Anyon Permutations in Quantum Double Models through Constant-depth Circuits

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.