Spectral Representation of Some Computably Enumerable Sets With an Application to Quantum Provability

We propose a new type of quantum computer which is used to prove a spectral representation for a class F of computable sets. When S in F codes the theorems of a formal system, the quantum computer produces through measurement all theorems and proofs of the formal system. We conjecture that the spectral representation is valid for all computably enumerable sets. The conjecture implies that the theorems of a general formal system, like Peano Arithmetic or ZFC, can be produced through measurement; however, it is unlikely that the quantum computer can produce the proofs as well, as in the particular case of F. The analysis suggests that showing the provability of a statement is different from writing up the proof of the statement.