Quantum Fast Poisson Solver: the algorithm and modular circuit design

TL;DR

This paper introduces a quantum algorithm and modular circuit design for efficiently solving the Poisson equation, leveraging advanced quantum functions and demonstrating practical implementation on a supercomputer-based quantum simulator.

Contribution

It presents a novel quantum Fast Poisson Solver with lower complexity and modular circuit design, based on the HHL algorithm and advanced function evaluation methods.

Findings

Circuit complexity is O(dlog2(ε^-1)) in qubits.

Circuit complexity is O(dlog3(ε^-1)) in operations.

Demonstrated on a quantum virtual system on a supercomputer.

Abstract

The Poisson equation has applications across many areas of physics and engineering, such as the dynamic process simulation of ocean current. Here we present a quantum Fast Poisson Solver, including the algorithm and the complete and modular circuit design. The algorithm takes the HHL algorithm as the template. The controlled rotation is performed based on the arc cotangent function which is evaluated by the Plouffe's binary expansion method. And the same method is used to compute the cosine function for the eigenvalue approximation in phase estimation. Quantum algorithms for solving square root and reciprocal functions are developed based on the non-restoring digit-recurrence method. These advances make the algorithm's complexity lower and the circuit-design more modular. The number of the qubits and operations used by the circuit are O(dlog2({\epsilon}-1)) and O(dlog3({\epsilon}-1)), respectively. We demonstrate our circuits on a quantum virtual computing system installed on the Sunway TaihuLight supercomputer. This is an important step toward practical applications of quantum Fast Poisson Solver in the near-term hybrid classical/quantum devices.