Protocols for Creating Anyons and Defects via Gauging

TL;DR

The paper develops a unifying, symmetry-guided framework to create and manipulate non-Abelian anyons and symmetry defects via gauging. By formalizing both the 2D group-gauging map $\mathsf{KW}_G$ and its dual $\mathsf{KW}_{\mathrm{Rep}(G)}$, it provides explicit PEPO representations and 1D circuits to generate precursor excitations, push them through gauging, and ungauge back to realize ribbon operators in $D(G)$. It demonstrates concrete constructions for $\mathbb{Z}_3$ toric code and $D(S_3)$, and extends to twisted quantum doubles by using SPT inputs, deriving efficient ribbon circuits for doubled semion, $D(D_4)$, and $D(Q_8)$. The approach yields unitary or finite-depth adaptive ribbons suitable for near-term devices and clarifies how symmetry membranes and gauging encode and transport the internal states of non-Abelian anyons and defects. Overall, it offers a practical, broadly applicable toolkit for ribbon operator construction across a wide class of group-based and twisted topological orders.

Abstract

Creating and manipulating anyons and symmetry defects in topological phases, especially those with a non-Abelian character, constitutes a primitive for topological quantum computation. We provide a physical protocol for implementing the ribbon operators of non-Abelian anyons and symmetry defects. We utilize dualities, in particular the Kramers-Wannier or gauging map, which have previously been used to construct topologically ordered ground states by relating them to simpler states. In this work, ribbon operators are implemented by applying a gauging procedure to a lower-dimensional region of such states. This protocol uses sequential unitary circuits or, in certain cases, constant-depth adaptive circuits. We showcase this for anyons and defects in the $\mathbb{Z}_3$ toric code and $S_3$ quantum double. The general applicability of our method is demonstrated by deriving unitary expressions for ribbon operators of various (twisted) quantum doubles.

Figures (11)

• Figure 1: Taking advantage of dualities: To create a given anyon or defect in a topological order, we apply a duality mapping in a finite region, "unzipping" to the simpler dual theory. Then, we create a "precursor" excitation, typically with a simple finite-depth circuit. Finally, we "zip" back up to the original theory, leaving behind the desired excitation.
• Figure 2: Creating a charge conjugation defect in $\mathbb{Z}_3$ toric code: a) A charge conjugation symmetry membrane reduces to a boundary circuit of controlled-$\mathcal{X}$ gates by using variables of the pre-gauged theory (See Eq. \ref{['eq:defect-circuit']}). b) To create charge conjugation defects: Map back to the necessary vertices using the 1D gauging map (affected degrees of freedom are highlighted in purple), then apply the derived boundary action (support highlighted in orange). Finally, map back to edges. c) The action of the measurement-based and unitary realizations of $\mathsf{KW}_{\mathbb{Z}_3}^{\text{1D}}$ differ by half a lattice translation. Here, solid circles mark edges, open circles mark vertices.
• Figure 3: Anyons and defects from the $G$-gauging map: a) A charge anyon ribbon maps to a pair of charges on top of the paramagnet. b) The automorphism symmetry membrane on edges pushes through the gauging map the vertices, now decorated with a boundary circuit (orange lines). If the initial state is a paramagnet, the right-hand membrane disappears, leaving only the boundary circuit.
• Figure S1: $\mathsf{KW}_G$projective entangled-pair operators: The tensors making up the PEPO expression for $\mathsf{KW}_G$ are pictured. We also specify the orientations we will use going forwards.
• Figure S2: Rep(G) gauging map PEPO tensors: The tensors making up the PEPO expression for $\mathsf{KW}_{Rep(G)}$ are pictured. The structure of the vertex and edge tensors have been reversed relative to the $\mathsf{KW}_G$ tensors.