Higher-dimensional quantum hypergraph-product codes

Abstract

We describe a family of quantum error-correcting codes which generalize both the quantum hypergraph-product (QHP) codes by Tillich and Z\'emor, and all families of toric codes on $m$-dimensional hypercubic lattices. Similar to the latter, our codes form $m$-complexes ${\mathcal{K}}_m$, with $m\ge 2$. These are defined recursively, with ${\mathcal{K}}_m$ obtained as a tensor product of a complex ${\mathcal{K}}_{m-1}$ with a $1$-complex parameterized by a binary matrix. Parameters of the constructed codes are given explicitly in terms of those of binary codes associated with the matrices used in the construction.