Classical deep networks are effective because depth enables adaptive geometric deformation of data representations. In quantum neural networks (QNNs), however, depth or state reachability alone does not guarantee this feature-learning capability. We study this question in the pure-state setting by viewing encoded data as an embedded manifold in $\mathbb{C}P^{2^n-1}$ and analysing infinitesimal unitary actions through Lie-algebra directions. We introduce Classical-to-Lie-algebra (CLA) maps and the criterion of almost Complete Local Selectivity (aCLS), which combines directional completeness with data-dependent local selectivity. Within this framework, we show that data-independent trainable unitaries are complete but non-selective, i.e. learnable rigid reorientations, whereas pure data encodings are selective but non-tunable, i.e. fixed deformations. Hence, geometric flexibility requires a non-trivial joint dependence on data and trainable weights. We further show that accessing high-dimensional deformations of many-qubit state manifolds requires parametrised entangling directions; fixed entanglers such as CNOT alone do not provide adaptive geometric control. Numerical examples validate that aCLS-satisfying data re-uploading models outperform non-tunable schemes while requiring only a quarter of the gate operations. Thus, the resulting picture reframes QNN design from state reachability to controllable geometry of hidden quantum representations.

翻译：经典深度网络之所以有效，是因为深度能够实现数据表示的自适应几何形变。然而，在量子神经网络（QNN）中，单纯的深度或状态可达性并不能保证这种特征学习能力。我们在纯态设定下研究这一问题，将编码数据视为$\mathbb{C}P^{2^n-1}$中的嵌入流形，并通过李代数方向分析无穷小酉作用。我们引入了经典到李代数（CLA）映射以及几乎完全局部选择性（aCLS）准则，该准则将方向完备性与数据依赖的局部选择性相结合。在此框架下，我们表明：与数据无关的可训练酉操作是完备但非选择性的（即可学习的刚性重定向），而纯数据编码是选择性但不可调节的（即固定形变）。因此，几何灵活性要求数据与可训练权重的非平凡联合依赖关系。我们进一步证明，访问多量子比特态流形的高维形变需要参数化纠缠方向；仅使用固定纠缠门（如CNOT）无法提供自适应几何控制。数值实验验证了满足aCLS准则的数据重上传模型优于不可调节方案，且其门操作数量仅为后者的四分之一。因此，所得结论将QNN设计从状态可达性重新定位为隐量子表示的可控几何特性。