Improved Quantum Lattice Boltzmann Method for Advection-Diffusion Equations with a Linear Collision Model

Abstract: Quantum computing has made tremendous progress in recent years, providing potentialities for breaking the bottleneck of computing power in the field of scientific computing, like computational fluid dynamics. To reduce computational costs and achieve an acceleration, we propose an ancilla free quantum lattice Boltzmann method for advection-diffusion equations that fully leverages the parallelism of quantum computing. More significantly, there is no need to perform quantum state tomography in each previous loop, if the macroscopic variables for a certain loop is needed. The non-unitary collision operators are replaced by the unique local unitary operations, and the removal of ancilla qubit greatly diminishes the complexity of the quantum circuit. The numerical simulations of the $D_1Q_3$ and $D_2Q_5$ models have confirmed the feasibility of the proposed algorithm.