Generalizing Quantum Tanner Codes

TL;DR

This paper generalizes quantum Tanner codes by extending their construction to broader mathematical structures, potentially enabling the discovery of new families of high-quality quantum error-correcting codes.

Contribution

It introduces a generalized framework for quantum Tanner codes using group actions, Schreier graphs, and larger square complexes, expanding their applicability.

Findings

Enlarged the class of quantum Tanner codes to include group actions on finite sets.

Demonstrated the potential for discovering new asymptotically good quantum codes.

Provided a theoretical foundation for future code constructions.

Abstract

In this work, we present a generalization of the recently proposed quantum Tanner codes by Leverrier and Z\'emor, which contains a construction of asymptotically good quantum LDPC codes. Quantum Tanner codes have so far been constructed equivalently from groups, Cayley graphs, or square complexes constructed from groups. We show how to enlarge this to group actions on finite sets, Schreier graphs, and a family of square complexes which is the largest possible in a certain sense. Furthermore, we discuss how the proposed generalization opens up the possibility of finding other families of asymptotically good quantum codes.