1-way quantum finite automata: strengths, weaknesses and generalizations

TL;DR

This paper analyzes the capabilities and limitations of 1-way quantum finite automata, comparing them with classical automata, demonstrating their space efficiency, and exploring potential generalizations for increased power.

Contribution

It provides a detailed comparison of 1-way QFAs with classical automata, introduces a space-efficient QFA construction, and investigates simplified models for enhanced quantum automata power.

Findings

QFAs match reversible automata when high accuracy is required

Quantum automata can be exponentially smaller than classical ones

Exploration of generalized models for more powerful quantum automata

Abstract

We study 1-way quantum finite automata (QFAs). First, we compare them with their classical counterparts. We show that, if an automaton is required to give the correct answer with a large probability (over 0.98), then the power of 1-way QFAs is equal to the power of 1-way reversible automata. However, quantum automata giving the correct answer with smaller probabilities are more powerful than reversible automata. Second, we show that 1-way QFAs can be very space-efficient. Namely, we construct a 1-way QFA which is exponentially smaller than any equivalent classical (even randomized) finite automaton. This construction may be useful for design of other space-efficient quantum algorithms. Third, we consider several generalizations of 1-way QFAs. Here, our goal is to find a model which is more powerful than 1-way QFAs keeping the quantum part as simple as possible.