Construction of quantum codes from new embeddings of graphs on compact surfaces

Abstract

In this paper we introduce a new class of sparse CSS quantum error-correcting codes based on new orientable self-dual embeddings of graphs. Each code in this class is associated with the embedding of the surface so that the qubits correspond to the edges of the embedding. The parameters of the new codes are $$[[\frac{m(m+1)}{2},\frac{m(m-3)}{2},3]]$$, where $$m=4s$$, $$s\ge 1$$. We also present a table of quantum codes whose parameters had not been shown before.