Quantum Solution for Nonlinear Differential Equations: Carleman and Liouville Linearization

Abstract

Nonlinear differential equations underlie many scientific fields, yet classical solvers scale poorly with increasing resolution. Quantum linear systems algorithms (QLSAs), such as Harrow-Hassidim-Lloyd (HHL), can offer asymptotic advantages for sparse, well-conditioned linear systems. However, they do not directly treat nonlinear dynamics. We study two linearization techniques—Carleman linearization and Liouville linearization—as preprocessors for prospective QLSA use. We assess the stability of the linearization step on two models with bounded phase spaces and single- or multi-attractor dynamics. Carleman linearization tracks single trajectories until truncation error dominates. The Liouville ensemble formulation, especially with narrow initial distributions, captures multi-basin splitting and trajectory statistics while providing built-in uncertainty quantification. These results suggest that Liouville may offer broader practical applicability as a QLSA preprocessor, particularly in regimes not covered by existing efficiency guarantees for Carleman linearization.