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.

中文翻译：

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二元量子有限自动机和亚对数空间量子图灵机运行时间的下界

Ambainis 和 Watrous 定义的具有量子态和经典态的双向有限自动机（2QCFA）是一种量子计算模型，其量子部分极其有限；然而，正如他们所展示的，2QCFA 出奇的强大：只有一个量子比特的 2QCFA 可以识别语言 $L_{pal}=\{w \in \{a,b\}^*:w \text{ 是一个回文}\}$ 在预期时间 $2^{O(n)}$ 内具有有界误差，输入长度为 $n$。我们证明他们的结果基本上无法改进：2QCFA（任何大小）无法识别 $L_{pal}$ 在预期时间 $2^{o(n)}$ 内的有界误差。据我们所知，这是第一个可以通过 2QCFA 在指数时间但不能在亚指数时间以有界误差识别的语言示例。而且，我们证明了在空间 $o(\log n)$ 和预期时间 $2^{n^{1-\Omega(1)}}$ 中运行的量子图灵机 (QTM) 无法识别有界的 $L_{pal}$错误; 同样，这是同类产品中的第一个下限。更一般地说，我们建立了任何 2QCFA 或 $o(\log n)$-space QTM 的运行时间的下限，这些 QTM 根据 $L$ 的自然“硬度度量”识别任何语言 $L$。这使我们能够展示一大类语言，对于这些语言，我们对任何此类 2QCFA 或 QTM 识别器的运行时间具有渐近匹配的下限和上限。