  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 $L(s_0,s_1)$ to show how open
ribbons can be used to teleport information between their endpoints $s_0,s_1$.
We give a self-contained account that builds on earlier work but emphasises
applications to quantum computing as surface code theory, including gates on
$D(S_3)$. We show how the theory reduces to a simpler theory for toric codes in
the case of $D( \Bbb Z_n)\cong \Bbb C\Bbb Z_n^2$, 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.
