Three Carleman routes to the quantum simulation of classical fluids

We discuss the Carleman approach to the quantum simulation of classical fluids, as applied to (i) lattice Boltzmann, (ii) Navier–Stokes, and (iii) Grad formulations of fluid dynamics. Carleman lattice Boltzmann shows excellent convergence properties, but it is plagued by nonlocality which results in an exponential depth of the corresponding circuit with the number of Carleman variables. The Carleman Navier–Stokes offers a dramatic reduction of the number Carleman variables, which might lead to a viable depth, provided locality can be preserved and convergence can be achieved with a moderate number of iterates also at sizeable Reynolds numbers. Finally, it is argued that Carleman Grad might combine the best of Carleman lattice Boltzmann and Carleman Navier–Stokes.