arXiv 2003.098772020-03cs.CC

双向量子有限自动机和亚空间量子图灵机运行时间的下界

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

Zachary Remscrim

备注：: In ITCS 2021

摘要：

由Ambainis和Watrous定义的具有量子和经典状态的双向有限自动机（2 QCFA）是一种量子计算模型，其量子部分非常有限;然而，正如他们所展示的那样，2 QCFA是惊人的强大：只有一个量子位的2 QCFA可以在预期时间内识别语言$L_{pal}=\{w \in \{a,b\}^*:w \text{ is a palindrome}\}$，并且在输入长度为$n$的$2^{O(n)}$上有界误差。我们证明，他们的结果基本上不能得到改善后：一个2 QCFA（任何大小）不能识别$L_{pal}$有界误差在预期的时间$2^{o(n)}$。据我们所知，这是第一个可以在指数时间而不是亚指数时间内通过2 QCFA识别有界误差的语言的例子。此外，我们证明了量子图灵机（QTM）运行在空间$o(\log n)$和预期时间$2^{n^{1-Ω(1)}}$不能识别$L_{pal}$有界误差，再次，这是第一个下界的同类。更一般地说，我们建立了一个运行时间的下限的任何2 QCFA或$o(\log n)$ -空间QTM，承认任何语言$L$在一个自然的“硬度措施”的$L$。这使我们能够展示一个大的语言家族，我们有渐近匹配的下限和上限的运行时间的任何这样的2 QCFA或QTM识别。

英文摘要：

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 $2^{n^{1-Ω(1)}}$ 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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