Topological quantum computation assisted by phase transitions

In this paper, we explore topological quantum computation augmented by subphases and phase transitions. We commence by investigating the anyon tunneling map, denoted as $\varphi$, between subphases of the quantum double model $\mathcal{D}(G)$ for any arbitrary finite group $G$. Subsequently, we delve into the relationship between $\varphi$ and the Floquet code, and extend the Abelian Floquet code to encompass non-abelian cases. We conclude by demonstrating how phase transitions in both the temporal and spatial directions can enhance the diversity of topological gates for general topological orders described by modular tensor categories.