  We present random quantum circuit models for non-unitary quantum dynamics of
free fermions in one spatial dimension. Numerical simulations reveal that the
dynamics tends towards steady states with logarithmic violations of the
entanglement area law and power law correlation functions. Moreover, starting
with a short-range entangled many-body state, the dynamical evolution of
entanglement and correlations quantitatively agrees with the predictions of
two-dimensional conformal field theory with a space-like time direction. We
argue that this behavior is generic in non-unitary free quantum dynamics with
time-dependent randomness, and show that the emergent conformal dynamics of
two-point functions arises out of a simple "nonlinear master equation".
