非RMM正则语言的指数型更简洁量子识别
Exponentially more concise quantum recognition of non-RMM regular languages
摘要：
我们表明，有量子设备，接受所有的正规语言，是指数比确定性有限自动机（DFA）更简洁。为此，我们提出了一种新的计算模型{ \it one-way quantum finite automata}（1 QFA），即{ \it one-way quantum finite automata with classical states}（1 QFAC），它自然地扩展了纯测量的1 QFA和DFA，并且其状态复杂度是由这两种模型限定的.本文的原始贡献如下。首先，我们证明了1 QFAC所接受的有界错误的语言集是由所有的正则语言组成的。其次，我们证明了1 QFAC最多是指数比DFA更简洁。第三，我们表明，以前的界限是紧的家庭的正规语言，不承认测量一次（RMO），测量多（RMM）和多字母1 QFA。更具体地说，我们展示了正则语言$L^0(m)$的$m$素数，使得：（i）$L^0(m)$不能被测量一次，测量多次和多字母1 QFA识别;（ii）接受$L^0(m)$的最小DFA具有$O(m)$状态;（iii）存在具有恒定经典状态和$O(\log(m))$量子基的1 QFAC，接受$L^0(m)$。第四，我们给出了一个多项式时间算法来确定是否任何两个1 QFAC是等价的。最后，我们证明了1 QFAC的状态最小化在EXPSPACE中是可判定的。最后，我们提出了一些开放的问题的文件。
英文摘要：
We show that there are quantum devices that accept all regular languages and that are exponentially more concise than deterministic finite automata (DFA). For this purpose, we introduce a new computing model of {\it one-way quantum finite automata} (1QFA), namely, {\it one-way quantum finite automata together with classical states} (1QFAC), which extends naturally both measure-only 1QFA and DFA and whose state complexity is upper-bounded by both. The original contributions of the paper are the following. First, we show that the set of languages accepted by 1QFAC with bounded error consists precisely of all regular languages. Second, we prove that 1QFAC are at most exponentially more concise than DFA. Third, we show that the previous bound is tight for families of regular languages that are not recognized by measure-once (RMO), measure-many (RMM) and multi-letter 1QFA. % More concretely we exhibit regular languages $L^0(m)$ for $m$ prime such that: (i) $L^0(m)$ cannot be recognized by measure-once, measure-many and multi-letter 1QFA; (ii) the minimal DFA that accepts $L^0(m)$ has $O(m)$ states; (iii) there is a 1QFAC with constant classical states and $O(\log(m))$ quantum basis that accepts $L^0(m)$. Fourth, we give a polynomial-time algorithm for determining whether any two 1QFAC are equivalent. Finally, we show that state minimization of 1QFAC is decidable within EXPSPACE. We conclude the paper by posing some open problems.
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。
