  We analyze the complexity of classically simulating continuous-time dynamics
of locally interacting quantum spin systems with a constant rate of
entanglement breaking noise. We prove that a polynomial time classical
algorithm can be used to sample from the state of the spins when the rate of
noise is higher than a threshold determined by the strength of the local
interactions. Furthermore, by encoding a 1D fault tolerant quantum computation
into the dynamics of spin systems arranged on two or higher dimensional grids,
we show that for several noise channels, the problem of weakly simulating the
output state of both purely Hamiltonian and purely dissipative dynamics is
expected to be hard in the low-noise regime.
