In this paper we introduce classically time-controlled quantum automata or
CTQA, which is a reasonable modification of Moore-Crutchfield quantum finite
automata that uses time-dependent evolution and a "scheduler" defining how long
each Hamiltonian will run. Surprisingly enough, time-dependent evolution
provides a significant change in the computational power of quantum automata
with respect to a discrete quantum model. Indeed, we show that if a scheduler
is not computationally restricted, then a CTQA can decide the Halting problem.
In order to unearth the computational capabilities of CTQAs we study the case
of a computationally restricted scheduler. In particular we showed that
depending on the type of restriction imposed on the scheduler, a CTQA can (i)
recognize non-regular languages with cut-point, even in the presence of
Karp-Lipton advice, and (ii) recognize non-regular languages with
bounded-error. Furthermore, we study the closure of concatenation and union of
languages by introducing a new model of Moore-Crutchfield quantum finite
automata with a rotating tape head. CTQA presents itself as a new model of
computation that provides a different approach to a formal study of "classical
control, quantum data" schemes in quantum computing.Comment: Long revisited version of LNCS 11324:266-278, 2018 (TPNC 2018