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)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 MATHERIALS - MATHematics for MatERIALS
3 CAS - Centre Automatique et Systèmes
4 QUANTIC - QUANTum Information Circuits

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.

Domaines: Automatique / Robotique; Physique Quantique [quant-ph]

Claude Le Bris, Pierre Rouchon, Julien Roussel. Adaptive low-rank approximation and denoised Monte Carlo approach for high-dimensional Lindblad equations. Physical Review, 2015, 92 (6), pp.062126. ⟨10.1103/PhysRevA.92.062126⟩. ⟨hal-01252664⟩