Lattice Boltzmann-Carleman quantum algorithm and circuit for fluid flows at moderate Reynolds number

We present a quantum computing algorithm for fluid flows based on the Carleman-linearization of the Lattice Boltzmann (LB) method. First, we demonstrate the convergence of the classical Carleman procedure at moderate Reynolds numbers, namely for Kolmogorov-like flows. Then we proceed to formulate the corresponding quantum algorithm, including the quantum circuit layout and analyze its computational viability. We show that, at least for moderate Reynolds numbers between 10 and 100, the Carleman-LB procedure can be successfully truncated at second order, which is a very encouraging result. We also show that the quantum circuit implementing the single time-step collision operator has a fixed depth, regardless of the number of lattice sites. However, such depth is of the order of ten thousands quantum gates, meaning that quantum advantage over classical computing is not attainable today, but could be achieved in the near-mid term future. The same goal for the multi-step version remains however an open topic for future research.