Lower Bounds on the Running Time of Two-Way Quantum Finite Automata and Sublogarithmic-Space Quantum Turing Machines

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 time $2^{O(n)}$, on inputs of length $n$. We prove that their result essentially cannot be improved upon: a 2QCFA (of any size) cannot recognize $L{pal}$ with bounded error in expected time $2^{o(n)}$. 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. Moreover, we prove that a quantum Turing machine (QTM) running in space $o(\log n)$ and expected time $2^{{n{1-\Omega(1)}}$} cannot recognize $L_{pal}$ with bounded error; again, this is the first lower bound of its kind. Far more generally, we establish a lower bound on the running time of any 2QCFA or $o(\log n)$-space QTM that recognizes any language $L$ in terms of a natural "hardness measure" of $L$. This allows us to exhibit a large family of languages for which we have asymptotically matching lower and upper bounds on the running time of any such 2QCFA or QTM recognizer.