Some algebraic properties of measure-once two-way quantum finite automata

Quantum finite automata (QFA) can be divided into four kinds depend upon the head-directions and the measure times. They are measure-once one way QFA (MO-1QFA) introduced by Moore and Crutchfield (Theor Comput Sci 237: 275–306, 2000); measure-many one way QFA (MM-1QFA) and measure-many two-way QFA (MM-2QFA) introduced by Kondacs and Watrous (Proceedings of the 38th IEEE annual symposium on 433 foundations of computer science, 66–75, 1997); and measure-once two-way QFA (MO-2QFA) which were not given until now. The purpose of this work is mainly to discuss one kind of QFA, which is called MO-2QFA for brief. First of all, the definition of MO-2QFA is given and the conditions for preserving unitary properties are shown. Then, we analysis the basic algebraic properties of the class of languages which can be recognized by MO-2QFA, such as the union, intersection, complement and reversal operations. As well, we consider the catenation operation on the class of quantum languages recognized by MO-2QFA is closed in the generalized conditions.

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