Abstract
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 Ω(N2/3/loglogN) which can improve Hastings’s recent
result [arXiv:2102.10030] by a polylogarithmic factor. Moreover, assisted by the Evra-Kaufman-
Zémor 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 quan-
tum concatenated codes with parameters Q = [[N,Ω(√N),Ω(√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(√N), and can correct any adversarial error of weight up to half the minimum distance
bound in O(√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.
A minimum distance bound for quasi-nD-cyclic codes
