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

TL;DR

This paper explores the duality relationships between hypermap-based quantum codes derived from a topological hypermap and its dual, revealing simplified duality connections among related codes.

Contribution

It introduces new hypermaps $H^ riangle$ and $H^ abla$ and demonstrates their associated quantum codes' duality simplifies the understanding of hypermap quantum code dualities.

Findings

Duality between hypermap quantum codes is simplified via related hypermaps.

The concepts of $H^ riangle$ and $H^ abla$ relate to complements of the dual hypermap.

Duality between $ ext{C}^ riangle$ and $ ext{C}^ abla$ is established.

Abstract

From a given topological hypermap $H$, we define two related hypermaps $H^\triangle$ and $H^\nabla$ as complements of the ordinary dual hypermap $H^*$ along with the concepts of their edge hypermap quantum codes $\mathcal{C}^\triangle$ 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 $\mathcal{C}^*$ from $H^*$ can be greatly simplified to the duality between $\mathcal{C}^\triangle$ and $\mathcal{C}^\nabla$.