What physical model does a Trotterized time evolution on a noisy quantum computer effectively simulate?

We consider the extent to which a noisy quantum computer is able to simulate the time evolution of a quantum spin system in a faithful manner. Given a reasonable set of assumptions regarding the manner in which noise acts on such a device, we argue for a circuit-level description of noise in terms of individual decoherence events following otherwise noise-free gates. With such a model, we further show how the effects of noise can be reinterpreted as a modification to the dynamics of the original system being simulated. We find that this modification corresponds to the introduction of static Lindblad terms, which act in addition to the original unitary dynamics. The form of these terms depends not only on the underlying noise processes occurring on the device, but also on the original unitary dynamics, as well as the manner in which these dynamics are simulated on the device, i.e., the choice of quantum algorithm. Our results are confirmed through numerical analysis. In addition to understanding the extent to which the result of a digital quantum simulation may differ from the intended calculation, our results may aide in tailoring quantum circuits to achieve the simulation of a given noisy spin system, as well as provide additional insight into Lindblad dynamics more broadly.