等变量量子学习的统计对称性释放
Statistical Symmetry Release for Equivariant Quantum Learning
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中文总结 AI 辅助
本文提出统计对称性释放方法，通过量子测量和置信界确定何时及如何放宽等变学习中的对称性约束，实现有保证的模型选择，并在八比特Ising模型上大幅减少所需测量次数。
中文摘要 AI 辅助
硬性对称性约束降低了模型复杂度，但也可能抹除标签信息。统计对称性释放决定了有限数据和量子测量何时证明放宽此类约束是合理的、应打开哪些方向以及移动多远。我们将全局信号检测与局部的、依赖于损失的改进联系起来。一个双副本twirl--swap门在配对态和群酉访问下，以与维度无关的副本数量估计对称性破缺补集中的任务信息；对相同记录重新加权可解析表示扇区。一个精确的对偶性将该Hilbert--Schmidt信号与有界输出读出可访问的更大信号区分开来。局部改进由释放梯度和损失修正的双对易子矩阵控制。同时置信界将经验方向选择转化为有保证的下降，使用共享的Pauli测量或具有状态无关截断界的标量探针。高斯检验下界量化了在校准的局部实验中搜索未知方向的成本。独立验证控制自适应生成的模型，一个快速的平方损失界保持了嵌套释放路径的逼近-估计速率。在八比特Ising模型上，共享测量在测试的预算网格上以比指定标量估计器少6300倍的射击次数认证释放。商量子自然梯度随后在训练期间控制参数冗余。总之，这些结果将对称性松弛转变为统计上合理的模型选择决策。
英文摘要
Hard symmetry constraints reduce model complexity, but can also erase label information. Statistical symmetry release determines when finite data and quantum measurements justify relaxing such a constraint, which directions to open, and how far to move. We connect global signal detection to local, loss-dependent improvement. A two-copy twirl--swap gate estimates task information in the symmetry-breaking complement with a dimension-independent copy count under paired-state and group-unitary access; reweighting the same records resolves representation sectors. An exact duality distinguishes this Hilbert--Schmidt signal from the larger signal accessible to bounded-outcome readouts. Local improvement is governed by the release gradient and a loss-corrected double-commutator matrix. Simultaneous confidence bounds convert empirical direction selection into certified descent, using either shared Pauli measurements or scalar probes with state-independent truncation bounds. Gaussian testing lower bounds quantify the cost of searching over unknown directions in the calibrated local experiment. Independent validation controls adaptively generated models, and a fast squared-loss bound preserves the approximation--estimation rate of a nested release path. On an eight-qubit Ising model, shared measurements certify release with 6300 times fewer shots than the specified scalar estimator on the tested budget grids. Quotient quantum natural gradient then controls parameter redundancy during training. Together, these results turn symmetry relaxation into a statistically justified model-selection decision.
