[3] A way of understanding the quantum Turing machine (QTM) is that it generalizes the classical Turing machine (TM) in the same way that the quantum finite automaton (QFA) generalizes the deterministic finite automaton (DFA).
In essence, the internal states of a classical TM are replaced by pure or mixed states in a Hilbert space; the transition function is replaced by a collection of unitary matrices that map the Hilbert space to itself.
In 1980 and 1982, physicist Paul Benioff published articles[5][6] that first described a quantum mechanical model of Turing machines.
A 1985 article written by Oxford University physicist David Deutsch further developed the idea of quantum computers by suggesting that quantum gates could function in a similar fashion to traditional digital computing binary logic gates.
[4] Iriyama, Ohya, and Volovich have developed a model of a linear quantum Turing machine (LQTM).