  We present a novel generic framework to approximate the non-equilibrium
steady states of dissipative quantum many-body systems. It is based on the
variational minimization of a suitable norm of the quantum master equation
describing the dynamics. We show how to apply this approach to different
classes of variational quantum states and demonstrate its successful
application to a dissipative extension of the Ising model, which is of
importance to ongoing experiments on ultracold Rydberg atoms. Finally, we
identify several advantages of the variational approach over previously
employed mean-field-like methods.

Quantum Physics;Statistical Mechanics;Atomic Physics;Computational Physics