Quantum B\"uchi Automata

TL;DR

This paper introduces quantum B"uchi automata (QBAs) to model infinite quantum system behaviors, exploring their language classes, properties, and decision problems, revealing a surprisingly limited number of language classes.

Contribution

It systematically defines QBAs, analyzes their recognized language classes, proves key properties, and clarifies their relation to classical $oldsymbol{ extomega}$-languages, including the discovery of a non-$oldsymbol{ extomega}$-context-free language.

Findings

At most four distinct classes of $oldsymbol{ extomega}$-languages recognized by QBAs.

Established pumping lemmas and closure properties for QBAs.

Identified an $oldsymbol{ extomega}$-language recognized by QBAs that is not $oldsymbol{ extomega}$-context-free.

Abstract

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.