Abstract
The time-evolution operator of open quantum systems is determined generally via a stochastic operator sum representation. Quantum trajectories originating from any quantum trajectory method applicable to the system can be used for the construction. The result does apply to non-Markovian systems as well. The computational time reducement is in the order of the dimension of the Hilbert space compared to conventional quantum trajectory methods combined with algebraic reconstruction of the map. To demonstrate the efficiency of the method, the effects of decoherence are analysed for a C-NOT gate realized by two atoms in a cavity.