The Quadratic State Cost of Classical Simulation of One-Way Quantum Finite Automata
This resolves a long-standing open problem in computational complexity theory by precisely quantifying the state cost of simulating quantum automata with classical probabilistic models, which is foundational for understanding quantum-classical trade-offs in automata theory.
The paper determined the worst-case state complexity of exactly simulating one-way quantum finite automata (1gQFA) with probabilistic finite automata (PFA), showing that every n-state 1gQFA can be simulated by a PFA with O(n^2) states, and for every n≥2, there exists an n-state 1gQFA requiring at least n^2-1 states in any equivalent PFA, establishing a tight Θ(n^2) bound.
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 \(Î(n^2)\).