AI 中文总结
该研究从动态李代数视角探讨量子机器学习中的李群模式连通性，构建可达酉李群上的模式连通性，给出几何解释，讨论过参数化作用，通过玩具数值实验验证，为理解QML模式连通性提供新视角。
AI 中文摘要
模式连通性在经典机器学习中作为参数空间低损耗区域的几何属性被广泛研究。在量子机器学习（QML）中，物理相关对象是由参数化量子电路实现的酉变换而非参数向量本身。本研究在由生成元的动态李代数生成的可达酉李群上构建模式连通性。在近最小连通性假设和低损耗带无临界值的情况下，可达李群上相应的低损耗子水平集是路径连通的。这为QML中的模式连通性提供了独立于特定参数化的几何解释。还讨论了过参数化如何将李群路径提升到参数空间，使李群连通性在参数空间实验中可观测。最后给出了玩具数值实验，训练酉之间的测地线插值显示出几乎为零的损耗障碍，与所提解释一致。
英文摘要
Mode connectivity has been widely studied in classical machine learning as a geometric property of low-loss regions in parameter space. In quantum machine learning (QML), however, the physically relevant object is not the parameter vector itself but the unitary transformation implemented by a parameterized quantum circuit. In this study, we formulate mode connectivity on the reachable unitary Lie group generated by the dynamical Lie algebra of the generators. We show that, under a near-minimum connectedness assumption and the absence of critical values in a low-loss band, the corresponding low-loss sublevel set on the reachable Lie group is path-connected. This provides a geometric interpretation of mode connectivity in QML that is independent of a particular parameterization. We further discuss how overparameterization can enable the lifting of Lie-group paths to parameter space, thereby making Lie-group connectivity observable in parameter-space experiments. Finally, we present toy numerical experiments in which geodesic interpolations between trained unitaries exhibit nearly zero loss barriers, consistent with the proposed interpretation.
