Universal Circuit Set with $S_3$ Quantum Double

In this paper, we address several key problems toward the ultimate goal of constructing an anyon quantum computer, capable of both universal quantum computation (UQC) and quantum error correction (QEC). We develop all the essential protocols to provide a comprehensive framework for harnessing the computational power of a specific non-Abelian topological order; that is the quantum double model $\mathcal{D}(S_3)$, based on the symmetric group $S_3$. Furthermore, we construct a family of error correction codes (ECCs) to encode each physical degree of freedom of $\mathcal{D}(S_3)$. This encoding suppresses the density and propagation of anyon errors, which is a necessary condition for continuous error correction. Under the hypothesis that a decoding scheme exists to leverage the intrinsic noise resilience of the $\mathcal{D}(S_3)$ topological order, our proposal enables a fault-tolerant, universal, gate set and offers a promising pathway toward realizing large-scale quantum computation.