Universal Quantum Computation with the $S_3$ Quantum Double: A Pedagogical Exposition

TL;DR

This paper explains how non-Abelian $S_3$ topological order can be used for universal quantum computation, detailing the braiding, measurement, and lattice model implementations to guide near-term quantum platforms.

Contribution

It provides a pedagogical review of $S_3$ topological order's role in universal quantum computation, including explicit models and operators for practical realization.

Findings

$S_3$ topological order supports universal quantum gates.

Explicit lattice model and ribbon operators are constructed.

Demonstrates practical pathways for near-term quantum platforms.

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.