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arXiv · 2112.01752

Qudit surface codes and hypermap codes

Zihan Lei
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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.

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Zihan Lei. 2021-12-03. Qudit surface codes and hypermap codes. https://arxiv.org/abs/2112.01752

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