Improved quantum hypergraph-product LDPC codes

TL;DR

This paper introduces improved quantum hypergraph-product LDPC codes, including rotated lattice toric codes and algebraic constructions that significantly increase code rates, enhancing quantum error correction efficiency.

Contribution

It presents novel techniques for enhancing quantum hypergraph-product codes, notably rotated lattices for toric codes and algebraic methods to nearly quadruple code rates.

Findings

Rotated lattice toric codes with specific parameters

Block length and distance formulas for minimal codes

Algebraic constructions nearly quadrupling code rate

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.