings quantum computer-aided techniques closer to future applications in engineering developments. Code is available at https://github.com/ToyotaCRDL/VQAPoisson.