Lifts of quantum CSS codes

TL;DR

This paper introduces a new concept of lifts for quantum CSS codes based on a canonical complex, enabling classification and construction of codes with improved parameters through covering spaces.

Contribution

It defines the Tanner cone-complex for CSS codes, classifies lifts of hypergraph product codes, and constructs new codes with enhanced parameters using covering maps.

Findings

Classification of lifts of hypergraph product codes

Equivalence with lifted product codes by Panteleev and Kalachev

Construction of new codes with improved parameters

Abstract

We propose a notion of lift for quantum CSS codes, inspired by the geometrical construction of Freedman and Hastings. It is based on the existence of a canonical complex associated to any CSS code, that we introduce under the name of Tanner cone-complex, and over which we generate covering spaces. As a first application, we describe the classification of lifts of hypergraph product codes (HPC) and demonstrate the equivalence with the lifted product code (LPC) of Panteleev and Kalachev, including when the linear codes, factors of the HPC, are Tanner codes. As a second application, we report several new non-product constructions of quantum CSS codes, and we apply the prescription to generate their lifts which, for certain selected covering maps, are codes with improved relative parameters compared to the initial one.