I'm off to Oxford this Thursday to see David Deutsch's talk on 'Quantum constructor theory', users.ox.ac.uk/~ppox/general/seminar.html (4:30pm, 10 Merton St, for those interested in popping along). Deutsch, of course, is the inventor of quantum computing, an advocate of the many-worlds interpretation of quantum theory, and a chap with some interesting thoughts on the relationship between physics, mathematics, and computation. Deutsch gives some hints about what quantum constructor theory might be in this interview at Edge.org (www.edge.org/3rd_culture/deutsch/deutsch_index.html):
What sorts of computations do physical processes correspond to; which of these 'computations' can be arranged with what resources? And which sorts can't be arranged at all? What little we know about this new subject consists of a few broad limitations such as the finiteness of the speed of light. The theory of computability and complexity theory give us more detail on the quantum side. But a big technological question in my field at the moment is, can useful quantum computers actually be built? The basic laws of physics seem to permit them. We can design them in theory. We know what physical operations they would have to perform. But there is still room for doubt about whether one can build them out of actual atoms and make them work in a useful way. Some people are still pessimistic about that, but either way, that debate is not really a scientific one at the moment, because there is no scientific theory about what can and can't be built. Similar questions are raised by the whole range of nanotechnology that has been proposed in principle. So that's where a quantum constructor theory is needed.