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

Claude Le Bris 1, 2 Pierre Rouchon 3, 4 Julien Roussel 1 
CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique, Inria Paris-Rocquencourt, ENPC - École des Ponts ParisTech
4 QUANTIC - QUANTum Information Circuits
ENS-PSL - École normale supérieure - Paris, Inria Paris-Rocquencourt, UPMC - Université Pierre et Marie Curie - Paris 6, MINES ParisTech - École nationale supérieure des mines de Paris, CNRS - Centre National de la Recherche Scientifique : UMR8551
Abstract : 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.
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Claude Le Bris, Pierre Rouchon, Julien Roussel. Adaptive low-rank approximation and denoised Monte Carlo approach for high-dimensional Lindblad equations. Physical Review, American Physical Society (APS), 2015, 92 (6), pp.062126. ⟨10.1103/PhysRevA.92.062126⟩. ⟨hal-01252664⟩



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