Topos-theoretic formalism of quantum artificial intelligence

Abstract: Based on ’t Hooft’s principle of superposition of states beyond the usual one as described by Dirac in the conventional quantum mechanics, we present a topos-theoretic formalism of quantum artificial intelligence. At first, according to Turing’s test, we interpret an artificial intelligence (AI) system as a physical system described by a topos (which is a new physical theory builded by Isham {\it et al} in 2008). Secondly, by using measure theory, we construct a topos-theoretic model for a classical AI system such as deep learning; while we construct a topos-theoretic model for a quantum AI system by operator theory. Finally, we give the topos-theoretic description of a quantum neural network.