一种通过对称性强制简并实现谱拟合的对易门
A commutant gate for spectral fitting through symmetry forced degeneracy
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中文总结 AI 辅助
该研究提出一种通过对称性强制简并实现谱拟合的对易门，可解决学习谱模型在对称简并扇区的失效问题，在噪声下分类精度优于能量聚类，拟合精度达机器精度。
中文摘要 AI 辅助
学习得到的谱模型在对称性强制简并扇区会失效，原因有二：当对称性强制能级完全重合时，原本用于拟合的单能级可观测量未被良好定义，因为该共享空间的每个单位向量都是本征向量；在对称性保护的交叉点附近，本征向量可观测量梯度带有因子1/(λ_i - λ_j)，当能级间隙闭合时该因子会真正奇异。通常的应对方法是对发散项进行正则化或对间隙设置阈值，但两种方法都存在实际代价：固定的间隙无法同时保护强制多重态并将两个真正不同的能级分开。我们展示了一种不同的解决方案，针对具有已知对称答案密钥的合成算子族：该门直接从观测到的算子中读取对称结构，即该族的线性对易子，这是Maehara和Murota同时块对角化后的一个奇异值分解零空间。块恒等性是从该对易子的中心读取的，而非从本征值聚类读取，这使得强制与偶然的区分是结构性的而非度量性的，目标函数在强制块的投影器迹与其他区域的单能级目标之间切换。在算子估计噪声下，该门的正确分类精度约为0.3，而能量聚类在0.02时就已失效。门控拟合在对称和对称性破缺区域均达到机器精度的真值，在后一区域收敛到非门控目标函数的奇点处的真值，可观测量偏差处于噪声水平。在正则表示之外的验证中，多重度与维度分离，排除了两个在所有正则表示测试中均通过的缺陷估计器。所展示的对象是一个门控估计器；完整的参数矩阵模型（矩阵需学习）是后续实验的目标。
英文摘要
Learned spectral models fail at symmetry forced degenerate sectors for two distinct reasons. Where symmetry forces levels to coincide exactly, the per level observable one would normally fit is not well defined, since every unit vector of that shared space is an eigenvector; and near a symmetry protected crossing, the eigenvector observable gradient carries a factor 1/(lambda_i - lambda_j) that is genuinely singular as the gap closes. The usual response is to regularize the divergence or threshold the gap, and both carry a real cost: a fixed gap cannot both protect a forced multiplet and keep two genuinely distinct levels apart. We show a different fix, on synthetic operator families with a known symmetry answer key. A gate reads the symmetry structure directly from the observed operators, as the linear commutant of the family, one singular value decomposition nullspace, following the simultaneous block diagonalization of Maehara and Murota. Block identity is read from the centre of that commutant rather than from eigenvalue clustering, which makes the forced versus accidental distinction structural rather than metric, and the objective switches between a projector trace through forced blocks and a per level target elsewhere. Under operator estimation noise the gate classifies correctly to epsilon of about 0.3, where energy clustering already fails by 0.02. Gated fitting reaches the truth at machine precision in both the symmetric and the symmetry breaking regime, in the latter converging to a truth that is a singularity of the ungated objective, and observable bias sits at the noise floor. Validation off the regular representation, where multiplicity and dimension separate, eliminated two defective estimators that all regular representation tests had passed. The demonstrated object is a gated estimator; a full parametric matrix model, with the matrices learned, is the next experiment.
