  We develop an efficient algorithm for a spatially inhomogeneous matrix-valued
quantum Boltzmann equation derived from the Hubbard model. The distribution
functions are $2 \times 2$ matrix-valued to accommodate the spin degree of
freedom, and the scalar quantum Boltzmann equation is recovered as special case
when all matrices are proportional to the identity. We use Fourier
discretization and fast Fourier transform to efficiently evaluate the collision
kernel with spectral accuracy, and numerically investigate periodic, Dirichlet
and Maxwell boundary conditions. Model simulations quantify the convergence to
local and global thermal equilibrium.
