So a universal computer could compute the result of itself being sucked into a black hole and having contact with the interior (singularity or whatever lies there). — Marchesk

We've got a pretty good idea of what it is like to fall into a black hole, so I don't see why you could not in principle do this - any finite system can be simulated to arbitrary accuracy by finite means.

Although, this seems unfair. I'm guessing the idea of simulating precludes the thing doing the simulating, otherwise we have regress and self-referential issues. I might as well ask if the computer can simulate itself itself being introduced to a really strong magnet, or whatever would disrupt a QC. — Marchesk

A quantum computer has got to be isolated from its environment. This can never be perfectly achieved, but there seems to be ways around the technical issues involving extra qubits and error correction. It's thus perhaps not too much of a stretch to think of the QC as a black box, with which there is no interaction save for setting inputs and reading the output.

Haven't thought this through, but this feature of the QC might solve any perceived problems with regression etc. The QC is actually in its own Hilbert space!

Maybe I'm missing something, but it seems to me that if the quantum computer can contain 2^1000 bits of information, and it is in the observable (from Earth) universe, then the observable-from-Earth universe contains more than 2^1000 bits of information, and hence cannot be described in 2^412 bits.

What am I missing?

It is undoubtedly counter-intuitive, particularly as there is a great deal of work that indicates that for a classical Turing machine, operating under classical physics this could not be possible. I think the culprit is classical physics.

As I mentioned, the Bekenstein bound cannot apply to what is going on in a quantum computer. This implies that the state of a quantum computer, even its output in general, is not an observable.