We utilize a concatenation scheme to construct new families of quantum error correction codes that include the Bacon-Shor codes. We show that our scheme can lead to asymptotically good quantum codes while Bacon-Shor codes cannot. Further, the concatenation scheme allows us to derive quantum LDPC codes of distance $\Omega(N^{2/3}/\log\log N)$ which can improve Hastings's recent result [arXiv:2102.10030] by a polylogarithmic factor. Moreover, assisted by the Evra-Kaufman-Z\'emor distance balancing construction, our concatenation scheme can yield quantum LDPC codes with non-vanishing code rates and better minimum distance upper bound than the hypergraph product quantum LDPC codes. Finally, we derive a family of fast encodable and decodable quantum concatenated codes with parameters ${Q}=[[N,\Omega(\sqrt{N}),\Omega( \sqrt{N})]]$ and they also belong to the Bacon-Shor codes. We show that ${Q}$ can be encoded very efficiently by circuits of size $O(N)$ and depth $O(\sqrt{N})$, and can correct any adversarial error of weight up to half the minimum distance bound in $O(\sqrt{N})$ time. To the best of our knowledge, they are the most powerful quantum codes for correcting so many adversarial errors in sublinear time by far.

翻译：我们用一种配方办法来构建包含培根-Shor 代码的量子错误校正代码的新组合。 我们显示我们的配方办法可以导致无模好的量子代码,而培根-Shor 代码则无法。 此外, 配方办法允许我们得出量子LDPC的距离代码$\Omega(N ⁇ 2/3}/\log\log\log N),这可以用一个多元对数系数来改进黑斯廷最近的结果[arXiv:2102.10030]。 此外, 在Evra-Kaufman- ⁇ N'emor 距离平衡构造的帮助下, 我们的配方方案可以产生量子LDPC的量子代码, 与非加速的代码相比, 更短的最小的距离。 最后, 我们的组合是快速可辨识和可解的量子聚合代码, 参数$[N, rqrt{N},\Oega (rent r) 最强的比重的值(right) 和最短的长度的值。