arXiv:2603.03071quant-phcs.LG2026-03

量子神经网络设计需兼顾数据与可调参数,才能实现有效特征学习。

From Reachability to Learnability: Geometric Design Principles for Quantum Neural Networks

  • 提出aCLS准则,结合方向完备性与数据依赖选择性。
  • 数据重上传模型仅用四分之一门操作即超越非可调方案。
  • 适合关注量子模型几何设计的科研人员参考。

经典深度网络因深度可实现数据表示的自适应几何变形而有效。但在量子神经网络(QNN)中,单纯增加深度或状态可达性并不能保证这种特征学习能力。本文在纯态设定下,将编码数据视为复射影空间 $\mathbb{C}P^{2^n-1}$ 中的嵌入流形,通过李代数方向分析无穷小酉操作。引入经典到李代数(CLA)映射及几乎完全局部选择性(aCLS)准则,该准则结合方向完备性与数据依赖的局部选择性。结果显示:数据无关的可训练酉算符具备完备性但无选择性,即仅能实现刚性重定向;而纯数据编码具有选择性但不可调,即固定变形。因此,几何灵活性要求数据与可调权重的非平凡联合依赖。进一步表明,实现多量子比特态流形的高维变形需参数化纠缠方向;固定纠缠门如CNOT无法提供自适应几何控制。数值实验验证:满足aCLS的数据重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量。该研究将QNN设计从状态可达性转向对隐藏量子表示可控几何的重构。

原文摘要 · Abstract (English)

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.

量子神经网络几何设计可学习性数据重上传

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