  The exact quantum state evolution of a fermionic gas with binary interactions
is obtained as the stochastic average of BCS-state trajectories. We find the
most general Ito stochastic equations which reproduce exactly the dynamics of
the system and we obtain some conditions to minimize the stochastic spreading
of the trajectories in the Hilbert space. The relation between the optimized
equations and mean-field equations is analyzed. The method is applied to a
simple two-site model. The simulations display effects that cannot be obtained
in the mean-field approximation.
