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

TL;DR

This paper precisely characterizes the quadratic state complexity involved in exactly simulating one-way quantum finite automata with probabilistic finite automata, establishing both upper and lower bounds.

Contribution

It provides the first exact quadratic bounds on the state cost for simulating one-way quantum automata with probabilistic automata, using explicit constructions and complexity arguments.

Findings

Every n-state 1gQFA can be exactly simulated by an O(n^2)-state PFA.

There exist n-state 1gQFA requiring at least n^2-1 states for equivalent PFA.

The worst-case state cost for simulation is Theta(n^2).

Abstract

Generalized finite automata (GFAs), probabilistic finite automata (PFAs), and one-way general quantum finite automata (1gQFA) recognize the same strict-cutpoint languages, but the state complexity of exact probabilistic simulation has remained unclear. This paper determines that worst-case cost exactly: every \(n\)-state 1gQFA admits exact strict-cutpoint simulation by a one-way PFA with \(O(n^2)\) states, via the standard \(n^2\)-dimensional mixed-state linearization together with an explicit alphabet-preserving construction that converts each \(k\)-state GFA into a one-way PFA with at most \(2k+6\) states; conversely, for every \(n\ge 2\), there exists an \(n\)-state 1gQFA for which every equivalent one-way PFA requires at least \(n^2-1\) states, obtained from a prepare--test construction and a Vapnik--Chervonenkis dimension argument. Hence the worst-case probabilistic state cost of exact strict-cutpoint simulation is \(\Theta(n^2)\).