  We demonstrate that substantial progress can be achieved in the study of the
phase structure of 4-dimensional compact QED by a joint use of hybrid Monte
Carlo and multicanonical algorithms, through an efficient parallel
implementation. This is borne out by the observation of considerable speedup of
tunnelling between the metastable states, close to the phase transition, on the
Wilson line. We estimate that the creation of adequate samples (with order 100
flip-flops) becomes a matter of half a year's runtime at 2 Gflops sustained
performance for lattices of size up to 24^4.
