Adaptive lattice-gas algorithm: Classical and quantum implementations

Lattice-gas algorithms (LGA) are a class of algorithms including, in chronological order, binary lattice-gas cellular automata, integer lattice-gas algorithms (ILGA), and lattice Boltzmann method (LBM). They are largely used for simulating nonlinear systems. Starting from one-dimensional ILGA, we design an algorithm where we carry out a fraction of the possible collisions. These fractions are then adapted to reproduce LBM equilibrium distributions, resulting in an adaptive lattice-gas algorithm that achieves the same simulation results of LBM. Considering this, we develop a quantum algorithm that involves a linear collision operator capable of simulating the same phenomena, while still using a measurement and reinitialization procedure. Multi-time-step implementation is possible in some specific cases, briefly discussed.