Qudit surface codes and hypermap codes

TL;DR

This paper generalizes surface and hypermap quantum codes to qudits of arbitrary dimension, establishing their properties through homological algebra and extending existing qubit-based frameworks.

Contribution

It introduces homological quantum codes for qudits on arbitrary 2-Complexes and generalizes hypermap-homology codes to higher dimensions.

Findings

Code dimension equals the first homology group size.

Codes share properties with qubit counterparts.

Hypermap codes are extended to qudits via homological constructions.

Abstract

In this article, we define homological quantum codes in arbitrary qudit dimensions $D\geq 2$ by directly defining CSS operators on a 2-Complex $\Sigma$. If the 2-Complex is constructed from a surface, we obtain a qudit surface code. We then prove that the dimension of the code we define always equals the size of the first homology group of $\Sigma$. We also define the distance of the codes in this setting, finding that they share similar properties with their qubit counterpart. Additionally, we generalize the hypermap-homology quantum code proposed by Martin Leslie to the qudit case. For every such hypermap code, we construct an abstract 2-Complex whose homological quantum code is equivalent to the hypermap code.