  We present a method to simulate the dynamics of large driven-dissipative
many-body open quantum systems using a variational encoding of the Wigner or
Husimi-Q quasi-probability distributions. The method relies on Monte-Carlo
sampling to maintain a polynomial computational complexity while allowing for
several quantities to be estimated efficiently. As a first application, we
present a proof of principle investigation into the physics of the
driven-dissipative Bose-Hubbard model with weak nonlinearity, providing
evidence for the high efficiency of the phase space variational approach.
