Topological Quantum Computation with Gapped Boundaries

TL;DR

This work develops a comprehensive framework for topological quantum computation using gapped boundaries in Kitaev's quantum double DW models. It combines exactly solvable lattice Hamiltonians with a rigorous algebraic (category-theoretic) model, showing how gapped boundaries and boundary defects realize a bordered topological order described by Lagrangian algebras and module categories, together with a bulk-to-boundary condensation formalism. The authors demonstrate a concrete surface-code implementation, outline topologically protected operations (tunneling, looping, braiding, and charge measurement), and show universal quantum computation using examples in $\mathfrak{D}(\mathbb{Z}_3)$ and $\mathfrak{D}(S_3)$, including a purely boundary-based universal gate set in the abelian $\mathfrak{D}(\mathbb{Z}_3)$ theory. The results provide a robust scheme for fault-tolerant quantum computation that leverages gapped boundaries to access computational power beyond bulk anyon braiding alone, with potential realizations in bilayer FQH systems and surface-code architectures.

Abstract

This paper studies fault-tolerant quantum computation with gapped boundaries. We first introduce gapped boundaries of Kitaev's quantum double models for Dijkgraaf-Witten theories using their Hamiltonian realizations. We classify the elementary excitations on the boundary, and systematically describe the bulk-to-boundary condensation procedure. We also provide a commuting Hamiltonian to realize defects between boundaries in any quantum double model. Next, we present the algebraic/categorical structure of gapped boundaries and boundary defects, which will be used to describe topologically protected operations and obtain quantum gates. To demonstrate a potential physical realization, we provide quantum circuits for surface codes that can perform all basic operations on gapped boundaries. Finally, we show how gapped boundaries of the abelian theory $\mathfrak{D}(\mathbb{Z}_3)$ can be used to perform universal quantum computation.

Figures (47)

• Figure 1.1: Pictorial summary of the Hamiltonian realization and algebraic model of gapped boundaries. In this picture, the hole $\mathfrak{h}$ is the inner boldfaced and multi-colored rectangle.
• Figure 2.1: Lattice for the Kitaev model. For simplicity of illustration and calculation, we use a square lattice, but in general, one can use an arbitrary lattice. If the group $G$ is nonabelian, it is necessary to define orientations on edges, as we have shown here. The edges $j$ and $j_1,...j_m$, used to obtain $A^g(v)$ and $B^h(p)$, are illustrated for this example of $v,p$.
• Figure 2.2: Illustration of Definitions \ref{['cilium-def']}-\ref{['ribbon-def']}. The direct lattice is shown as before, and the dual lattice is shown in dotted lines. $s=(v,p)$ is a cilium. $\tau$ is a dual triangle, and $\tau'$ is a direct triangle. $\rho = \rho_1 \rho_2$ is a composite ribbon, formed by gluing the last site of $\rho_1$ to the first site of $\rho_2$. $\rho$ is an open ribbon, and $\sigma$ is a closed ribbon.
• Figure 2.3: Lattice for the Kitaev model with boundary. For any fixed group $G$, there can be multiple ways to define projection operators at the boundary such that all terms in the new Hamiltonian still commute. These are studied in Section \ref{['sec:bd-hamiltonian']}.
• Figure 2.4: Example: defining the Hamiltonian (\ref{['eq:gapped-bds-hamiltonian']}), in the case of two holes on an infinite lattice. The new Hamiltonians $H^{(K_1,1)}_{(G,1)}$ ($H^{(K_2,1)}_{(G,1)}$) are applied to all vertices, plaquettes, and edges within the blue (red) shaded region, and all vertices (blue or red dots) on the boldfaced lines. Specifically, we note that the vertices on the boldfaced lines are part of the holes, while the edges are not (black lines vs. blue or red dots). The bulk Hamiltonian $H_{(G,1)}$ is applied to all other vertices and plaquettes (white region).

Theorems & Definitions (54)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Theorem 2.6
• proof
• Remark 2.7
• Remark 2.8
• Remark 2.9