  We present a twofold contribution to the numerical simulation of Lindblad
equations. First, an adaptive numerical approach to approximate Lindblad
equations using low-rank dynamics is described: a deterministic low-rank
approximation of the density operator is computed, and its rank is adjusted
dynamically, using an on-the-fly estimator of the error committed when reducing
the dimension. On the other hand, when the intrinsic dimension of the Lindblad
equation is too high to allow for such a deterministic approximation, we
combine classical ensemble averages of quantum Monte Carlo trajectories and a
denoising technique. Specifically, a variance reduction method based upon the
consideration of a low-rank dynamics as a control variate is developed.
Numerical tests for quantum collapse and revivals show the efficiency of each
approach, along with the complementarity of the two approaches.
