Quantum Double Aspects of Surface Code Models

This work takes a fresh view of Kitaev models, which are discrete lattice models for topological phases of matter. Such exotic many-body highly-entangled quantum states host anyonic quasiparticles which exhibit interesting exchange statistics, which, besides being of fundamental interest, can enable topological quantum information processing. The approach taken by the authors is in terms of Hopf algebras, which clarifies existing literature on Kitaev models. The novelty lies in the more abstract generalisations involving Hopf algebras with no group structure. Such cases were previously lacking descriptions of some lattice model features, which have the benefit over the categorical picture of staying grounded and concrete.
This work is primarily of theoretical interest; the authors observe that many of the facts taken for granted about surface codes, which are special cases of Kitaev models, break down or take a different form. Such general models could be used to implement Fibonacci anyons or similar non-Abelian anyons,which have no implementation on the regular Kitaev model, although the useful topological properties required for error  correction are contingent on the structure of the underlying Hopf algebra. In principle, this widens the range of implementable quantum error-correcting codes, quantum memories, and topological quantum computing models.

Alexander Cowtan, Shahn Majid

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