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

2026-04-08

AI Summary

This work investigates the state complexity required for classical probabilistic finite automata (PFAs) to exactly simulate one-way generalized quantum finite automata (1gQFAs) recognizing strict cut-point languages. By employing a mixed-state linearization technique together with an alphabet-preserving construction, the authors show that any n-state 1gQFA can be exactly simulated by a PFA with O(n²) states. Conversely, leveraging the prepare-and-measure framework and a VC-dimension-based lower bound argument, they establish the existence of languages for which any exact classical simulation necessitates at least n²−1 states. This paper thus establishes, for the first time, a tight Θ(n²) bound on the state cost of classically simulating 1gQFAs exactly, revealing a quadratic state-complexity gap between quantum and classical models for this task.

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 \(Θ(n^2)\).

Problem

• quantum finite automata
• probabilistic simulation
• state complexity
• strict-cutpoint languages
• exact simulation

Innovation

• quantum finite automata
• state complexity
• probabilistic simulation
• strict-cutpoint languages
• VC dimension