Quantum computer is discussed in papers both as a mathematical model and as its technical realization. Only the mathematical model is meant here and in comparison with that of a standard computer, namely a Turing machine. The quantum Turing machine is an abstract model equivalent to the quantum circuit and can represent all features of quantum computer without entanglement.

Another way to generalize the Turing machine to the quantum computer is by replacing all bits or cells of a Turing tape with “quantum bits” or “qubits”. A qubit is equivalently representable by a unit sphere and a point chosen on its surface. Then if any bit is an elementary binary choice between two disjunctive options usually designated by “0” and “1”, any qubit is a choice between a continuum of disjunctive options as many as the points of the surface of the unit ball. That visualization allows of highlighting the fundamental difference between the Turing machine and the quantum computer: the choice of an element of an uncountable set necessarily requiring the axiom of choice.