从可达性到可学习性：量子神经网络的几何设计原则
From Reachability to Learnability: Geometric Design Principles for Quantum Neural Networks
AI总结：
本文研究量子神经网络中几何变形能力的实现，提出CLA映射和aCLS准则，展示数据与可训练权重的联合依赖性，证明参数化纠缠方向对高维变形的重要性。
摘要：
经典深度网络有效是因为深度能实现数据表示的自适应几何变形。在量子神经网络中，深度或状态可达性本身不足以保证这种特征学习能力。本文通过将编码数据视为嵌入在ℂP^{2^n-1}中的流形，并分析通过李代数方向的微小幺正作用，研究这一问题。引入经典到李代数（CLA）映射和几乎完全局部选择性（aCLS）准则，结合方向完整性和数据依赖的局部选择性。在该框架下，证明数据无关的可训练幺正操作是完整的但非选择性的，即可学习的刚性重定向；而纯数据编码是选择性的但非可调的，即固定变形。因此，几何灵活性需要数据和可训练权重的非平凡联合依赖。进一步证明，访问多量子比特态流形的高维变形需要参数化纠缠方向；固定纠缠器如CNOT单独无法提供自适应几何控制。数值示例验证了满足aCLS的数据重新上传模型在性能上优于非可调方案，且仅需四分之一的门操作。因此，最终的图景将QNN设计从状态可达性转向可控的隐藏量子表示几何。
英文摘要：
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.
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