We study a generalization of Kitaev’s abelian toric code model defined on CW complexes. In this model, qudits are attached to [Formula: see text]-dimensional cells and the interaction is given by generalized star and plaquette operators. These are defined in terms of coboundary and boundary maps in the locally finite cellular cochain complex and the cellular chain complex. We find that the set of energy-minimizing ground states and the types of charges carried by certain localized excitations depends only on the proper homotopy type of the CW complex. As an application, we show that the homological product of a CSS code with the infinite toric code has excitations with abelian anyonic statistics.

中文翻译：

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同调码和阿贝尔任意子

我们研究了在 CW 复合体上定义的 Kitaev 的阿贝尔复曲面码模型的推广。在这个模型中，qudits 附加到 [公式：见文本] 维单元格，并且相互作用由广义星形和斑块算子给出。这些是根据局部有限细胞共链复合体和细胞链复合体中的共边界和边界图来定义的。我们发现能量最小化基态的集合和某些局部激发携带的电荷类型仅取决于 CW 复合体的正确同伦类型。作为一个应用，我们展示了一个 CSS 代码与无限复曲面代码的同调积具有与阿贝尔任意子统计的激励。