Abstract
We propose a generalization of the color codes based on finite groups $G$. For non-abelian groups, the resulting model supports non-abelian anyonic quasiparticles and topological order. We examine the properties of these models such as their relationship to Kitaev quantum double models, quasiparticle spectrum, and boundary structure.
Highlights
Topological codes are a promising avenue to achieve robust quantum memories [1] or implement fault-tolerant quantum computation [2]
We will define a quantum double model for a group G on an particular directed square lattice (Fig. 4), with qudits of dimension |G| placed at every edge, noting that transforming the qudit on an edge by g → g−1 is equivalent to reversing the direction of the edge
We have defined a generalization of the color codes to finite group G, and explored many of their basic properties
Summary
Topological codes are a promising avenue to achieve robust quantum memories [1] or implement fault-tolerant quantum computation [2]. The color codes [6,7] are a family of topological codes with Abelian anyonic excitations They may be used to perform computation by code deformation, but are notable for having a large class of transversal gates [6], giving rise to high fault-tolerance thresholds [8]. This is motivated in analogy to the generalization of the toric code to the quantum double models [2] These generalized color codes support non-Abelian anyons for nonAbelian groups G and so may in general be used for topological quantum computation by braiding these quasiparticles.
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