Hamiltonian simulation for nonlinear partial differential equation by Schr\"{o}dingerization

TL;DR

This paper introduces a novel Hamiltonian simulation method for nonlinear PDEs by combining Carleman linearization and Schr"odingerization, enabling efficient quantum simulation of nonlinear classical systems.

Contribution

It proposes the CLS method that transforms nonlinear PDEs into Schr"odinger equations, extending Hamiltonian simulation to nonlinear systems for the first time.

Findings

Successfully applied to nonlinear reaction--diffusion equations

Demonstrates potential for exponential speedup in quantum simulations

Extends Hamiltonian simulation applicability beyond linear systems

Abstract

Hamiltonian simulation is a fundamental algorithm in quantum computing that has attracted considerable interest owing to its potential to efficiently solve the governing equations of large-scale classical systems. Exponential speedup through Hamiltonian simulation has been rigorously demonstrated in the case of coupled harmonic oscillators. The question arises as to whether Hamiltonian simulations in other physical systems also accelerate exponentially. Schr\"odingerization is a technique that transforms the governing equations of classical systems into the Schr\"odinger equation. However, since the Schr\"odinger equation is a linear equation, Hamiltonian simulation is often limited to linear equations. The research on Hamiltonian simulation methods for nonlinear governing equations remains relatively limited. In this study, we propose a Hamiltonian simulation method for nonlinear partial differential equations (PDEs). The proposed method is named Carleman linearization + Schr\"odingerization (CLS), which combines Carleman linearization (CL) and warped phase transformation (WPT). CL is first applied to transform a nonlinear PDE into a linear differential equation. This linearized equation is then mapped to the Schr\"odinger equation via WPT. The original nonlinear PDE can be solved efficiently by the Hamiltonian simulation of the resulting Schr\"odinger equation. By applying this method, we transform the original governing equation into the Schr\"odinger equation. Solving the transformed Schr\"odinger equation then enables the analysis of the original nonlinear equation. As a specific application, we apply this method to the nonlinear reaction--diffusion equation to demonstrate that Hamiltonian simulations are applicable to nonlinear PDEs.