量子学习中的对称性发现：有限测量下的可观测量级与任务级推断
Symmetry Discovery in Quantum Learning: Observable-Level and Task-Level Inference from Finite Measurements
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中文总结 AI 辅助
本文建立有限测量下推断量子学习模型对称群的统计理论，提出无偏阴影统计量及任务验证方法，确定测量支持的对称性并量化其成本与收益。
中文摘要 AI 辅助
对称性降低了量子学习模型的容量，但所施加的群必须同时匹配测量信息和标签变换。我们建立了从候选变换中推断该群的有限测量理论。核心结构结果将可观测不可见变换识别为投影态的稳定化子，当探针跨度不变时。它将恢复的生成元转化为有效的子群，并将连续不可见空间与其李代数等同。对于有限字典，无偏阴影统计量以逆间隙测量速率区分零与正平方期望差异，优于均匀差异估计的逆平方间隙速率。交换性量子比特下界证明在固定快照尺度下间隙依赖最优，同时区间支持数据相关的容差。任务验证随后通过特征核测试联合分布，或通过经典-量子差异测试其编码均值。精确的群平均恒等式将后者与联合态不对称性关联，并指定其转换为二元任务破坏质量。投影偏差量化过度对称的成本，而ℓ1读出界量化放松对称性所获得的容量。在不变纯态骨干上，保留和非平凡破坏扇区是Fisher正交的。伊辛链计算连接有限快照恢复、标签依赖对称性和物理扇区漂移。这些结果确定测量支持哪种对称性，并为后续发布决策提供统计和几何基础。
英文摘要
Symmetry reduces the capacity of a quantum learning model, but the imposed group must match both the measured information and the label transformation. We establish a finite-measurement theory for inferring this group from candidate transformations. The central structural result identifies observable-invisible transformations with the stabilizer of a projected state whenever the probe span is invariant. It turns recovered generators into a valid subgroup and identifies the continuous invisible space with its Lie algebra. For finite dictionaries, an unbiased shadow statistic distinguishes zero from positive squared expectation discrepancies with an inverse-gap measurement rate, improving the inverse-square-gap rate of uniform discrepancy estimation. A commuting qubit lower bound proves the gap dependence optimal at fixed snapshot scale, and simultaneous intervals support data-dependent tolerances. Task validation then tests either the joint distribution through a characteristic kernel or its encoded mean through a classical--quantum discrepancy. An exact group-average identity relates the latter to joint-state asymmetry and specifies its conversion to binary task breaking mass. Projection bias quantifies the cost of excessive symmetry, while an $\ell_1$ readout bound quantifies the capacity gained by relaxing it. At an invariant pure-state backbone, retained and nontrivial breaking sectors are Fisher-orthogonal. Ising-chain calculations connect finite-shot recovery, label-dependent symmetry, and physical sector drift. These results determine which symmetry the measurements support and provide the statistical and geometric basis for a subsequent release decision.
