Advanced Quantum Lattice Boltzmann Simulations: Incorporating Boundary Conditions and Non-Linear Dynamics via Quantum Carleman Linearization

This study extends the applicability of the quantum lattice Boltzmann method (QLBM) to non-linear fluid dynamics with complex boundaries, a persistent challenge in quantum computing. We address the non-linear LBM collision term by implementing quantum Carleman linearization (QCL) and introduce a novel quantum circuit for bounce-back boundary condition. The QCL technique linearizes the collision operator for its representation as a unitary quantum gate, while the bounce-back boundary condition is modeled using a circuit that applies controlled quantum gates to reverse velocity states at pre-defined boundary locations. The presented framework's versatility and performance are demonstrated through two benchmarks: a Gaussian hill simulation for the linear advection-diffusion equation (ADE), and the non-linear Kármán vortex street, which is handled by the QCL-enhanced collision operator and our boundary implementation. The framework's performance is benchmarked against classical LBM, while the QLBM itself is evaluated on two distinct quantum backends: the Qiskit Aer simulator and ibm_yonsei real quantum hardware. For the Gaussian hill simulation, ibm_yonsei hardware reproduced the velocity field with a 0.92 correlation coefficient compared to the classical LBM, and for the Kármán vortex, it qualitatively captured periodic vortex shedding. These findings demonstrate the practical feasibility of applying our advanced QLBM framework to a range of flow problems on current quantum hardware.