The state of a Markovian open quantum system is completely determined by its density matrix which evolves according to a Lindblad master equation. When the system is composed by many interacting particles, the complexity arising from the many-body problem merges with the necessity to represent mixed states. In this work  we exploit a variational ans¨atz described by a neural network to represent a generic nonequilibrium density matrix . By deriving a variational principle, we show that it is possible to deﬁne an iterative procedure where the network parameters are varied in order to minimize a cost function quantifying the distance from the asymptotic steady-state. Such a procedure, similar in spirit to supervised learning, can be performed eﬃciently by means of a Montecarlo sampling of the cost function . As a ﬁrst application and proof-of-principle, we apply the method to the dissipative quantum transverse Ising model .
 F. Vicentini, A. Biella, N. Regnault, and C. Ciuti, arXiv:1902.10104 [quant-ph] (2019)
 G. Torlai and R. G. Melko, Phys. Rev. Lett. 120, 240503 (2018)
 G. Carleo and M. Troyer, Science 355, 602 (2017)
 J. Jin, A. Biella, O. Viyuela, C. Ciuti, R. Fazio, and D. Rossini, Phys. Rev. B 98(R), 241108 (2018).
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