Upper Bounds on the Rate of Low Density Stabilizer Codes for the Quantum Erasure Channel

Using combinatorial arguments, we determine an upper bound on achievable rates of stabilizer codes used over the quantum erasure channel. This allows us to recover the no-cloning bound on the capacity of the quantum erasure channel, R is below 1-2p, for stabilizer codes: we also derive an improved upper bound of the form : R is below 1-2p-D(p) with a function D(p) that stays positive for 0 < p < 1/2 and for any family of stabilizer codes whose generators have weights bounded from above by a constant - low density stabilizer codes. We obtain an application to percolation theory for a family of self-dual tilings of the hyperbolic plane. We associate a family of low density stabilizer codes with appropriate finite quotients of these tilings. We then relate the probability of percolation to the probability of a decoding error for these codes on the quantum erasure channel. The application of our upper bound on achievable rates of low density stabilizer codes gives rise to an upper bound on the critical probability for these tilings.