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

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

Problem Statement
The Lattice Boltzmann Method (LBM) is a widely used computational fluid dynamics (CFD) technique that discretizes space, time, and velocity. While the streaming step of LBM is linear and amenable to quantum algorithms, the local collision step introduces non-linear dependencies that challenge pure quantum solvers, which rely on unitary operations. Previous quantum approaches to LBM have either required reformulating the problem as binary optimization, using adiabatic annealing, or employing Variational Quantum Circuits (VQCs) that are limited to fixed relaxation parameters (e.g., $\tau=1$), necessitating retraining for different flow regimes. Furthermore, existing methods often struggle with the non-unitary nature of the Bhatnagar-Gross-Krook (BGK) collision operator and the high gate counts required for linearization techniques like Carleman expansion.

Methodology
The authors propose a hybrid quantum-classical surrogate model designed for Noisy Intermediate-Scale Quantum (NISQ) hardware to approximate the non-linear BGK collision operator. The core of the methodology involves decoupling the collision process into two distinct components:

1. Equilibrium Prediction via VQC: A shallow, parameterized quantum circuit predicts the post-collision equilibrium distribution. The circuit utilizes an Angle Encoding (AE) scheme where classical macroscopic variables (density $\rho$, velocity $\mathbf{u}$) are encoded as rotation angles. The architecture employs a Parallel Ansatz (PA) with $N=3$ qubits to handle the three velocity components. The circuit structure includes data re-uploading (repeating encoding and entangling layers) to expand the accessible frequency spectrum, allowing the model to represent higher-order non-linearities.
2. BGK Relaxation via Circuit Extension: Instead of learning the full relaxation dynamics, the authors embed the BGK relaxation step directly into the circuit structure. The relaxation parameter $\omega$ is encoded as a quantum probability amplitude controlling a Fredkin (controlled-SWAP) gate. This gate mixes the pre-collision state with the predicted equilibrium state. This design allows the model to accept any relaxation parameter $\omega \in [0.5, 2)$ as an input without retraining.
3. Post-Processing Constraints: To address approximation errors inherent in the quantum surrogate, the model applies classical post-processing to enforce:
   • Mass and Momentum Conservation: Using an algebraic correction derived from Corbetta et al.
   • Rotational Equivariance: Enforcing symmetry via group averaging over the octahedral group $D_{48}$ (for 3D) or dihedral group $D_8$ (for 2D).
   • Directional Orbit Compression: Reducing the number of trained circuits by grouping velocity directions into symmetry orbits, sharing parameters across physically equivalent directions.

Key Contributions

• Variable Relaxation Parameter: The paper introduces the first hybrid quantum-classical surrogate that explicitly encodes the relaxation parameter $\omega$ as a circuit input, enabling the recovery of the complete BGK post-collision state across the full physically admissible range without retraining.
• Architectural Analysis: The authors critically reassess standard VQC metrics (expressibility, entanglement, effective dimension) in the context of fluid dynamics. They identify that the number of encoding repetitions ($L_{SW}$) is the primary driver of model capacity, while achieving Haar-level entanglement is not a prerequisite for competitive accuracy.
• 3D High-Fidelity Simulation: The work presents the first three-dimensional high-fidelity quantum LBM surrogate simulation, validated on the Taylor–Green vortex benchmark.
• Orbit Compression: The implementation of directional orbit compression significantly reduces training and inference overhead while maintaining accuracy and stabilizing the simulation against stochastic training variance.

Results
The surrogate was validated against the classical BGK operator using established benchmark problems:

• Taylor–Green Vortex (3D): Simulated at $Re \approx 1$ with relaxation parameters $\omega = 0.75$ and $\omega = 1.25$. The surrogate accurately reproduced the kinetic energy decay and vortex dynamics with median relative errors around $10^{-4}$. The inclusion of conservation and symmetry corrections was shown to be necessary for long-term stability.
• Double Shear Layer (2D): Simulated at Reynolds numbers $Re = 100, 400,$ and $1000$. The model successfully captured the Kelvin-Helmholtz instability, vortex pairing, and transition to turbulence. Pointwise errors remained low ($\sim 10^{-4}$ to $10^{-3}$), concentrated in regions of high non-linearity (vortex cores).
• Generalization: The model demonstrated robust generalization across different Reynolds numbers and relaxation parameters without retraining, confirming the efficacy of the variable $\omega$ encoding.

Significance and Claims
The paper claims that this hybrid strategy offers a practical path toward realizing the potential of quantum computing in fluid engineering. By offloading the non-unitary BGK relaxation to a closed-form circuit extension and using a shallow VQC only for the non-linear equilibrium prediction, the approach mitigates the limitations of current NISQ hardware (noise, limited qubits, shallow depth). The authors emphasize that their approach avoids the exponential gate costs of linearization methods and the retraining requirements of previous VQC-based LBM surrogates. The work suggests that hybrid quantum-classical strategies can achieve high accuracy and generalizability, closely replicating classical solutions for moderate Reynolds number flows, thereby validating the feasibility of quantum-enhanced CFD surrogates.