Exponentially strongly fragmented systems equilibrate in exponential time

We investigate the robustness of Hilbert-space fragmentation in kinetically constrained models when the system is coupled to maximally depolarizing noise at the boundary. By mapping the thermalization process to a random walk on a Krylov graph, we are able to bound the thermalization time in terms of the connectivity of the graph. Specifically, we find that it is important to distinguish between cases when the largest Krylov sector is an exponentially small versus polynomially small fraction of the total Hilbert space. We conjecture that exponential fragmentation leads to exponentially slow thermalization. We further categorize systems with exponentially slow dynamics into three distinct classes and demonstrate that the conjecture holds in numerous exponentially fragmented systems, each corresponding to one of these three classes.