Topological quantum color code model on infinite lattice

TL;DR

This work develops a rigorous, infinite-lattice theory for the color code by employing a quasi-local $C^*$-algebra framework and cone-local Doplicher-Haag-Roberts analysis to classify topological excitations. It shows the anyon content matches the representation category $\,\mathsf{Rep}(D(\mathbb{Z}_2 \times \mathbb{Z}_2))$, equivalently two copies of the toric code, and proves Haag duality for the ground state. The authors construct explicit string operators, derive fusion and braiding data, and establish an equivalence of braided tensor categories with the Drinfeld double; they also prove Haag duality, providing a solid foundational basis for the topological phase in the thermodynamic limit. The results connect algebraic quantum field theory methods with topological quantum error correction, with implications for robust quantum memory and fault-tolerant quantum computation.

Abstract

The color code model is a crucial instance of a Calderbank--Shor--Steane (CSS)-type topological quantum error-correcting code, which notably supports transversal implementation of the full Clifford group. Its robustness against local noise is rooted in the structure of its topological excitations. From the perspective of quantum phases of matter, it is essential to understand these excitations in the thermodynamic limit. In this work, we analyze the color code model on an infinite lattice within the quasi-local $C^{*}$-algebra framework, using a cone-localized Doplicher-Haag-Roberts (DHR) analysis. We classify its irreducible anyon superselection sectors and construct explicit string operators that generate anyonic excitations from the ground state. We further examine the fusion and braiding properties of these excitations. Our results show that the topological order of the color code is described by $\mathsf{Rep}(D(\mathbb{Z}_2 \times \mathbb{Z}_2)) \simeq \mathsf{Rep}(D(\mathbb{Z}_2)) \boxtimes \mathsf{Rep}(D(\mathbb{Z}_2))$, which is equivalent to a double layer of the toric code and consistent with established analyses on finite lattices.

Figures (6)

• Figure 1: Two equivalent colored trivalent lattices used in color code model. Each vertex (represented by a black dot) hosts a physical qubit. The lattice is composed of hexagon (left) or square (right) faces, each face colored red, green, or blue, in such a way that no two adjacent faces share the same color. Each vertex is trivalent, meaning exactly three edges meet at each vertex.
• Figure 2: An illustration of three strings: $\gamma^{\mathtt{r}}$ (red), $\gamma^{\mathtt{g}}$ (green), and $\gamma^{\mathtt{b}}$ (blue), arranged from left to right. The blue string $\gamma^{\mathtt{b}}$ on the right is a closed string.
• Figure 3: A cone with apex $a$. The black bullet vertices are considered to be inside the cone.
• Figure 4: (Left) Three vertical half-infinite strings $\gamma_1$, $\gamma_2$, and $\gamma_3$ are shown, along with the closed winding strings $\zeta_1$, $\zeta_2$, and $\zeta_3$ that encircle their respective initial faces. The distinct colors of the strings are also indicated. (Right) Deformation of string operators from $S^{\mathtt{rx}}S^{\mathtt{bz}}$ to $S^{\mathtt{ry}}S^{\mathtt{gz}}$. The product of the matrix on the right of the bullet with the matrix at the bullet yields the matrix on the left. Note that the new string $\tilde{\gamma}_2^N$ has a new starting face $f_1^2$.
• Figure 5: The cone $\Lambda$ is to the left of $\Lambda'$.

Theorems & Definitions (68)

• Lemma 2.1: alicki2007statistical
• Lemma 2.2
• proof
• Proposition 2.3: naaijkens2017quantum
• Definition 2.4
• Theorem 2.5
• proof
• Definition 3.1: String
• Definition 3.2: String operator
• Lemma 3.3