On the state complexity of semi-quantum finite automata

TL;DR

This paper investigates the state complexity of semi-quantum finite automata, demonstrating they can recognize certain languages more efficiently than classical automata, even with minimal quantum resources, and explores trade-offs and simple language examples.

Contribution

It introduces new results on semi-quantum automata with minimal quantum components, showing they outperform classical automata and analyzing resource trade-offs with simple languages.

Findings

Semi-quantum automata recognize certain languages more efficiently.

Trade-offs between classical and quantum states are demonstrated.

Results hold for simple, well-studied languages and promise problems.

Abstract

Some of the most interesting and important results concerning quantum finite automata are those showing that they can recognize certain languages with (much) less resources than corresponding classical finite automata \cite{Amb98,Amb09,AmYa11,Ber05,Fre09,Mer00,Mer01,Mer02,Yak10,ZhgQiu112,Zhg12}. This paper shows three results of such a type that are stronger in some sense than other ones because (a) they deal with models of quantum automata with very little quantumness (so-called semi-quantum one- and two-way automata with one qubit memory only); (b) differences, even comparing with probabilistic classical automata, are bigger than expected; (c) a trade-off between the number of classical and quantum basis states needed is demonstrated in one case and (d) languages (or the promise problem) used to show main results are very simple and often explored ones in automata theory or in communication complexity, with seemingly little structure that could be utilized.