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Surrogate Quantum Circuit Design for the Lattice Boltzmann Collision Operator

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.

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