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Anyon Permutations in Quantum Double Models through Constant-depth Circuits

Yabo Li, Zijian Song

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.

Anyon Permutations in Quantum Double Models through Constant-depth Circuits

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 quantum double correspond to compositions of three classes of one-dimensional self-dualities, including gauging of certain subgroups of , stacking with symmetry-protected topological phases, and outer automorphisms of the group . We construct circuits realizing the first class by employing self-dual unitary gauging maps, and present transversal circuits for the latter two classes.
Paper Structure (15 sections, 89 equations, 3 figures, 1 table)

This paper contains 15 sections, 89 equations, 3 figures, 1 table.

Table of Contents

  1. Structure of the anyon permutation group
  2. Review of the electric-magnetic duality in the toric code model
  3. Anyon permutation: class I
  4. Gauging map and self-duality
  5. $G = N \rtimes Q$-quantum double model
  6. Operator maps
  7. Hamiltonian maps
  8. Ribbon operator maps
  9. Example: generalized electric-magnetic duality in $D_4$ quantum double
  10. Hamiltonian maps
  11. Ribbon operators maps
  12. Example: Anyon permutation from twisted gauging in $\mathbb{Z}_4 \times \mathbb{Z}_4$ quantum double model
  13. Hamiltonian maps
  14. Ribbon operators maps
  15. Anyon permutation: class II

Figures (3)

  • Figure 1: (a) Illustration of the electric--magnetic duality circuit. After applying the gauging maps on all three colors of triangles, the corresponding domain wall is effectively swept across the entire system. (b) The lattice deformation from the original triangular lattice to a hexagonal lattice. (c) The global controlled-conjugation circuit applied after $\Gamma_1$ and $\Gamma_0$ for the $D(G)$ quantum double. A single arrow denotes a controlled-conjugation gate. The local gates are applied in the order: green (solid), blue (dashed), and yellow (dash-dotted). The $CC^{(2)}$ circuit after $\Gamma_2$ is given by on-site controlled-conjugation gates.
  • Figure 2: The deformed 3-colorable triangular lattice with periodic boundary condition. Qubits (Qudits) are placed on the edges of this hexagonal lattice. Generalized $X$ terms are defined on each vertex, while generalized $Z$ terms are defined on each plaquette.
  • Figure 3: Orientation of qudits on the three triangles. Each vertex hosts two qudits: the bottom (blue) qudit is associated with the first $\mathbb{Z}_4$ group labeled by $(n,q)$, while the top (red) qudit is associated with the second $\mathbb{Z}_4$ group labeled by $(n',q')$.