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 (CLB), ii) Navier-Stokes (CNS) and iii) Grad (CG) formulations of fluid dynamics. CLB 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 CNS 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 CG might combine the best of CLB and CNS.