Quantum double aspects of surface code models

We revisit the Kitaev model for fault tolerant quantum computing on a square lattice with underlying double $D(G)$ symmetry, where $G$ is finite group. provide projection operators its quasiparticles content as irreducible representations of and combine this $D(G)$-bimodule properties open ribbon excitation spaces $L(s_0,s_1)$ to show how ribbons can be used teleport information between their endpoints $s_0,s_1$. give self-contained account that builds earlier work but emphasises applications surface code theory, including gates $D(S_3)$. theory reduces simpler toric codes in case $D( \Bbb Z_n)\cong C\Bbb Z_n^2$, braiding. In other direction, we our constructions generalise $D(H)$ models based finite-dimensional Hopf algebra $H$, site actions partial results equivariance even when not semisimple.