发表机构
University of Pittsburgh（匹兹堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究以量子信号处理（QSP）为可解量子表示学习模型，推导了其量子神经正切核特性，证明了稀疏数据下的收敛保证，为量子数据几何提供了受控训练动力学理论。
AI 中文摘要
表示学习始于训练改变定义数据间相似性的特征。冻结核模型仅对固定几何进行重加权。我们确立量子信号处理（QSP）作为表示学习 regime 的可解量子模型。在任意深度，我们计算其量子神经正切核的精确均值与方差，揭示出依赖输入的角几何，即使 underlying 幺正矩阵趋近 Haar 随机性，其对角仍保持非自平均性。我们还证明了全非线性梯度流的稀疏数据保证，无需冻结或对核进行集合平均：实现的动力学收敛为具有时变核闭合性和显式收敛时间的可积标量流。对于每个数据集和轨迹，存在有限深度的速度极限。在更高数据密度下，数值结果显示超出标量和冻结核描述的耦合演化。这些结果为学习到的量子数据几何提供了受控理论，其可证训练动力学超出冻结极限。
英文摘要
Representation learning begins when training changes the features that define similarity between data. A frozen-kernel model only reweights a fixed geometry. We establish quantum signal processing (QSP) as a solvable quantum model of the representation-learning regime. At arbitrary depth, we compute the exact mean and variance of its quantum neural tangent kernel, revealing an input-dependent angular geometry whose diagonal remains non-self-averaging even when the underlying unitary approaches Haar randomness. We also prove a sparse-data guarantee for the full nonlinear gradient flow without freezing or ensemble-averaging the kernel: the realized dynamics converges to an integrable scalar flow with a time-dependent kernel closure and explicit convergence times. A finite-depth speed limit holds for every data set and trajectory. At higher data density, numerical results show coupled evolution beyond both the scalar and frozen-kernel descriptions. These results give a controlled theory of learned quantum data geometry with provable training dynamics beyond the frozen limit.
