  Understanding dissipation in 2D quantum many-body systems is a remarkably
difficult open challenge. Here we show how numerical simulations for this
problem are possible by means of a tensor network algorithm that approximates
steady-states of 2D quantum lattice dissipative systems in the thermodynamic
limit. Our method is based on the intuition that strong dissipation kills
quantum entanglement before it gets too large to handle. We test its validity
by simulating a dissipative quantum Ising model, relevant for dissipative
systems of interacting Rydberg atoms, and benchmark our simulations with a
variational algorithm based on product and correlated states. Our results
support the existence of a first order transition in this model, with no
bistable region. We also simulate a dissipative spin-1/2 XYZ model, showing
that there is no re-entrance of the ferromagnetic phase. Our method enables the
computation of steady states in 2D quantum lattice systems.
