Quantum finite automata (QFAs) have been extensively studied in the literature. In this paper, we define and systematically study quantum B\"uchi automata (QBAs) over infinite words to model the long-term behavior of quantum systems, which extend QFAs. We introduce the classes of $\omega$-languages recognized by QBAs in probable, almost sure, strict and non-strict threshold semantics. Several pumping lemmas and closure properties for QBAs are proved. Some decision problems for QBAs are investigated. In particular, we show that there are surprisingly only at most four substantially different classes of $\omega$-languages recognized by QBAs (out of uncountably infinite). The relationship between classical $\omega$-languages and QBAs is clarified using our pumping lemmas. We also find an $\omega$-language recognized by QBAs under the almost sure semantics, which is not $\omega$-context-free.

翻译：量子有限自动机（QFAs）已在文献中得到广泛研究。本文定义并系统研究了无限词上的量子Büchi自动机（QBAs），以模拟量子系统的长期行为，这是对QFAs的扩展。我们引入了QBAs在概率、几乎必然、严格与非严格阈值语义下识别的$\omega$-语言类。证明了QBAs的若干泵引理与封闭性质。研究了QBAs的一些判定问题。特别地，我们证明QBAs识别的$\omega$-语言类（在不可数无限中）最多仅有四个本质不同的类。利用我们的泵引理阐明了经典$\omega$-语言与QBAs之间的关系。我们还发现了一个在几乎必然语义下由QBAs识别但非$\omega$-上下文无关的$\omega$-语言。