Quantum Reversibility and a New Model of Quantum Automaton

This article is an attempt to generalize the classical theory of reversible computing, principally developed by Bennet [IBM J. Res. Develop., 17(1973)] and by Fredkin and Toffoli [Internat. J. Theoret. Phys., 21(1982)], to the quantum case. This is a fundamental step towards the construction of a quantum computer because a time efficient quantum computation is a reversible physical process. The paper is organized as follows. The first section reviews the classical theory of reversible computing. In the second section it is showed that the designs used in the classical framework to decrease the consumption of space cannot be generalized to the quantum case; it is also suggested that quantum computing is generally more demanding of space than classical computing. In the last section a new model of fully quantum and reversible automaton is proposed. The computational power of this automaton is at least equal to that of classical automata. Some conclusion are drawn in the last section. 1 Reversible computing: a glance The classical theory of reversible computing has been analyzed extensively ([2, 3]). The main results are the following: 1. Every irreversible computation f : x → f(x) can be effectively enclosed in a reversible computation F : (x, 0) → (f(x), x). F is reversible because its input (x, 0) is uniquely determined by its output (f(x), x). 2. The time and space required to compute F are linearly dependent on those required to compute f . These results imply that there are no effects on the complexity hierarchy when the time and space required to compute f are those required to compute its reversible realization F (reversible space equal irreversible space). To investigate whether is it possible to extend the above classical results to the quantum case, we first revisit the ideas behind classical reversible computing.