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 \(\mathcal S\) of computable sets. When \(S\in \mathcal S\) 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 \(\mathcal S\). The analysis suggests that showing the provability of a statement is different from writing up the proof of the statement.
Unable to display preview. Download preview PDF.
- 1.Aigner, M., Schmidt, V.A.: Good proofs are proofs that make us wiser: interview with Yu. I. Manin. The Berlin Intelligencer, 16–19 (1998), http://www.ega-math.narod.ru/Math/Manin.html
- 2.Berndt, B.C.: Ramanujan’s Notebooks, Part V. Springer, Heidelberg (2005)Google Scholar
- 3.Buzek, V., Hillery, M.: Quantum cloning. Physics World 14(11), 25–29 (2001)Google Scholar
- 4.Calude, C.S., Calude, E., Marcus, S.: Proving and programming. In: Calude, C.S. (ed.) Randomness & Complexity, from Leibniz to Chaitin, pp. 310–321. World Scientific, Singapore (2007)Google Scholar
- 6.Hiai, F., Yanagi, K.: Hilbert Spaces and Linear Operators. Makino-Shoten (1995) (in Japanese)Google Scholar
- 8.Mahan, G.D.: Many-Particle Physics, 3rd edn. Kluwer Academic/Plenum Publishers, New York (2010)Google Scholar
- 9.Matijasevič, Y.V.: Hilbert’s Tenth Problem. The MIT Press, Cambridge (1993)Google Scholar
- 10.Penrose, R., Hameroff, S.: Consciousness in the universe: Neuroscience, quantum space-time geometry and Orch OR theory. Journal of Cosmology 14 (2011), http://journalofcosmology.com/Consciousness160.html