On injectivity of quantum finite automata

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