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

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