Finite state and finite stop quantum languages

We propose the concept of finite stop quantum automata (ftqa) based on Hilbert space and compare it with the finite state quantum automata (fsqa) proposed by Moore and Crutchfield (Theoretical Computer Science 237(1-2), 2000, 275-306). The languages accepted by fsqa form a proper subset of the languages accepted by ftqa. In addition, the fsqa form an infinite hierarchy of language inclusion with respect to the dimensionality of unitary matrices. We introduce complex-valued acceptance degrees and two types of finite stop quantum automata based on them: the invariant ftqa (icftq) and the variant ftqa (vcftq). The languages accepted by icftq form a proper subset of the languages accepted by vcftq. In addition, the icftq form an infinite hierarchy of language inclusion with respect to the dimensionality of unitary matrices. In this way, we establish two proper inclusion relations L (fsqa) subset of L (ftqa) and L (icftq) subset of L (vcftq), where the symbol L means languages, and two infinite language hierarchies L-n (fsqa) subset of Ln+1 (fsqa), L-n (icftq) subset of Ln+1 (icftq).