Generating logical magic states with the aid of non-Abelian topological order

In fault-tolerant quantum computing with the surface code, non-Clifford gates are crucial for universal computation. However, implementing these gates using methods like magic state distillation and code switching requires significant resources. In this work, we propose a new protocol that combines magic state preparation and code transformation to realize logical non-Clifford operations with the potential for fault tolerance. Our approach begins with a special logical state in the $\mathbb{Z}_4$ surface code. By applying a sequence of transformations, the system goes through different topological codes, including the non-Abelian $D_4$ quantum double model. This process ultimately produces a magic state encoded in the $\mathbb{Z}_{2}$ surface code. A logical $T$ gate can be implemented in the standard $\mathbb{Z}_2$ surface code by gate teleportation. In our analysis, we employ a framework where the topological codes are represented by their topological orders and all the transformations are considered as topological manipulations such as gauging symmetries and condensing anyons. This perspective is particularly useful for understanding transformations between topological codes.