  We explore the Carlemann linearization of the collision term of the lattice
Boltzmann formulation, as a first step towards formulating a quantum lattice
Boltzmann algorithm. Specifically, we deal with the case of a single,
incompressible fluid with the Bhatnagar Gross and Krook equilibrium function.
Under this assumption, the error in the velocities is proportional to the
square of the Mach number. Then, we showcase the Carlemann linearization
technique for the system under study. We compute an upper bound to the number
of variables as a function of the order of the Carlemann linearization. We
study both collision and streaming steps of the lattice Boltzmann formulation
under Carlemann linearization. We analytically show why linearizing the
collision step sacrifices the exactness of streaming in lattice Boltzmann,
while also contributing to the blow up in the number of Carlemann variables in
the classical algorithm. The error arising from Carlemann linearization has
been shown analytically and numerically to improve exponentially with the
Carlemann linearization order. This bodes well for the development of a
corresponding quantum computing algorithm based on the Lattice Boltzmann
equation.
