Topological quantum color code model on infinite lattice

• URL: http://arxiv.org/abs/2601.12409v1
• Date: Sun, 18 Jan 2026 13:53:18 GMT
• Title: Topological quantum color code model on infinite lattice
• Abstract summary: The color code model is a crucial instance of a Calderbank--Shor--Steane (CSS)-type topological quantum error-correcting code.<n>From the perspective of quantum phases of matter, it is essential to understand these excitations in the thermodynamic limit.<n>We show that the order of the color code is described by $mathsfRep(D(mathbbZ_2)) simeq mathsfRep(D(mathbbZ_2)$, which is equivalent to a double layer of the toric code
• Score: 1.715412829836753
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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