AI 中文总结
本文研究带经典态的量子有限自动机（QFACs）的空问题，证明其可定义语言包含正则语言与量子有限自动机（QFA）可定义语言，将其空问题归约为QFA与有限自动机的语言交问题，并针对平面QFACs、单循环QFACs分别给出相关结论与验证方法。
AI 中文摘要
带经典态的量子有限自动机（QFACs）是一类基于有限量子操作字母表的非确定有限自动机。我们研究该模型在有限字上的表达能力及对应的空问题。我们证明正则语言与量子有限自动机（QFAs）可定义的语言不可比，且两者均被QFAC可定义的语言严格包含。我们表明QFAC的空问题可归约为QFA与有限自动机的语言交的空问题，该交问题在严格阈值下可判定，但非严格情形下不可判定。此外，我们考虑平面QFACs（底层自动机无嵌套循环的限制情形）的该问题，并将其与高维轨道问题（动力系统中的长期开放挑战）关联。最后，我们提出一种可靠且半完备的证据搜索过程，用于验证单循环QFACs的非空性，这类QFACs具有足够的表达能力来表示一些著名量子算法，如Grover搜索和量子随机游走。
英文摘要
Quantum Finite Automata with Classical states (QFACs) are nondeterministic finite automata over a finite alphabet of quantum operations. We study expressiveness of this model on finite words and the corresponding emptiness problem. We show that regular languages are incomparable with those definable by Quantum Finite Automata (QFAs) and that both are strictly subsumed by QFAC-definable languages. We show that the emptiness problem for a QFAC can be reduced to the emptiness of the language intersection of a QFA and a finite automaton. This intersection is known to be decidable for strict thresholds but undecidable for non-strict cases. Furthermore, we consider the problem for flat QFACs, a restriction where the underlying automata contain no nested loops, and relate it to the higher-dimensional orbit problem, a long-standing open challenge in dynamical systems. Finally, we propose a sound and semi-complete witness searching procedure to verify the non-emptiness of one-loop QFACs, which are sufficiently expressive to represent some prominent quantum algorithms, such as Grover's search and quantum random walks.
