Variational Neural-Network Ansatz for Steady States in Open Quantum Systems

Abstract

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...

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