New Classes of Quantum Codes Associated with Surface Maps

TL;DR

This paper introduces thirteen new classes of quantum codes derived from semi-equivelar and equivelar maps on various surfaces, enhancing the topological quantum code landscape with novel constructions and asymptotic encoding rates.

Contribution

The paper presents thirteen new quantum code classes linked to specific surface maps, expanding the types of topological quantum codes with new geometric and homological insights.

Findings

Thirteen new quantum code classes introduced.

Encoding rate approaches 1 for certain classes as size increases.

Other classes have a bounded encoding rate less than 1.

Abstract

If the cyclic sequences of {face types} {at} all vertices in a map are the same, then the map is said to be a semi-equivelar map. In particular, a semi-equivelar map is equivelar if the faces are the same type. Homological quantum codes represent a subclass of topological quantum codes. In this article, we introduce {thirteen} new classes of quantum codes. These codes are associated with the following: (i) equivelar maps of type $ [k^k]$, (ii) equivelar maps on the double torus along with the covering of the maps, and (iii) semi-equivelar maps on the surface of \Echar{-1}, along with {their} covering maps. The encoding rate of the class of codes associated with the maps in (i) is such that $ \frac{k}{n}\rightarrow 1 $ as $ n\rightarrow\infty $, and for the remaining classes of codes, the encoding rate is $ \frac{k}{n}\rightarrow \alpha $ as $ n\rightarrow \infty $ with $ \alpha< 1 $.