  We provide $poly\log$ sparse quantum codes for correcting the erasure channel
arbitrarily close to the capacity. Specifically, we provide $[[n, k, d]]$
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 $\langle S_1, S_2, ... S_{n-k} \rangle$ such that $|S_i|\leq
\log^{2+\zeta}(n)$ for all $i$ and for any $\zeta > 0$ with high probability.
In this work we show that the result of Delfosse et al. is tight: one can
construct capacity approaching codes with weight almost $O(1)$.

Quantum Physics