Improved quantum hypergraph-product LDPC codes

Abstract: We suggest several techniques to improve the toric codes and the finite-rate generalized toric codes (quantum hypergraph-product codes) recently introduced by Tillich and Z\'emor. For the usual toric codes, we introduce the rotated lattices specified by two integer-valued periodicity vectors. These codes include the checkerboard codes, and the family of minimal single-qubit-encoding toric codes with block length $n=t^2+(t+1)^2$ and distance $d=2t+1$, $t=1,2,...$. We also suggest several related algebraic constructions which nearly quadruple the rate of the existing hypergraph-product codes.