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.

翻译：我们提出了一种量子CSS码的提升概念，其灵感来源于Freedman和Hastings的几何构造。该概念基于与任何CSS码相关联的一个典范复形——我们称之为Tanner锥复形——并在此复形上生成覆盖空间。作为第一个应用，我们描述了超图乘积码(HPC)提升的分类，并证明了其与Panteleev和Kalachev提出的提升乘积码(LPC)的等价性，包括当线性码（HPC的因子）为Tanner码的情形。作为第二个应用，我们报道了若干新型非乘积结构的量子CSS码，并应用该构造生成其提升——对于某些特定的覆盖映射，这些提升码具有相较于初始码更优的相对参数。