From Generalization of Bacon-Shor Codes to High Performance Quantum LDPC Codes

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.