Beyond conventional mathematical knowledge

Among the many ramifications of quantum computation for apparently distant fields of study are its implications for the notion of mathematical proof. When we think about mathematical proof we usually mean a physical record that represents “a sequence of propositions each of which is either an axiom or follows from earlier propositions in the sequence by the given rules of inference”. Quantum computation forced us to leave that definition behind. Henceforward, a proof must be regarded as a process — the computation itself — for we must accept that in future, quantum computers will prove theorems by methods that neither a human brain nor any other arbiter will ever be able to check step-by-step, since if the `sequence of propositions’ corresponding to such a proof were printed out, the paper would fill the observable universe many times over.