  We introduce in this work an efficient numerical method for the simulation of
the quantum Liouville-BGK equation, which models the diffusive transport of
quantum particles. The corner stone to the model is the BGK collision operator,
obtained by minimizing the quantum free energy under the constraint that the
local density of particles is conserved during collisions. This leads to a
large system of coupled nonlinear nonlocal PDEs whose resolution is
challenging. We then define a splitting scheme that separates the transport and
the collision parts, which, exploiting the local conservation of particles,
leads to a fully linear collision step. The latter involves the resolution of a
constrained optimization problem that is is handled with the nonlinear
conjugate gradient algorithm. We prove that the time semi-discrete scheme is
convergent, and as an application of our numerical scheme, we validate the
quantum drift-diffusion model that is obtained as the diffusive limit of the
quantum Liouville-BGK equation.
