Carleman-VQLS for Nonlinear Dynamics  

We introduce a hybrid quantum-classical pipeline for solving the Duffing equation that leverages Carleman linearization and the Variational Quantum Linear Solver (VQLS). First, we demonstrate that Carleman linearization accurately approximates the Duffing equation, with errors diminishing as the truncation order increases. Next, across IBM and Xanadu platforms, we deploy VQLS to block-tridiagonal examples to implement symmetry-grouped cost functions under global and local frameworks, compare Hermitian processes in the same cost framework, and evaluate different ansatz architectures with the same Hermitianization. The optimization examples achieve near-unity fidelities and probability distributions, as well as vanishing relative residual and final cost. Our results highlight the effectiveness of topology-agnostic ansatz, Hermitianization, and cost function in approximating quantum states that are proportional to classical solutions for nonlinear dynamics.