Well, this isn't just going to keep inert matter cold, this will do massive amounts of computation, so it makes sense that it would need to dump a lot of heat.
It is, IMO, a bit dubious whether or not anything is truly flowing in a superconductor at constant current. Electrons don't have identity, so the 'constant flow of electrons' can be rephrased as 'the physical system isn't changing'... and the degree to which you can tell that there are electrons moving about is also the degree to which the superconductor isn't truly zero-resistance.
Of course it isn't actually constant -- there are multiple electrons, and you can tell that the electron field is quantized. But the degree to which that is visible, is the exact degree to which the superconductor nevertheless doesn't superconduct!