On the Simulation Cost of Quantum Finite Automata

2026-05-11

AI Summary

This work investigates the classical exact probabilistic simulation cost of quantum finite automata under strict cutpoint conditions to rigorously quantify their quantum advantage. By introducing a "prepare-and-test" framework and integrating analyses of real operator degrees of freedom, finite sign-rank matrix theory, and Forster’s spectral method, the study establishes simulation cost as a natural metric for quantum advantage for the first time. The main contributions include proving that the simulation cost for one-way automata with $c$ classical states and a $q$-dimensional quantum register is $\Theta(cq^2)$, and that the worst-case cost for $n$-dimensional one-query quantum automata is $\Theta(n^2)$. These results delineate a clear hierarchy of quantum advantage and uncover deep connections between simulation cost, sign rank, and spectral techniques.

Abstract

This paper identifies exact probabilistic simulation cost as the natural quantitative measure of quantum advantage for finite automata under strict cutpoints. It gives sharp simulation laws for two representative models. A one-way finite automaton with $c$ classical states and a $q$-dimensional quantum register has exact probabilistic simulation cost $Θ(cq^2)$, while an $n$-dimensional measure-once one-way quantum finite automaton has worst-case cost $Θ(n^2)$. The proofs develop a prepare--test framework, in which prefixes generate the relevant real operator degrees of freedom and suffixes convert them into strict-cutpoint tests. The same obstruction is recast through finite sign-rank matrices, clarifying the role of Forster's spectral method. Placed beside the surrounding two-way separations, these results give a clean hierarchy of finite-automata quantum advantage.

Problem

• quantum finite automata
• simulation cost
• quantum advantage
• strict cutpoints
• probabilistic simulation

Innovation

• quantum finite automata
• exact probabilistic simulation cost
• prepare–test framework
• sign-rank
• quantum advantage