  For a digital quantum simulator (DQS) imitating a target system, we ask the
following question: Under what conditions is the simulator dynamics similar to
that of the target in the presence of coupling to a bath? In this paper, we
derive conditions for close simulation for three different physical regimes,
replacing previous heuristic arguments on the subject with rigorous statements.
In fact, we find that the conventional wisdom that the simulation cycle time
should always be short for good simulation need not always hold up. Numerical
simulations of two specific examples strengthen the evidence for our analysis,
and go beyond to explore broader regimes.
