Quantum machine learning for the quantum lattice Boltzmann method: Trainability of variational quantum circuits for the nonlinear collision operator across multiple time steps

This study investigates the application of quantum machine learning (QML) to approximate the nonlinear component of the collision operator within the quantum lattice Boltzmann method (QLBM). To achieve this, we train a variational quantum circuit (VQC) to construct an operator $U$. When applied to the post-linear-collision quantum state $\ket{\Psi_i}$, this operator yields a final state $\ket{\Psi_f} = U\ket{\Psi_i}$ that successfully replicates the nonlinear collision dynamics derived from the Bhatnagar-Gross-Krook (BGK) approximation. Within this framework, we present two distinct architectures: the R1 model and the R2 model. The R1 model is designed for quantum simulations that involve multiple time steps without intermediate measurements, focusing on accurately capturing nonlinear dynamics in continuous evolution. In contrast, the R2 model is tailored to achieve the high-precision reconstruction of the nonlinear operator for a single time step with an unitary operator.