Given their potential for fault-tolerant operations, topological quantum states are currently a focus of intense activity. Of particular interest are topological quantum error correction codes, such as the surface and planar stabilizer codes that are equivalent to the celebrated toric code. While every stabilizer state maps to a graph state under local Clifford operations, the graphs associated with topological stabilizer codes remain unknown. We show that the toric code graph is composed of only two kinds of subgraphs: star graphs (which encode Greenberger-Horne-Zeilinger states) and half graphs. The topological order is identified with the existence of multiple star graphs, which reveals a connection between the repetition and toric codes. The graph structure readily yields a log-depth quantum circuit for state preparation, assuming geometrically nonlocal gates, which can be reduced to a constant depth including ancillae and measurements at the cost of increasing the circuit width. The results provide a graph-theoretic framework for the investigation of topological order and the development of topological error correction codes.

中文翻译：

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复曲面代码的图形状态表示

鉴于其容错操作的潜力，拓扑量子态目前是激烈活动的焦点。特别令人感兴趣的是拓扑量子纠错码，例如与著名的复曲面码等效的表面和平面稳定器码。虽然每个稳定器状态都映射到局部 Clifford 操作下的图状态，但与拓扑稳定器代码相关的图仍然未知。我们表明复曲面代码图仅由两种子图组成：星图（编码 Greenberger-Horne-Zeilinger 状态）和半图。拓扑顺序被确定为多个星图的存在，这揭示了重复码和复曲面码之间的联系。图结构很容易产生用于状态准备的对数深度量子电路，假设几何非局部门，可以减少到恒定深度，包括辅助和测量，但代价是增加电路宽度。结果为拓扑序的研究和拓扑纠错码的开发提供了图论框架。