  We propose a quantum algorithm for solving physical problems represented by
the lattice Boltzmann formulation. Specifically, we deal with the case of a
single phase, incompressible fluid obeying the Bhatnagar-Gross-Krook model. We
use the framework introduced by Kowalski that links the nonlinear dynamics of a
system to the evolution of bosonic modes, assigning a Carleman linearization
order to the truncation in the bosonic Fock space of the bosons. The streaming
and collision steps are both achieved via unitary operators. A quantized
version of the nonlinear collision term has been implemented, without
introducing variables of discrete densities coupled from neighbouring sites,
unlike the classical Carleman technique. We use the compact mapping of the
bosonic modes to qubits that uses a number of qubits which scales
logarithmically with the size of truncated bosonic Fock space. The work can be
readily extended to the multitude of multiphysics problems which could adapt
the lattice Boltzmann formulation.
