Turing’s thinking machine and ’t Hooft’s principle of superposition of states

Abstract: In his 1950 paper  cite{Turing1950}, Turing proposed the notion of a thinking machine, namely a machine that can think. But a thinking machine has to follow a certain law of physics, provided it is realized physically. In this paper, we show that Turing’s thinking machine necessrily obeys ’t Hooft’s principle of superposition of states, which was presented by ’t Hooft  cite{Hooft2016} in 2016 beyond the usual one as described by Dirac  cite{Dirac1958} in the conventional quantum mechanics. Precisely, Turing’s thinking machine must be a quantum machine, while ’t Hooft’s principle characterizes its thinking behavior in a probabilistic way.