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

## TL;DR

This paper introduces polylogarithmic sparse quantum codes that nearly achieve the capacity of the noisy quantum erasure channel, with low-weight generators and high probability success.

## Contribution

It constructs capacity-approaching quantum codes with polylogarithmic weight generators, demonstrating the tightness of previous bounds and advancing quantum error correction.

## Key findings

- Codes correct erasures close to capacity at erasure probability 0.33
- Generator weights are bounded by polylogarithmic functions of n
- The results show tightness of previous bounds on code weight

## 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 $\langle S_1, S_2, ... S_{n-k} \rangle$ 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 $O(1)$.