What Exactly Are Quantum Computations? Classical and Quantum Turing Machines

Abstract

After Feynman’s pioneering work, the abstract mathematical model for quantum computers has been often represented in terms of the notion of quantum Turing machine (the quantum counterpart of the classical notion of Turing machine). But what exactly are quantum Turing machines? So far, the literature has not provided a rigorous “institutional” concept of quantum Turing machine. Some definitions seem to be based on a kind of “imitation” of the classical definition of Turing machine, by referring to a tape (where the symbols are written) and to a moving head (which changes its position on the tape). These concepts, however, seem to be hardly applicable to physical quantum computers, where quantum uncertainties should also concern the behavior of moving heads. We introduce the more abstract notion of quantum computing machine, whose computations can be linguistically described by formulas of quantum computational logics. To what extent can abstract quantum computing machines be simulated by classical Turing machines? Does quantum computation give rise to possible violations of the Church-Turing thesis? These hard questions did not find, so far, a definite answer.