Quantum algorithm and circuit design solving the Poisson equation

TL;DR

This paper introduces a quantum algorithm and circuit design that efficiently approximates solutions to the Poisson equation in multiple dimensions, leveraging quantum superposition and achieving scalability with respect to dimension and precision.

Contribution

The work presents a scalable quantum algorithm and circuit design for solving the Poisson equation, with performance guarantees and modules applicable to other problems.

Findings

Quantum algorithm encodes solutions in quantum states.

Circuit complexity is nearly linear in dimension and polylogarithmic in precision.

Performance guarantees demonstrate scalability and efficiency.

Abstract

The Poisson equation occurs in many areas of science and engineering. Here we focus on its numerical solution for an equation in d dimensions. In particular we present a quantum algorithm and a scalable quantum circuit design which approximates the solution of the Poisson equation on a grid with error \varepsilon. We assume we are given a supersposition of function evaluations of the right hand side of the Poisson equation. The algorithm produces a quantum state encoding the solution. The number of quantum operations and the number of qubits used by the circuit is almost linear in d and polylog in \varepsilon^{-1}. We present quantum circuit modules together with performance guarantees which can be also used for other problems.