## Abstract

In this work we introduce a new notion of expansion in higher dimensions that is stronger than the well studied cosystolic expansion notion, and is termed Collective-cosystolic expansion. We show that tensoring two cosystolic expanders yields a new cosystolic expander, assuming one of the complexes in the product, is not only cosystolic expander, but rather a collective cosystolic expander. We then show that the well known bounded degree cosystolic expanders, the Ramanujan complexes are, in fact, collective cosystolic expanders. This enables us to construct new bounded degree cosystolic expanders, by tensoring of Ramanujan complexes. Using our new constructed bounded degree cosystolic expanders we construct explicit quantum LDPC codes of distance ?n logk n for any k, improving a recent result of Evra et. al. [FOCS, 2020], and setting a new record for distance of explicit quantum LDPC codes. The work of Evra et. al. [FOCS, 2020] took advantage of the high dimensional expansion notion known as cosystolic expansion, that occurs in Ramanujan complexes. Our improvement is achieved by considering tensor product of Ramanujan complexes, and using their newly derived property, the collective cosystolic expansion.

Original language | English |
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Title of host publication | STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing |

Editors | Samir Khuller, Virginia Vassilevska Williams |

Publisher | Association for Computing Machinery |

Pages | 1317-1329 |

Number of pages | 13 |

ISBN (Electronic) | 9781450380539 |

DOIs | |

State | Published - 15 Jun 2021 |

Event | 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021 - Virtual, Online, Italy Duration: 21 Jun 2021 → 25 Jun 2021 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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ISSN (Print) | 0737-8017 |

### Conference

Conference | 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021 |
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Country/Territory | Italy |

City | Virtual, Online |

Period | 21/06/21 → 25/06/21 |

### Bibliographical note

Publisher Copyright:© 2021 Owner/Author.

### Funding

R. T. (incumbent of the Lillian and George Lyttle Career Development Chair) was supported by the ISF grant No. 335/19 and by a research grant from the Center for New Scientists of Weizmann Institute. T. K. was supported by ERC.

Funders | Funder number |
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European Research Council | |

Israel Science Foundation | 335/19 |

## Keywords

- Cosystolic Expanders
- Quantum codes
- Ramanujan Complexes}

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