On the duality between homological quantum codes of a hypermap and its dual hypermap

Abstract

From a given topological hypermap $H$, we define two related hypermaps $H^{△}$ and $H^{\nabla }$ as complements of the ordinary dual hypermap $H^{\ast }$ along with the concepts of their edge hypermap quantum codes ${\mathcal{C}}^{△}$ and ${\mathcal{C}}^{\nabla }$. We then show that, when the sets of special darts are naturally corresponded, the duality between the ordinary hypermap quantum code $\mathcal{C}$ from $H$ and the one ${\mathcal{C}}^{\ast }$ from $H^{\ast }$ can be greatly simplified to the duality between ${\mathcal{C}}^{△}$ and ${\mathcal{C}}^{\nabla }$.