Fault-tolerant non-Clifford gates within stabilizer codes via non-Abelian topological order

We propose a protocol to implement fault-tolerant logical gates between stabilizer codes by entangling into a non-Abelian topological order. Generalizing a proposal put forth by Davydova et al. involving the twisted quantum double of Z_2^3, we show that gauging the topological symmetry of symmetry-enriched topological orders (SET) provides a framework that generates a large class of non-Clifford logical gates between qudit surface codes. Among other more general examples, we concretely detail a protocol that utilizes the quantum double of S_3 to generate a controlled-charge conjugation gate between a qubit and qutrit surface code, as well as an analogous protocol that utilizes the quantum double of D_4 to generate a controlled-SWAP gate between a three qubit surface codes. The preparation of non-Abelian states as well as logical state injection between the Abelian and non-Abelian codes are executed via finite-depth quantum circuits with transversal measurement and local non-Clifford gates. This work serves as a step towards classifying the computational power of non-Abelian quantum phases beyond the paradigm of anyon braiding on near-term quantum devices.