  In this paper we derive accurate numerical methods for the quantum Boltzmann
equation for a gas of interacting bosons. The schemes preserve the main
physical features of the continuous problem, namely conservation of mass and
energy, the entropy inequality and generalized Bose-Einstein distributions as
steady states. These properties are essential in order to develop schemes that
are able to capture the energy concentration behavior of bosons. In addition we
develop fast algorithms for the numerical evaluation of the resulting
quadrature formulas which allow the final schemes to be computed only in O(N^2
log N) operations instead of O(N^3).
