  We introduce state-independent, non-perturbative Hamiltonian quantum speed
limits for population leakage and fidelity loss, for a gapped open system
interacting with a reservoir. These results hold in the presence of initial
correlations between the system and the reservoir, under the sole assumption
that their interaction and its commutator with the reservoir Hamiltonian are
norm-bounded. The reservoir need not be thermal and can be time-dependent. We
study the significance of energy mismatch between the system and the local
degrees of freedom of the reservoir which directly interact with the system. We
demonstrate that, in general, by increasing the system gap we may reduce this
energy mismatch, and consequently drive the system and the reservoir into
resonance, which can accelerate fidelity loss, irrespective of the thermal
properties or state of the reservoir. This implies that quantum error
suppression strategies based on increasing the gap are not uniformly
beneficial. Our speed limits also yield an elementary lower bound on the
relaxation time of spin systems.
