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arXiv:2604.07058

The State Cost of Classical Simulation of One-Way General Quantum Finite Automata

Zeyu Chen, Junde Wu

Published Apr 8, 2026• 0 citations
arXivHTMLPDF

Abstract

Under strict cutpoints, probabilistic finite automata (PFAs) and one-way general quantum finite automata (1gQFAs) recognize the same stochastic languages, shifting the theoretical focus to the state cost required for a classical PFA to simulate a 1gQFA. For an $n$-state ($n\geq 2$) 1gQFA, the state cost upper bound of classical simulation was previously known to be $n^2+3$, while the lower bound remained an open problem widely conjectured to be quadratic. After establishing a well-defined notion of classical simulation, we improve the existing state cost upper bound from $n^2+3$ to $n^2+1$. Subsequently, we introduce the concepts of shattering and dynamic shattering, which are used to determine the memory required for a PFA to recognize a language. Using these techniques, we prove that the state cost lower bound of the simulation reaches $n^2+1$ over a four-letter alphabet. With the upper and lower bounds thus matching, the problem is fully resolved.

DOI
10.48550/arxiv.2604.07058

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We construct zero-error quantum finite automata (QFAs) for promise problems which cannot be solved by bounded-error probabilistic finite automata (PFAs). Here is a summary of our results: - There is a promise problem solvable by an exact two-way QFA in exponential expected time, but not by any bounded-error sublogarithmic space probabilistic Turing machine (PTM). - There is a promise problem solvable by an exact two-way QFA in quadratic expected time, but not by any bounded-error $ o(\log \log n) $-space PTMs in polynomial expected time. The same problem can be solvable by a one-way Las Vegas (or exact two-way) QFA with quantum head in linear (expected) time. - There is a promise problem solvable by a Las Vegas realtime QFA, but not by any bounded-error realtime PFA. The same problem can be solvable by an exact two-way QFA in linear expected time but not by any exact two-way PFA. - There is a family of promise problems such that each promise problem can be solvable by a two-state exact realtime QFAs, but, there is no such bound on the number of states of realtime bounded-error PFAs solving the members this family. Our results imply that there exist zero-error quantum computational devices with a \emph{single qubit} of memory that cannot be simulated by any finite memory classical computational model. This provides a computational perspective on results regarding ontological theories of quantum mechanics \cite{Hardy04}, \cite{Montina08}. As a consequence we find that classical automata based simulation models \cite{Kleinmann11}, \cite{Blasiak13} are not sufficiently powerful to simulate quantum contextuality. We conclude by highlighting the interplay between results from automata models and their application to developing a general framework for quantum contextuality.
Abstract. We prove that several variants of the quantum finite automaton (QFA) model are strictly superior to their probabilistic counterparts, in terms of both language recognition power and succinctness, in the onesided unbounded error setting. This is the first case where one-way QFA models have been shown to outperform the corresponding probabilistic automata in both these respects. We give a characterization of the class of languages recognized by these QFA’s with cutpoint 0, demonstrating that it is equal to the class of exclusive stochastic languages. We prove several closure properties of the related classes. 1