Complete characterization of finite commutative rings admitting quantum fractional revival

Complete the characterization of finite commutative rings whose unitary Cayley graphs admit quantum fractional revival, including cases such as the unitary Cayley graph of the ring \(\mathbb{Z}_{30}\) to which the approach developed in the paper does not apply.

Background

The paper characterizes quantum fractional revival for unitary Cayley graphs over finite local rings, proving that it occurs precisely for the rings $\mathbb{F}_2$, $\mathbb{Z}_4$, and $\mathbb{Z}_2[x]/(x^2)$. For finite commutative rings, which decompose as products of finite local rings, the authors establish several necessary conditions and construct families that admit quantum fractional revival, but they do not obtain a complete classification.

The authors explicitly identify $G_{\mathbb{Z}_{30}}$ as a case not covered by their method. Resolving this case and the broader classification problem requires further techniques capable of handling the remaining products of finite local rings.