Spread of Entanglement in Generalized Kicked Ising Chain

We investigate the dynamics of entanglement in a generalized version of the kicked Ising chain, extending the model from the standard qubit case (local dimension $q=2$) to higher local dimensions ($q > 2$). We identify the existence of ''dual-unitary'' points where the model's space-time duality allows for exact analytical solutions. Our analysis reveals that while a few unique dual-unitary points exist analytically for systems with local dimensions $q=3$ and $q=4$, such points do not exist for $q \ge 5$ due to the lack of a unique kicking strength that satisfies the required matrix element conditions. Utilizing the transfer matrix method and a replica trick specifically adapted for higher dimensions, we derive exact expressions for the growth of entanglement entropy in the $q=3$ (kicked Potts-type) model starting from a class of solvable initial states. Our results demonstrate that at the dual-unitary point, both R\'enyi and von Neumann entanglement entropies grow linearly with time until reaching a maximum value determined by the subsystem size.