  We consider continuous structures which are obtained from finite dimensional
Hilbert spaces over $\mathbb{C}$ by adding some unitary operators. Quantum
automata and circuits are naturally interpretable in such structures. We
consider appropriate algorithmic problems concerning continuous theories of
natural classes of these structures. We connect them with the topic of
approximations by metric groups. This paper extends and corrects the paper A.
Ivanov, "Continuous structures of quantum circuits", arXiv: 1406.4635.
Replacing version contains a new section (Section 4.4).
