The two-way finite automaton with quantum and classical states (2QCFA), defined by Ambainis and Watrous, is a model of quantum computation whose quantum part is extremely limited; however, as they showed, 2QCFA are surprisingly powerful: a 2QCFA with only a single-qubit can recognize the language $L_{pal}=\{w \in \{a,b\}^*:w \text{ is a palindrome}\}$ with bounded-error in expected exponential time. We prove that their result essentially cannot be improved upon: a 2QCFA (of any finite size) cannot recognize $L_{pal}$ with bounded-error in expected time $2^{o(n)}$, on inputs of length $n$. To our knowledge, this is the first example of a language that can be recognized with bounded-error by a 2QCFA in exponential time but not in subexponential time. A key tool in our result is a generalization to 2QCFA of a technical lemma that was used by Dwork and Stockmeyer to prove a lower bound on the expected running time of any two-way probabilistic finite automaton that recognizes a non-regular language with bounded-error.

Furthermore, we prove strong lower bounds on the expected running time of any 2QCFA that recognizes a group word problem with bounded-error. In a recent paper, we showed that 2QCFA can recognize, with bounded-error, a broad class of group word problems in expected exponential time, and a more narrow class of group word problems in expected polynomial time. As a consequence, we can now exhibit a large family of natural languages that can be recognized with bounded-error by a 2QCFA in expected exponential time, but not in expected subexponential time. Moreover, we obtain significant progress towards a precise classification of those group word problems that can be recognized with bounded-error in expected polynomial time by a 2QCFA.

We also consider the one-way measure-once quantum finite automaton (1QFA), defined by Moore and Crutchfield, as well as a natural generalization to one-way measure-once finite automata with quantum and classical states (1QCFA). We precisely classify those groups whose word problem may be recognized with positive one-sided error (for both the bounded-error and unbounded-error cases) by a 1QFA or 1QCFA with any particular number of quantum states and any particular number of classical states; we also obtain partial results in the negative one-sided error case. As an immediate corollary, we show that allowing a 1QFA or 1QCFA to have even a single additional quantum or classical state enlarges the class of languages that may be recognized with positive one-sided error (of either type).