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We present a twofold contribution to the numerical simulation of Lindbladequations. First, an adaptive numerical approach to approximate Lindbladequations using low-rank dynamics is described: a deterministic low-rankapproximation of the density operator is computed, and its rank is adjusteddynamically, using an on-the-fly estimator of the error committed when reducingthe dimension. On the other hand, when the intrinsic dimension of the Lindbladequation is too high to allow for such a deterministic approximation, wecombine classical ensemble averages of quantum Monte Carlo trajectories and adenoising technique. Specifically, a variance reduction method based upon theconsideration of a low-rank dynamics as a control variate is developed.Numerical tests for quantum collapse and revivals show the efficiency of eachapproach, along with the complementarity of the two approaches.

Adaptive low-rank approximation and denoised Monte-Carlo approach for high-dimensional Lindblad equations