Abelian multi-cycle codes for single-shot error correction

We construct a family of quantum low-density parity-check codes

generalizing both the (abelian) two-block group-algebra codes and the

higher-dimensional quantum hypergraph-product (QHP) codes. Similarly

to QHP codes, the proposed codes have highly redundant sets of

low-weight stabilizer generators, which improves decoding accuracy in

a fault-tolerant regime and gives them single-shot properties. The

advantage of the new construction is that it gives shorter codes. We

derive simple expressions for the code dimension, establish bounds on

the distance, and explicitly construct some relatively short

codes. Circuit simulations for codes locally equivalent to

4-dimensional toric codes show an error-threshold close to 1.1%,

better than for toric or surface codes with a similar noise model.