Measurement-Efficient Variational Quantum Linear Solver for Carleman-Linearized Nonlinear Dynamics

Abstract

We present hybrid quantum-classical pipelines for solving the Duffing equation that leverage Carleman linearization and the Variational Quantum Linear Solver (VQLS). First, we demonstrate that Carleman linearization accurately approximates the weakly nonlinear Duffing equation, with errors diminishing as the truncation order increases. Next, across IBM and Xanadu platforms, we deploy VQLS with symmetry-grouped Hadamard Test evaluations under both global and local cost formulations, compare distinct Hermitianization within a common cost framework, and benchmark hardware-efficient ansatz architectures under a fixed Hermitianization. Across block-banded test cases, each method achieves near-unity fidelity and vanishing relative residuals. These results show that topology-agnostic ansatz, optimized Hermitianization, and efficient cost formulation enable VQLS to recover quantum states proportional to classical solutions for Carleman-structured systems, providing a portable recipe for quantum-in-the-loop simulation of nonlinear dynamics.