A Quantum-Classical Surrogate Model for the Collision Operator of the Lattice Boltzmann Method

We introduce a hybrid approach utilising a quantum machine learning surrogate model to approximate the non-linear collision dynamics of the LBM. It effectively offloads the non-unitary operations that challenge pure quantum solvers. The expressivity of the surrogate is built on the ability of parameterised quantum circuits to implement partial Fourier series, with data re-uploading extending the spectrum of representable frequencies. Unlike previous approaches with a fixed relaxation parameter, the surrogate recovers the complete Bhatnagar-Gross-Krook (BGK) collision dynamics across the full physically admissible range of relaxation without retraining. We reassess the relevance of standard variational quantum circuit (VQC) metrics, including expressibility, entanglement, and effective dimension, by relating them directly to task-specific surrogate performance and identifying the key architectural parameters that determine approximation accuracy. The proposed surrogate is validated against the classical BGK collision operator using established benchmark problems, including the Taylor-Green vortex for evaluating energy dissipation and the double shear layer for assessing shear-driven instabilities and nonlinear flow evolution. Our results demonstrate that the hybrid model achieves high accuracy and generalisability while closely replicating classical solutions. These findings suggest that hybrid quantum-classical strategies offer a practical path toward realising the potential of quantum computing in fluid engineering.