  Effective cavities can be optically-induced in atomic media and employed to
strengthen optical nonlinearities. Here we study the integration of induced
cavities with a photonic quantum gate based on Rydberg blockade. Accounting for
loss in the atomic medium, we calculate the corresponding finesse and gate
infidelity. Our analysis shows that the conventional limits imposed by the
blockade optical depth are mitigated by the induced cavity in long media, thus
establishing the total optical depth of the medium as a complementary resource.
