Quantum Circuits for partial differential equations via Schr\"odingerisation

TL;DR

This paper develops quantum circuits based on Schr"odingerisation techniques to efficiently solve general PDEs, demonstrating quantum advantages in high-dimensional problems through detailed implementation and complexity analysis.

Contribution

It provides the first explicit quantum circuit implementation for general PDEs using Schr"odingerisation, expanding quantum simulation capabilities beyond Schr"odinger-type equations.

Findings

Effective quantum algorithms for heat and advection equations

Quantum advantage demonstrated in high-dimensional PDEs

Complexity analysis shows potential speedup over classical methods

Abstract

Quantum computing has emerged as a promising avenue for achieving significant speedup, particularly in large-scale PDE simulations, compared to classical computing. One of the main quantum approaches involves utilizing Hamiltonian simulation, which is directly applicable only to Schr\"odinger-type equations. To address this limitation, Schr\"odingerisation techniques have been developed, employing the warped transformation to convert general linear PDEs into Schr\"odinger-type equations. However, despite the development of Schr\"odingerisation techniques, the explicit implementation of the corresponding quantum circuit for solving general PDEs remains to be designed. In this paper, we present detailed implementation of a quantum algorithm for general PDEs using Schr\"odingerisation techniques. We provide examples of the heat equation, and the advection equation approximated by the upwind scheme, to demonstrate the effectiveness of our approach. Complexity analysis is also carried out to demonstrate the quantum advantages of these algorithms in high dimensions over their classical counterparts.