Moderate-length lifted quantum Tanner codes

TL;DR

This paper introduces new quantum Tanner codes constructed from classical Tanner codes with geometrical graph structures, providing bounds on their parameters and presenting explicit examples with improved distance properties.

Contribution

It develops a framework for analyzing lifted quantum Tanner codes using cellular homology, establishing bounds on code parameters and presenting explicit moderate-length codes with enhanced distances.

Findings

Distance bounds are established for lifted codes.

Explicit examples of codes with distances surpassing the square root of length.

Existence of a $[[96,2,12]]$ quantum code with specific weight properties.

Abstract

We introduce new families of quantum Tanner codes, a class of quantum codes that first appeared in the work of Leverrier and Z\'emor (FOCS 2022). These codes are built from two classical Tanner codes, for which the underlying graphs are extracted from coverings of 2D geometrical complexes, and the local linear codes are tensor-products of cyclic or double-circulant linear codes. The advantage of code lifting is that, for any lift of odd index $t$ of an $[[n,k,d]]$-code, we can adapt the study of the transfer homomorphism arising in cellular homology to describe symmetries of its logical operators and to establish that its dimension is lower bounded by $k$, and its distance is upper bounded by $t\cdot d$. Moreover, when the dimension of the lifted code is equal to $k$, its distance is lower bounded by $d$. These parameter bounds also apply to the previous methods of code lifting of Gu\'emard (IEEE Trans. Inf. Theory, 2025). Finally, We present several explicit families, and identify instances of moderate length quantum codes which are degenerate, have low check weight, and whose distance surpasses the square root of the code length. Among them, we report the existence of a $[[96,2,12]]$-code whose distance growth saturates our bound, and for which half of the checks are of weight 8 and the other half of weight 4.