Research report: State complexity of operations on two-way quantum finite automata

TL;DR

This paper investigates the size complexity of two-way quantum finite automata for various language operations, providing upper bounds, examples of non-regular languages accepted efficiently, and discussing the tightness of these bounds.

Contribution

It introduces new upper bounds for state complexity of operations on 2qfa's and presents non-regular languages accepted with linear time, highlighting areas for bound improvements.

Findings

Upper bounds for state complexity of intersection, union, reversal, and catenation.

Existence of non-regular languages accepted by 2qfa's with linear time and one-sided error.

Bounds for simulating one-way deterministic automata by reversible automata are not tight.

Abstract

This paper deals with the size complexity of minimal {\it two-way quantum finite automata} (2qfa's) necessary for operations to perform on all inputs of each fixed length. Such a complexity measure, known as state complexity of operations, is useful in measuring how much information is necessary to convert languages. We focus on intersection, union, reversals, and catenation operations and show some upper bounds of state complexity of operations on 2qfa's. Also, we present a number of non-regular languages and prove that these languages can be accepted by 2qfa's with one-sided error probabilities within linear time. Notably, these examples show that our bounds obtained for these operations are not tight, and therefore worth improving. We give an instance to show that the upper bound of the state number for the simulation of one-way deterministic finite automata by two-way reversible finite automata is not tight in general.