Quantum double aspects of surface code models

We revisit the Kitaev model for fault tolerant quantum computing on a square lattice with underlying quantum double D(G) symmetry, where G is a finite group. We provide projection operators for its quasiparticles content as irreducible representations of D(G) and combine this with D(G)-bimodule properties of open ribbon excitation spaces L(s0, s1) to show how open ribbons can be used to teleport information between their endpoints s0, s1. We give a self-contained account that builds on earlier work but emphasises applications to quantum computing as surface code theory, including gates on D(S3). We show how the theory reduces to a simpler theory for toric codes in the case of D(Zn)≅CZn, including toric ribbon operators and their braiding. In the other direction, we show how our constructions generalise to D(H) models based on a finite-dimensional Hopf algebra H, including site actions of D(H) and partial results on ribbon equivariance even when the Hopf algebra is not semisimple.

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