发表机构
Friedrich-Alexander-Universität Erlangen-Nürnberg; University of Vienna（埃尔朗根-纽伦堡大学; 维也纳大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过副本交换蒙特卡洛模拟，研究二维环面码在多种晶格上的最大似然错误阈值，发现对偶晶格阈值围绕方形晶格分裂，且平均配位数与顶点排列共同决定阈值。
AI 中文摘要
我们研究了晶格几何如何影响二维环面码的错误阈值。在存在比特翻转或相位翻转错误的情况下，环面码映射到二维随机键伊辛模型（RBIM）。我们使用副本交换蒙特卡洛模拟确定了RBIM的临界行为。虽然先前的研究在次优解码器（如最小权重完美匹配）下探讨了一般晶格几何的阈值，但近最优解码器的快速发展使得解析最终的、最大似然码容量变得相关。我们在Nishimori线上计算了方形、蜂窝、三角形、骰子和kagome晶格的最优错误阈值。由于晶格对偶性，给定晶格上的相位翻转阈值等同于其对偶晶格上的比特翻转阈值。我们发现，对偶晶格对的最优阈值表现出围绕自对偶方形晶格的特征性对偶驱动分裂，反映了在次优解码器中观察到的定性行为。虽然平均配位数对错误阈值的影响最大，但我们的结果表明，顶点和plaquette的详细排列在确定其精确值方面也起着重要作用。
英文摘要
We investigate how the lattice geometry influences the error thresholds of two-dimensional toric codes. In the presence of bit- or phase-flip errors, the toric code maps onto the two-dimensional random-bond Ising model (RBIM). We determine the critical behaviour of the RBIM using replica-exchange Monte Carlo simulations. While previous studies have explored thresholds for general lattice geometries under suboptimal decoders such as minimum-weight perfect matching, the rapid development of near-optimal decoders makes resolving the ultimate, maximum-likelihood code capacity relevant. We compute these optimal error thresholds on the Nishimori line for the square, honeycomb, triangular, dice, and kagome lattices. Owing to lattice duality, the phase-flip threshold on a given lattice is equivalent to the bit-flip threshold on its dual. We find that the optimal thresholds of dual-lattice pairs display a characteristic duality-driven splitting around the self-dual square-lattice, mirroring the qualitative behaviour observed for suboptimal decoders. While the average coordination number has the strongest impact on the error threshold, our results demonstrate that the detailed arrangement of vertices and plaquettes also plays a significant role in determining its precise value.
Comments23 pages, 14 figures
