We present a general variational approach to determine the steady state of open quantum lattice systems via a neural network approach. The steady-state density matrix of the lattice system is constructed via a purified neural network ansatz in an extended Hilbert space with ancillary degrees of freedom. The variational minimization of cost functions associated to the master equation can be performed using a Markov chain Monte Carlo sampling. As a first application and proof-of-principle, we apply the method to the dissipative quantum transverse Ising model.In spite of the tremendous experimental progress in the isolation of quantum systems, a finite coupling to the environment [1] is unavoidable and certainly plays a crucial role in the practical implementation of quantum information and quantum simulation protocols [2]. Moreover, through an active control of the environment via the so-called reservoir engineering, an open quantum manybody system can be prepared in non-trivial phases [3][4][5] with also possible quantum applications [6, 7]. The theoretical description of open quantum manybody systems is in general out-of-the equilibrium and much less developed than for equilibrium systems. A mixed state with a finite entropy can be described by a density matrix, whose evolution is described by a master equation. Recently, a few theoretical methods have been developed to solve the master equation of open quantum manybody systems, including analytical approaches based on the Keldysh formalism [8,9], numerical algorithms based on matrix product operator and tensor-network techniques [10][11][12][13][14], cluster mean-field methods [15,16], corner-space renormalization [17][18][19], Gutzwiller mean-field [20], full configuration-interaction Monte Carlo [21], permutationinvariant solvers [22] or efficient stochastic unravelings for disordered systems [23]. The research in the field is very active, since the different methods are optimal for different specific regimes. For example, the corner-space renormalization method is best suited for systems with moderate entropy, while matrix product operator techniques to systems with short-range quantum correlations.In the last decade, the field of artificial neural networks has enjoyed a dramatic expansion and success thanks to remarkable applications in the recognition of complex patterns such as visual images or human speech (for a recent review see, e.g., [24]). The optimization (supervised learning) of the network is obtained by tuning the weights quantifying the connections between neural units via a variational minimization of a properly defined cost function. The wavefunction of a manybody system is in general a complex quantity, which is hard to be recognized. Recent works have proposed to exploit arti-ficial neural networks to construct trial wavefunctions, where the connection weights in the network play the role of variational parameters [25,26]. Neural network approaches have already been succesffuly applied to a wide number (see e.g. [27][28][29][30][31]) of clos...

We theoretically explore the dynamical properties of a first-order dissipative phase transition in coherently driven Bose-Hubbard systems, describing, e.g., lattices of coupled nonlinear optical cavities. Via stochastic trajectory calculations based on the truncated Wigner approximation, we investigate the dynamical behavior as a function of system size for 1D and 2D square lattices in the regime where mean-field theory predicts nonlinear bistability. We show that a critical slowing down emerges for increasing number of sites in 2D square lattices, while it is absent in 1D arrays. We characterize the peculiar properties of the collective phases in the critical region.