  In this paper the notion of quantum finite one-counter automata (QF1CA) is
introduced. Introduction of the notion is similar to that of the 2-way quantum
finite state automata by A.Kondacs and J.Watrous. The well-formedness
conditions for the automata are specified ensuring unitarity of evolution. A
special kind of QF1CA, called simple, that satisfies the well-formedness
conditions is introduced. That allows to specify rules for constructing such
automata more naturally and simpler than in general case. Possible models of
language recognition by QF1CA are considered. The recognition of some languages
by QF1CA is shown and compared with recognition by probabilistic counterparts.
