Comparing active-site and operator spreading in dark-state circuits

The universal dynamics of information spreading is a key question in quantum many-body physics. This spread can be quantified using out-of-time-order correlators (OTOCs), which reveal causal structures in the growth of initially local operators. Measuring such correlators in experiments, including analog quantum simulators and digital quantum computers, is challenging as it requires the ability to explicitly reverse time, or access to auxiliary degrees of freedom with high connectivity. However, causal structures similar to those revealed by OTOCs also appear in many-body systems with dark states that do not participate in the dynamics. In this work, we focus on stochastic automaton circuits as proxies for such quantum dynamics and make detailed comparisons between the spreading of active sites on initially dark backgrounds and operator spreading as probed by OTOCs. We also ask how both probes are affected by power-law long-range interactions, which are known to modify the behavior of OTOCs as a function of the interaction range.