Surrogate Quantum Circuit Design for the Lattice Boltzmann Collision Operator

TL;DR

This work develops a low-depth surrogate quantum circuit (SQC) to approximate the nonlinear, dissipative BGK collision operator within the $D_2Q_9$ lattice Boltzmann method. By embedding mass conservation, $D_8$ and scale equivariance in the quantum architecture and training the circuit with a momentum-penalized loss, the authors achieve a compact, unitary representation whose effects emerge in the measured population space. The 15-block SQC, implemented with a root-density encoding and Ising-based entanglers, attains large reductions in circuit depth (compared to prior quantum collision circuits) while reproducing key BGK features, including partial nonlinearity and dissipation, and generalizes to Reynolds numbers and flow regimes beyond its training data. Validation on Taylor–Green vortex decay and lid-driven cavity demonstrates accurate vortex decay, recirculation, and reasonable dissipation across $Re=10$ and $Re=50$, with measured limitations in strong gradient regions and a clear path toward measurement-free multi-step formulations.

Abstract

This study introduces a framework for learning a low-depth surrogate quantum circuit (SQC) that approximates the nonlinear, dissipative, and hence non-unitary Bhatnagar-Gross-Krook (BGK) collision operator in the lattice Boltzmann method (LBM) for the D2Q9 lattice. By appropriately selecting the quantum state encoding, circuit architecture, and measurement protocol, non-unitary dynamics emerge naturally within the physical population space. This approach removes the need for probabilistic algorithms relying on any ancilla qubits and post-selection to reproduce dissipation, or for multiple state copies to capture nonlinearity. The SQC is designed to preserve key physical properties of the BGK operator, including mass conservation, scale equivariance, and D8 equivariance, while momentum conservation is encouraged through penalization in the training loss. When compiled to the IBM Heron quantum processor's native gate set, assuming all-to-all qubit connectivity, the circuit requires only 724 native gates and operates locally on the velocity register, making it independent of the lattice size. The learned SQC is validated on two benchmark cases, the Taylor-Green vortex decay and the lid-driven cavity, showing accurate reproduction of vortex decay and flow recirculation. While integration of the SQC into a quantum LBM framework presently requires measurement and re-initialization at each timestep, the necessary steps towards a measurement-free formulation are outlined.

Figures (18)

• Figure 1: Action of the dihedral group D$_8$ on the D$_2$Q$_9$ populations, highlighting the transformations of the axial $f_1$ (blue) and diagonal $f_5$ (red) under each group element. Here $r$ is a 90$\degree$ anti-clockwise rotation about the origin, and $s$ is a reflection across the horizontal axis.
• Figure 2: Mapping of lattice symmetries to permutations of qubits in the velocity register: (center) the original 4 qubit basis state $\ket{Q_3Q_2Q_1Q_0}$, (left) the reflection induced by $U_s$ across the horizontal axis permutes qubits $3 \leftrightarrow 1$ (leaving $2$ and $0$ fixed) giving $\ket{Q_1Q_2Q_3Q_0}$, (right) the 90$\degree$ anti-clockwise rotation induced by $U_r$ cyclically permutes the qubits $0 \rightarrow 1 \rightarrow 2 \rightarrow 3 \rightarrow 0$, giving $\ket{Q_2Q_1Q_0Q_3}$.
• Figure 3: D$_8$-invariant coupling sets on the square qubit configuration: (left) axial coupling pattern $O_{\text{axial}} = \{\{0,1\}, \{1,2\}, \{2,3\}, \{3,0\}\}$, corresponding to the nearest-neighbor pairs along the edge of the square; (right) diagonal coupling pattern $O_{\text{diag}} = \{ \{0,2\}, \{1,3\}\}$, connecting opposite corners.
• Figure 4: End-to-end training loop for the SQC. The pre-collision populations $f_i$ are first encoded into a quantum state. The SQC then acts on this state to generate the post-collision state, from which the post-collision predictions $\hat{f}_i^{\text{eq}}$ are obtained after measurement in the computational basis. These predictions are evaluated using the combined loss $L = \mathrm{MSE} + \alpha L_m$, and the circuit parameters $\theta_i$ are updated via classical gradient descent.
• Figure 5: Average test‐set MSE loss for each Ising entangling‐layer configuration, showing only the rotation‐order variant ($X \to Z$ or $Z \to X$) that achieved the lowest loss.