Advanced Quantum Poisson Solver in the NISQ era

TL;DR

This paper introduces an advanced quantum algorithm for solving the Poisson equation with high accuracy and scalable problem size, suitable for NISQ devices, by improving eigenvalue handling and circuit design.

Contribution

It presents a novel quantum Poisson solver that enhances accuracy and scalability through eigenvalue amplification and dynamic problem size control in NISQ hardware.

Findings

Improved solution accuracy via eigenvalue amplification.

Demonstrated scalability with dynamic problem size.

Experimental results show hardware errors dominate performance.

Abstract

The Poisson equation has many applications across the broad areas of science and engineering. Most quantum algorithms for the Poisson solver presented so far, either suffer from lack of accuracy and/or are limited to very small sizes of the problem, and thus have no practical usage. Here we present an advanced quantum algorithm for solving the Poisson equation with high accuracy and dynamically tunable problem size. After converting the Poisson equation to the linear systems through the finite difference method, we adopt the Harrow-Hassidim-Lloyd (HHL) algorithm as the basic framework. Particularly, in this work we present an advanced circuit that ensures the accuracy of the solution by implementing non-truncated eigenvalues through eigenvalue amplification as well as by increasing the accuracy of the controlled rotation angular coefficients, which are the critical factors in the HHL algorithm. We show that our algorithm not only increases the accuracy of the solutions, but also composes more practical and scalable circuits by dynamically controlling problem size in the NISQ devices. We present both simulated and experimental solutions, and conclude that overall results on the quantum hardware are dominated by the error in the CNOT gates.