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

Abstract

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 $\ket{Ψ_i}$, this operator yields a final state $\ket{Ψ_f} = U\ket{Ψ_i}$ 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.

Figures (23)

• Figure 1: Scheme of channels for D2Q9. Each number represents the index used to map each channel.
• Figure 4: Schematic representation of the one-register Variational Quantum Circuit (R1 model) for the Quantum Lattice Boltzmann Method with one layer $B=1$. The architecture follows the same principles of previous research lactatus2025surrogate to conserve dihedral symmetry as introduced in Sec \ref{['sec:dihedral_d2q9']}. It also includes three different gates for each qubit, following the Euler angles principle for general rotations.
• Figure 5: a) Velocity field for an LBM at second order $O(u^2)$ 2D TGV simulation with $u_{max}=0.05$ and $\tau=1$ with $L=64$ lattice sites per dimension and $T=50$ time-steps. b) Relative error between the velocity field of the nonlinear LBM and the linear LBM using only $O(u)$ to calculate the equilibrium distribution function.
• Figure 6: Relative error $\epsilon_{rel}$ of a TGV simulation using a QML model. The simulation parameters are: $64\times64$ lattice sites, $\tau=1$, $T=50$ time steps and $k_x=\frac{\pi}{N_X}$ and $u_{max}=0.05$. The total collision operator is composed of a first step with the linear collision operator and a second step involving the learned unitary from the QML model using $L_{\rho}$ as cost function.
• Figure 7: Comparison of the MSE of the predicted distribution functions $|f_i^{vqc}|$ with a QML when targeting the post-nonlinear-collision distribution $|f_i^{ref}|$ , compared with the mean square error between $|f_i^{ref}|$ and $|f_i^{lin}|$. The training used the data from 50 time-steps with different maximum velocities from a Taylor-Green vortex simulation. The ratio between the predicted MSE and the MSE with respect to the linear distribution is also shown. The cost function used by the QML is $L=10^{-4}L_{\phi}+L_{A}$, with each loss representing the phase and amplitude MSE, respectively, from \ref{['eq:loss_amp_phase']}.