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 CLS-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征的自适应几何形变。然而，在量子神经网络中，仅凭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此问题，将编码数据视为嵌入在 $\mathbb{C}P^{2^n-1}$ 中的流形，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映射以及几乎完全局部选择性准则，该准则结合了方向完备性与数据依赖的局部选择性。在此框架内，我们证明数据无关的可训练酉算符具有完备性但缺乏选择性，即可学习的刚性重定向；而纯数据编码具有选择性但不可调节，即固定的形变。因此，几何灵活性要求对数据和可训练权重具有非平凡的联合依赖性。我们进一步证明，要实现对多量子比特状态流形的高维形变，需要参数化的纠缠方向；仅使用固定纠缠器如CNOT无法提供自适应的几何控制。数值算例验证了满足CLS准则的数据重上传模型优于不可调节方案，同时仅需四分之一的门操作。因此，所得结论将QNN设计从状态可达性重新定位为对隐藏量子表征的可控几何形态。