HAM-Schr\"{o}dingerisation: a generic framework of quantum simulation for any nonlinear PDEs

TL;DR

This paper extends the Schr"{o}dingerisation quantum simulation method to nonlinear PDEs by integrating it with the homotopy analysis method, enabling quantum solutions for complex nonlinear problems.

Contribution

It introduces a novel framework combining Schr"{o}dingerisation with HAM to simulate any nonlinear PDEs on quantum computers, expanding the scope of quantum simulation techniques.

Findings

Framework successfully extends quantum simulation to nonlinear PDEs.

Guarantees convergence of the series solution for nonlinear problems.

Potential to simulate turbulent flows efficiently using quantum computing.

Abstract

Recently, Jin et al. proposed a quantum simulation technique for ANY linear partial differential equations (PDEs), called Schr\"{o}dingerisation [1,2,3]. In this paper, the Schr\"{o}dingerisation technique for quantum simulation is expanded to ANY nonlinear PDEs by combining it with the homotopy analysis method (HAM). The HAM can transfer a nonlinear PDE into a series of linear PDEs with guaranteeing convergence of the series. In this way, ANY nonlinear PDEs can be solved by quantum simulation using a quantum computer. For simplicity, we call the procedure ``HAM-Schr\"{o}dingerisation quantum algorithm''. Quantum computing is a groundbreaking technique. Hopefully, the ``HAM-Schr\"{o}dingerisation quantum algorithm'' can open a door to highly efficient simulation of complicated turbulent flows by means of quantum computing in future.