The Quadratic State Cost of Classical Simulation of One-Way Quantum Finite Automata
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908947726729216 |
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| author | Chen, Zeyu Wu, Junde |
| author_facet | Chen, Zeyu Wu, Junde |
| contents | 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 \(Θ(n^2)\). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_07058 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Quadratic State Cost of Classical Simulation of One-Way Quantum Finite Automata Chen, Zeyu Wu, Junde Formal Languages and Automata Theory 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 \(Θ(n^2)\). |
| title | The Quadratic State Cost of Classical Simulation of One-Way Quantum Finite Automata |
| topic | Formal Languages and Automata Theory |
| url | https://arxiv.org/abs/2604.07058 |