HAM-Schrödingerisation: a generic framework of quantum simulation for any nonlinear PDEs

TL;DR

This work addresses solving nonlinear PDEs on quantum computers by extending the Schrödingerisation approach from linear PDEs to nonlinear problems. It combines Schrödingerisation with the Homotopy Analysis Method (HAM) to convert a nonlinear PDE of the form $\partial_t \psi = \mathcal{N}[\psi] + g$ into a convergent sequence of linear subproblems via an embedding parameter $q \in [0,1]$ and a convergence-control parameter $c_0$, yielding a series $\Psi(\mathbf{r},t,q) = \psi_0(\mathbf{r},t) + \sum_{m=1}^{\infty} \psi_m(\mathbf{r},t) q^m$. Each $\psi_m$ is obtained from a linear deformation equation and can be solved with Schrödingerisation; the auxiliary operator $\mathcal{L}^\\star$ and the MDDiM (method of directly defining inverse mapping) provide convergence guarantees and an efficient quantum-computation pipeline. The authors anticipate quantum-speedup for nonlinear PDEs and turbulence-related simulations, with future work needed to validate performance and address practical noise and runtime considerations.

Abstract

Recently, Jin et al. proposed a quantum simulation technique for ANY linear partial differential equations (PDEs), called Schrödingerisation [1,2,3]. In this paper, the Schrödingerisation 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-Schrödingerisation quantum algorithm''. Quantum computing is a groundbreaking technique. Hopefully, the ``HAM-Schrödingerisation quantum algorithm'' can open a door to highly efficient simulation of complicated turbulent flows by means of quantum computing in future.