Chiu Fan Bowen Lo, Anasuya Lyons, Ruben Verresen, Ashvin Vishwanath, Nathanan Tantivasadakarn (Feb 24 2025).

Abstract: Non-Abelian topological order (TO) enables topologically protected quantum computation with its anyonic quasiparticles. Recently, TO with $S_3$ gauge symmetry was identified as a sweet spot -- simple enough to emerge from finite-depth adaptive circuits yet powerful enough to support a universal topological gate-set. In these notes, we review how anyon braiding and measurement in $S_3$ TO are primitives for topological quantum computation and we explicitly demonstrate universality. These topological operations are made concrete in the $S_3$ quantum double lattice model, aided by the introduction of a generalized ribbon operator. This provides a roadmap for near-term quantum platforms.

Arxiv: https://arxiv.org/abs/2502.14974