  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.
