# On Injectivity of Quantum Finite Automata

## TL;DR

This paper investigates the injectivity problem of quantum finite automata, proving its undecidability for certain configurations and exploring related mathematical properties and problems.

## Contribution

It establishes the undecidability of the injectivity problem for MO-QFA with specific state and matrix conditions, using advanced mathematical tools and reductions.

## Key findings

- Injectivity problem is undecidable for 8-state MO-QFA with rational matrices.
- Undecidability persists with rational initial vectors but requires more states.
- New results on rational polynomial packing functions may have broader implications.

## 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.