## Abstract

This work provides the first explicit and non-random family of [[N,K,D]] LDPC quantum codes which encode K in Theta left({N{frac {4}{5}}}right) logical qubits with distance D in Omega left({N{frac {3}{5}}}right). The family is constructed by amalgamating classical codes and Ramanujan graphs via an operation called balanced product. Recently, Hastings-Haah-O'Donnell and Panteleev-Kalachev were the first to show that there exist families of LDPC quantum codes which break the mathrm {polylog}(N)sqrt {N} distance barrier. However, their constructions are based on probabilistic arguments which only guarantee the code parameters with high probability whereas our bounds hold unconditionally. Further, balanced products allow for non-abelian twisting of the check matrices, leading to a construction of LDPC quantum codes that can be shown to have Kin Theta (N) and that we conjecture to have linear distance Din Theta (N).

Original language | English |
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Pages (from-to) | 6653 - 6674 |

Number of pages | 22 |

Journal | IEEE Transactions on Information Theory |

Volume | 67 |

Issue number | 10 |

Early online date | 19 Jul 2021 |

DOIs | |

Publication status | Published - 19 Jul 2021 |

### Bibliographical note

Funding Information:Manuscript received January 13, 2021; revised May 17, 2021; accepted June 16, 2021. Date of publication July 19, 2021; date of current version September 15, 2021. The work of Nikolas P. Breuckmann was supported in part by the University College London Quantum Science and Technology Institute (UCLQ) Fellowship and in part by the Engineering and Physical Sciences Research Council (EPSRC) Prosperity Partnership in Quantum Software for Simulation and Modelling under Grant EP/S005021/1. (Corresponding author: Nikolas P. Breuckmann.) Nikolas P. Breuckmann is with the Department of Physics and Astronomy and the Department of Computer Science, University College London, London WC1E 6BT, U.K. (e-mail: nikobreu@gmail.com).

Publisher Copyright:

© 1963-2012 IEEE.

## Keywords

- Quantum codes
- quantum error-correction
- quantum fault-tolerance