Minimal distances for certain quantum product codes and tensor products of chain complexes

TL;DR

This paper establishes lower bounds for minimal homological distances in tensor products of chain complexes, with explicit formulas for certain cases, advancing quantum error correction theory.

Contribution

It introduces a method to bound and explicitly compute homological distances in tensor product complexes, generalizing known quantum code families.

Findings

Derived lower bounds for homological distances.

Provided explicit formulas when one complex is a linear map.

Generalized several known quantum error-correcting code families.

Abstract

We use a map to quantum error-correcting codes and a subspace projection to get lower bounds for minimal homological distances in a tensor product of two chain complexes of vector spaces over a finite field. Homology groups of such a complex are described by the K\"unneth theorem. We give an explicit expression for the distances when one of the complexes is a linear map between two spaces. The codes in the construction, subsystem product codes and their gauge-fixed variants, generalize several known families of quantum error-correcting codes.