  We present a quantum algorithm for computational fluid dynamics based on the
Lattice-Boltzmann method. Our approach involves a novel encoding strategy and a
modified collision operator, assuming full relaxation to the local equilibrium
within a single time step. Our quantum algorithm enables the computation of
multiple time steps in the linearized case, specifically for solving the
advection-diffusion equation, before necessitating a full state measurement.
Moreover, our formulation can be extended to compute the non-linear equilibrium
distribution function for a single time step prior to measurement, utilizing
the measurement as an essential algorithmic step. However, in the non-linear
case, a classical postprocessing step is necessary for computing the moments of
the distribution function. We validate our algorithm by solving the one
dimensional advection-diffusion of a Gaussian hill. Our results demonstrate
that our quantum algorithm captures non-linearity.
