For a quantum digital simulator (QDS) simulating a closed many-body quantum system, we ask the following questions: if both systems interacting with the same bath, in the same way, will the dynamics of the target system and simulator behave somewhat close to each other? In this paper, we study the open system dynamics for both target system and its digital simulator and compare their channels acting on the same initial states. Analytically, we give conditions when their stroboscopic behavior are close to each other, which shows that the fast gates alone are sufficient for QDS to possess the same decoherence process as the target. Numerical simulation of the open system dynamics for a simplified Kitaev toric code model and its QDS in a thermal boson bath verifies the analytical results. This results may shed light on developing new methods of quantum error correction and Gibbs state preparation.