Digital quantum simulator in the presence of a bath*

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.