## Abstract

Constructing quantum LDPC codes with a minimum distance that grows faster than a square root of the length has been a major challenge of the field. With this challenge in mind, we investigate constructions that come from high-dimensional expanders, in particular Ramanujan complexes. These naturally give rise to very unbalanced quantum error correcting codes that have a large X-distance but a much smaller Z-distance. However, together with a classical expander LDPC code and a tensoring method that generalises a construction of Hastings and also the Tillich-Zemor construction of quantum codes, we obtain quantum LDPC codes whose minimum distance exceeds the square root of the code length and whose dimension comes close to a square root of the code length. When the ingredient is a 3-dimensional Ramanujan complex, we show that its 2-systole behaves like a square of the log of the complex size, which results in an overall quantum code of minimum distance n{1/2} log n, and sets a new record for quantum LDPC codes. When we use a 2-dimensional Ramanujan complex, or the 2-skeleton of a 3-dimensional Ramanujan complex, we obtain a quantum LDPC code of minimum distance n{1/2} log{1/2}n. We then exploit the expansion properties of the complex to devise the first polynomial time algorithm that decodes above the square root barrier for quantum LDPC codes.

Original language | English |
---|---|

Title of host publication | Proceedings - 2020 IEEE 61st Annual Symposium on Foundations of Computer Science, FOCS 2020 |

Publisher | IEEE Computer Society |

Pages | 218-227 |

Number of pages | 10 |

ISBN (Electronic) | 9781728196213 |

DOIs | |

State | Published - Nov 2020 |

Event | 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020 - Virtual, Durham, United States Duration: 16 Nov 2020 → 19 Nov 2020 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
---|---|

Volume | 2020-November |

ISSN (Print) | 0272-5428 |

### Conference

Conference | 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020 |
---|---|

Country/Territory | United States |

City | Virtual, Durham |

Period | 16/11/20 → 19/11/20 |

### Bibliographical note

Publisher Copyright:© 2020 IEEE.

### Funding

SE is supported by NSF grant number DMS-1638352. TK Aknowledges support by ERC, and GZ is supported by ANR project ANR-18-CE47-0010 (QUDATA).

Funders | Funder number |
---|---|

ANR-18-CE47-0010 | |

National Science Foundation | DMS-1638352 |

Engineering Research Centers | |

Agence Nationale de la Recherche |

## Keywords

- Quantum code
- Ramanujan Complex
- expander