Quantum computation has emerged as a powerful computational medium of our time, having demonstrated the efficiency in solving the integer factoring and searching a database faster than any currently known classical computer algorithm. Adiabatic evolution of quantum systems have been studied as a potential means that physically realizes such quantum computation. Up to now, all the research on adiabatic quantum systems has dealt with polynomial time-bounded computation and little attention has been paid to, for example, adiabatic quantum systems consuming only constant memory space. Such quantum systems can be modeled in a form similar to quantum finite automata. This exposition dares to ask a bold question of how to make adiabatic quantum computation fit into the rapidly progressing framework of quantum automata theory. As our answer to this eminent but profound question, we first lay out a basic framework of adiabatic evolutionary quantum systems (AEQSs) with limited computational resources and then establish their close connection to quantum finite automata. We also explore fundamental structural properties of languages solved quickly by such adiabatic evolutionary quantum systems.