Anyon Permutations in Quantum Double Models through Constant-depth Circuits

TL;DR

This paper constructs explicit constant-depth local unitary circuits to realize anyon permutations in Kitaev's quantum double models, linking topological order symmetries with one-dimensional self-dualities and their transformations.

Contribution

It introduces a holographic framework connecting anyon permutations to self-dualities, and provides explicit circuit constructions for these transformations in quantum double models.

Findings

Explicit constant-depth circuits for anyon permutations

Connection between topological symmetries and 1D self-dualities

Circuit constructions for gauging and automorphisms

Abstract

We provide explicit constant-depth local unitary circuits that realize general anyon permutations in Kitaev's quantum double models. This construction can be naturally understood through a correspondence between anyon permutation symmetries of two-dimensional topological orders and self-dualities in one-dimensional systems, where local gates implement self-duality transformations on the boundaries of microscopic regions. From this holographic perspective, general anyon permutations in the $D(G)$ quantum double correspond to compositions of three classes of one-dimensional self-dualities, including gauging of certain subgroups of $G$, stacking with $G$ symmetry-protected topological phases, and outer automorphisms of the group $G$. We construct circuits realizing the first class by employing self-dual unitary gauging maps, and present transversal circuits for the latter two classes.