Quantum Büchi Automata
Qisheng Wang, Mingsheng Ying
TL;DR
This work defines and analyzes quantum Büchi automata (QBAs) for infinite words, extending measure-once quantum finite automata (MO-QFAs) to capture long-term quantum behavior with Büchi-like acceptance. It formalizes the acceptance function f_A^{QBA}(w) and studies four threshold semantics, showing that only four ω-language classes—$\\mathbb{L}_{>0}(QBA)$, $\mathbb{L}_{>\lambda}(QBA)$, $\mathbb{L}_{=1}(QBA)$, and $\mathbb{L}_{\geq \lambda}(QBA)$—are substantially distinct, with precise inclusion and non-inclusion relations. The paper proves pumping lemmas, establishes closure properties (e.g., union closure, but not intersection or complementation under certain thresholds), and demonstrates decidability of emptiness and intersection emptiness across semantics. It also clarifies the relationship with classical ω-languages, showing QBAs can recognize languages beyond ω-regular and even ω-context-free, while some classical ω-languages remain unrecognizable by QBAs under the studied semantics. Overall, QBAs provide a rigorous quantum framework for modeling long-run quantum behavior, with implications for quantum model checking and verification, and open avenues for minimization and equivalence results.
Abstract
Quantum finite automata (QFAs) have been extensively studied in the literature. In this paper, we define and systematically study quantum Büchi automata (QBAs) over infinite words to model the long-term behavior of quantum systems, which extend QFAs. We introduce the classes of $ω$-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 $ω$-languages recognized by QBAs (out of uncountably infinite). The relationship between classical $ω$-languages and QBAs is clarified using our pumping lemmas. We also find an $ω$-language recognized by QBAs under the almost sure semantics, which is not $ω$-context-free.