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arXiv:2608.29319

The Emptiness Problem for Quantum Finite Automata with Classical States

Jyun-Ao Lin, Patrick Totzke, Yun Chen Tsai, Di-De Yen

Published Aug 29, 2026• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2608.29319

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Let L>λ(A)L_{>\lambda}(\mathcal{A})L>λ​(A) and L≥λ(A)L_{\geq\lambda}(\mathcal{A})L≥λ​(A) be the languages recognized by {\em measure many 1-way quantum finite automata (MMQFA)} (or,{\em enhanced 1-way quantum finite automata(EQFA)}) A\mathcal{A}A with strict, resp. non-strict cut-point λ\lambdaλ. We consider the languages equivalence problem, showing that \begin{itemize} \item {both strict and non-strict languages equivalence are undecidable;} \item {to do this, we provide an additional proof of the undecidability of non-strict and strict emptiness of MMQFA(EQFA), and then reducing the languages equivalence problem to emptiness problem;} \item{Finally, some other Propositions derived from the above results are collected.} \end{itemize}
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