Polylog-LDPC Capacity Achieving Codes for the Noisy Quantum Erasure Channel

We provide polylog sparse quantum codes for correcting the Erasure channel arbitrarily close to the capacity. Specifically, we provide the [[n, k, d]] quantum stabilizer codes that correct for the erasure channel arbitrarily close to the capacity if the erasure probability is at least 0.33 and with a generating set < S-1, S-2,... Sn-k > such that vertical bar S-i vertical bar <= log(2+zeta) (n) for all i and for any zeta > 0 with high probability. In this paper, we show that the result of Delfosse et al. is tight: one can construct capacity approaching codes with weight almost O(1).