Quantum erasure-correcting codes and percolation on regular tilings of the hyperbolic plane

We are interested in percolation for a family of self-dual tilings of the hyperbolic plane. We achieve an upper bound on the critical probability for these tilings by taking appropriate finite quotients and associating them with a family of quantum CSS codes. We then relate the probability of percolation to the probability of a decoding error for these codes on the quantum erasure channel.