## Section: New Results

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

Participants: Pierre Rouchon

The results of this section were published in [17] .

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 variable is developed. Numerical tests for quantum collapse and revivals show the efficiency of each approach, along with the complementarity of the two approaches.

This work results from a collaboration with Claude Le Bris of the Matherials project-team and in the framework of the ANR-project EMAQS entitled "Evaluation and Manipulation At Quantum Scale" coordinated by Karine Beauchard from ENS-Rennes.