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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.Comment: 32 page
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