发表机构
University of California, Berkeley; Simons Institute for the Theory of Computing, University of California at Berkeley; Instituto de Física Teórica UAM-CSIC, Universidad Autónoma de Madrid; University of California, Santa Barbara; Technion; Lawrence Berkeley National Laboratory（加州大学伯克利分校; 加州大学伯克利分校西蒙斯理论计算研究所; 马德里自治大学-西班牙国家科学研究委员会理论物理研究所; 加州大学圣塔芭芭拉分校; 以色列理工学院; 劳伦斯伯克利国家实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出非阿贝尔混合态中强到弱对称性破缺的解码框架，发现强对称性破缺至中心，中心排序转变阈值随层级不降，并用张量网络验证S_3和SU(2)模型。
AI 中文摘要
我们发展了一种处理非阿贝尔对称群G的强到弱自发对称性破缺（sw-SSB）的操作性方法。我们将其表述为区分两个强对称态的解码任务，这两个态在插入一对不可约表示后不同，并且都受到强对称退相干的影响。对于有限群G，该任务与在纯电荷噪声下区分量子双D(G)的逻辑态是等价的。我们推导了一个统计力学模型层级，描述基于在逐渐增大的嵌套区域上测量的融合结果的最优解码。对于阿贝尔码的最大似然解码器，测量的不可约表示充当服从Nishimori类条件的淬火无序。引人注目的是，由于局部不可约表示不能确定它们全局融合的方式，强G×G对称性仅非局域地实现，并且在Z(G)（G的中心）下平凡带电的不可约表示可以隐藏在噪声产生的不可约表示中。因此，强对称性自发破缺为Z(G)乘以弱对称性。在非零错误率下唯一的转变是中心的排序，其阈值不会随着层级的上升而降低。该结论对于非幂零群在层级的任何有限级别都成立。我们通过张量网络计算对S_3和SU(2)确认了这些预测。在第一级，中心平凡的S_3模型没有转变，所有不可约表示的保真度相关性可证明衰减到非零值。SU(2)模型表现出与Nishimori-Ising普适类一致的Z_2转变，其中自旋-1/2不可约表示的保真度相关性变为长程。阈值从层级的第一级上升到第二级增加了16%。我们的结果为研究非阿贝尔sw-SSB和为非阿贝尔拓扑码构建解码器提供了系统框架。
英文摘要
We develop an operational approach to strong-to-weak spontaneous symmetry breaking (sw-SSB) of a non-Abelian symmetry group G. We formulate it as the decoding task of distinguishing two strongly symmetric states that differ by a pair of inserted irreps, after both are corrupted by strongly symmetric decoherence. For finite G, this task is dual to distinguishing logical states of the quantum double D(G) under pure-charge noise. We derive a hierarchy of statistical mechanics models describing optimal decoding from the fusion outcomes measured on successively larger nested regions. As for maximum-likelihood decoders of Abelian codes the measured irreps act as quenched disorder obeying a Nishimori-like condition. Strikingly, because local irreps do not determine how they fuse globally, the strong $G\times G$ symmetry is realized only nonlocally, and irreps trivially charged under Z(G) (the center of G) can hide inside noise-generated ones. As a result, the strong symmetry spontaneously breaks down to Z(G), times the weak symmetry. The only transition at nonzero error rate is then an ordering of the center, whose threshold cannot decrease as the hierarchy is ascended. This conclusion holds at any finite level of the hierarchy for non-nilpotent groups. We confirm these predictions with tensor-network calculations for $S_3$ and SU(2). At the first level, the $S_3$ model, whose center is trivial, shows no transition, and the fidelity correlations of all irreps provably decay to non-zero values. The SU(2) model exhibits a $Z_2$ transition consistent with the Nishimori-Ising universality class, where the fidelity correlations of spin-$1/2$ irreps become long-range. The threshold rises from the first to the second level of the hierarchy by $16\%$. Our results provide a systematic framework for studying non-Abelian sw-SSB and for constructing decoders for non Abelian topological codes.
Comments24 + 15 pages, 10 figures
