Quantum finite automata with pairwise-commuting unitary operators

The structure of the sets of languages recognized by Quantum Finite Automata is characterized in terms of the quotient sets defined by the mappings of the input semigroup into the set of associated unitary operators. The case when unitary operators are pairwise-commuting is analyzed. The sets of languages recognized by the 1-qubit QFA models with the single measurement at the final instant under the supposition that the associated unitary operators are rotations of the Bloch sphere around the fixed coordinate axis are characterized. The structure of the set of all finitely generated commutative semigroups of unitary operators acting in the 2-dimensional complex space is investigated.