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

TL;DR

This paper introduces a quantum computing approach to represent and potentially generate theorems of formal systems through spectral methods, highlighting differences between provability and proof construction.

Contribution

It presents a spectral representation for computably enumerable sets using quantum computers and explores its implications for formal system theorems.

Findings

Quantum computer produces all theorems and proofs for sets in class F.

Spectral representation conjectured to apply to all computably enumerable sets.

Provability may be distinguishable from proof generation in quantum systems.

Abstract

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.