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On injectivity of quantum finite automata

Bell, PC and Hirvensalo, M (2021) On injectivity of quantum finite automata. Journal of Computer and System Sciences, 122. pp. 19-33. ISSN 0022-0000

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We consider notions of freeness and ambiguity for the acceptance probability of Moore-Crutchfield Measure Once Quantum Finite Automata (MO-QFA). We study the injectivity problem of determining if the acceptance probability function of a MO-QFA is injective over all input words, i.e., giving a distinct probability for each input word. We show that the injectivity problem is undecidable for 8 state MO-QFA, even when all unitary matrices and the projection matrix are rational and the initial state vector is real algebraic. We also show undecidability of this problem when the initial vector is rational, although with a huge increase in the number of states. We utilize properties of quaternions, free rotation groups, representations of tuples of rationals as linear sums of radicals and a reduction of the mixed modification of Post's correspondence problem, as well as a new result on rational polynomial packing functions which may be of independent interest.

Item Type: Article
Uncontrolled Keywords: 0802 Computation Theory and Mathematics, 0805 Distributed Computing, 0806 Information Systems
Subjects: Q Science > QA Mathematics
Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Divisions: Computer Science & Mathematics
Publisher: Elsevier
Date Deposited: 01 Jul 2021 10:30
Last Modified: 28 May 2022 00:50
DOI or ID number: 10.1016/j.jcss.2021.05.002
URI: https://researchonline.ljmu.ac.uk/id/eprint/15201
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