On Injectivity of Quantum Finite Automata

Bell, Paul

Abstract

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.

Citation

Bell, P. (2021). On Injectivity of Quantum Finite Automata. Journal of Computer and System Sciences, 19-33. https://doi.org/10.1016/j.jcss.2021.05.002

• Acceptance Date: May 10, 2021
• Publication Date: Dec 1, 2021
• Publicly Available Date: Jun 30, 2021
• Journal: Journal of Computer and System Sciences
• Print ISSN: 0022-0000
• Publisher: Elsevier
• Pages: 19-33
• DOI: https://doi.org/10.1016/j.jcss.2021.05.002
• Public URL: https://keele-repository.worktribe.com/output/423617
• Publisher URL: https://www.sciencedirect.com/science/article/abs/pii/S0022000021000532#:~:text=On%20injectivity%20of%20quantum%20finite%20automata%E2%98%86&text=We%20study%20the%20injectivity%20problem,probability%20for%20each%20input%20word.