  A novel scheme to simulate the evolution of a restricted set of observables
of a quantum system is proposed. The set comprises the spectrum-generating
algebra of the Hamiltonian. The idea is to consider a certain open-system
evolution, which can be interpreted as a process of weak measurement of the
distinguished observables performed on the evolving system of interest. Given
that the observables are "classical" and the Hamiltonian is moderately
nonlinear, the open system dynamics displays a large time-scales separation
between the dephasing of the observables and the decoherence of the evolving
state in the basis of the generalized coherent states (GCS), associated with
the spectrum-generating algebra. The time scale separation allows the unitary
dynamics of the observables to be efficiently simulated by the open-system
dynamics on the intermediate time-scale.The simulation employs unraveling of
the corresponding master equations into pure state evolutions, governed by the
stochastic nonlinear Schroedinger equantion (sNLSE). It is proved that GCS are
globally stable solutions of the sNLSE, if the Hamilonian is linear in the
algebra elements.
