Quantum Algorithms for Nonlinear Dynamics: Revisiting Carleman Linearization with No Dissipative Conditions

TL;DR

This paper extends the analysis of Carleman linearization for nonlinear dynamical systems by establishing error bounds and convergence properties under resonance conditions, broadening quantum simulation applications beyond dissipative regimes.

Contribution

It introduces a new resonance-based regime for Carleman linearization, proving linear convergence without dissipative conditions, supported by numerical experiments on various nonlinear PDEs.

Findings

Error bounds established beyond dissipative conditions

Linear convergence under resonance conditions

Numerical validation on Burgers', FPU, and KdV models

Abstract

In this paper, we explore the embedding of nonlinear dynamical systems into linear ordinary differential equations (ODEs) via the Carleman linearization method. Under dissipative conditions, numerous previous works have established rigorous error bounds and linear convergence for Carleman linearization, which have facilitated the identification of quantum advantages in simulating large-scale dynamical systems. Our analysis extends these findings by exploring error bounds beyond the traditional dissipative condition, thereby broadening the scope of quantum computational benefits to a new class of dynamical regimes. This novel regime is defined by a resonance condition, and we prove how this resonance condition leads to a linear convergence with respect to the truncation level $N$ in Carleman linearization. We support our theoretical advancements with numerical experiments on a variety of models, including the Burgers' equation, Fermi-Pasta-Ulam (FPU) chains, and the Korteweg-de Vries (KdV) equations, to validate our analysis and demonstrate the practical implications.