Structure-Preserving Quantum Algorithms for Sparse Polynomial Nonlinear Dynamical Systems via Compressed Carleman–Koopman Lifting and Schrödingerisation

This paper presents a rigorous derivation pipeline for the quantum simulation of sparse polynomial nonlinear ordinary differential equations. The framework proceeds in three stages: (i) mapping the nonlinear dynamics to an exact infinite-dimensional linear system via Carleman–Koopman lifting; (ii) applying a graph-theoretic truncation and compression scheme that exploits the sparsity and locality of the dependency graph; and (iii) implementing Schr\"odingerisation for the resulting class of dissipative, Hermitian positive definite generators to yield a unitary-overlap formula compatible with standard quantum primitives. The study provides a comprehensive mathematical construction of these steps, establishing the algebraic and analytical foundations required for structure-preserving quantum simulation.