AI 中文总结
该研究针对任意拓扑序，提出基于整数线性规划的译码器，其在三类拓扑序的纠错任务中性能优于多数现有译码器，可应用于容错量子计算。
AI 中文摘要
拓扑序（TO）被广泛用作量子纠错码，其中任意子激发充当错误症候群。对于某些阿贝尔TO，译码可通过独立匹配每种的粒子-反粒子对来实现。然而，基于匹配的译码器无法处理阿贝尔或非阿贝尔TO中更通用的融合规则，也无法解释关联不同任意子种类的噪声。尽管聚类译码器的适用范围更广，但它们通常忽略任意子数据和融合特性，导致实际性能较差。在这项工作中，我们引入了一种基于整数线性规划（ILP）的、适用于任意TO的根本不同的译码器。ILP公式通过引入辅助变量将纠错问题线性化，并将融合规则编码为线性约束，随后经典优化可识别最小权重错误构型。作为具体示例，我们确定了三种TO的纠错阈值：存在电荷与通量错误关联的退极化噪声下的阿贝尔$\boldsymbol{\text{Z}_2}$ TO、不支持成对匹配译码器的阿贝尔$\boldsymbol{\text{Z}_3}$ TO，以及可生成所有任意子种类的噪声通道下的非阿贝尔$\boldsymbol{\text{D}_4}$ TO。我们通过展示该ILP译码器在这三种情况下均优于大多数现有译码器，证明了其通用性。我们进一步扩展该方法以纳入有噪声的症候测量，并提出了适用于连续纠错的即时变体。我们的结果确立了ILP作为处理关联错误和通用任意子融合规则的自然框架，以及作为适用于任意TO中不相干任意子噪声的强大灵活通用译码器，可应用于容错量子计算。
英文摘要
Topological orders (TOs) are widely used as quantum error-correcting codes, with anyon excitations serving as error syndromes. For certain Abelian TOs, decoding can be performed by independently matching particle-antiparticle pairs of each species. However, matching-based decoders cannot handle more general fusion rules in either Abelian or non-Abelian TOs, nor account for noise that correlates different anyon species. While clustering decoders are more broadly applicable, they typically neglect anyon data and fusion properties, leading to poor performance in practice. In this work, we introduce a fundamentally different decoder for arbitrary TOs based on integer linear programming (ILP). The ILP formulation linearizes the error-correction problem through the introduction of auxiliary variables and encodes fusion rules as linear constraints. Classical optimization then identifies the minimum-weight error configuration. As concrete examples, we determine error-correction thresholds for three TOs: the Abelian $\mathbb{Z}_2$ TO under depolarizing noise, where charge and flux errors are correlated; the Abelian $\mathbb{Z}_3$ TO, which does not admit a pairwise matching decoder; and the non-Abelian $D_4$ TO under noise channels that generate all anyon species. We demonstrate the versatility of the ILP decoder by showing a clear performance advantage over most existing decoders in all three cases. We further extend the method to incorporate noisy syndrome measurements and propose a just-in-time variant for continuous error correction. Our results establish ILP as a natural framework for handling correlated errors and general anyon fusion rules, and as a powerful and flexible general-purpose decoder for incoherent anyon noise in arbitrary TOs, with applications to fault-tolerant quantum computation.
