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

Abstract

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 $⟨S_1,S_2,...S_{n-k}⟩$ such that $|S_i|\le {\log }^{2+\hat{I}\P }(n)$ for all $i$ and for any $\hat{I}\P >0$ 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 $O(1)$.