Quantum counter automata

TL;DR

This paper demonstrates that quantum real-time one-counter automata can recognize certain languages more effectively than classical probabilistic automata, establishing quantum superiority in this computational model.

Contribution

It introduces quantum real-time one-counter automata and proves their superiority over probabilistic models in recognizing specific languages with perfect soundness.

Findings

Quantum rtQ1CAs recognize non-context-free languages.

Quantum models outperform classical probabilistic automata.

Introduces generalized quantum automata models.

Abstract

The question of whether quantum real-time one-counter automata (rtQ1CAs) can outperform their probabilistic counterparts has been open for more than a decade. We provide an affirmative answer to this question, by demonstrating a non-context-free language that can be recognized with perfect soundness by a rtQ1CA. This is the first demonstration of the superiority of a quantum model to the corresponding classical one in the real-time case with an error bound less than 1. We also introduce a generalization of the rtQ1CA, the quantum one-way one-counter automaton (1Q1CA), and show that they too are superior to the corresponding family of probabilistic machines. For this purpose, we provide general definitions of these models that reflect the modern approach to the definition of quantum finite automata, and point out some problems with previous results. We identify several remaining open problems.