Quantum machine learning research has expanded rapidly due to potential computational advantages over classical methods. Angle encoding has emerged as a popular choice as feature map (FM) for embedding classical data into quantum models due to its simplicity and natural generation of truncated Fourier series, providing universal function approximation capabilities. Efficient FMs within quantum circuits can exploit exponential scaling of Fourier frequencies, with multi-dimensional inputs introducing additional exponential growth through mixed-frequency terms. Despite this promising expressive capability, practical implementation faces significant challenges. Through controlled experiments with white-box target functions, we demonstrate that training failures can occur even when all relevant frequencies are theoretically accessible. We illustrate how two primary known causes lead to unsuccessful optimization: insufficient trainable parameters relative to the model's frequency content, and limitations imposed by the ansatz's dynamic lie algebra dimension, but also uncover an additional parameter burden: the necessity of controlling non-unique frequencies within the model. To address this, we propose near-zero weight initialization to suppress unnecessary duplicate frequencies. For target functions with a priori frequency knowledge, we introduce frequency selection as a practical solution that reduces parameter requirements and mitigates the exponential growth that would otherwise render problems intractable due to parameter insufficiency. Our frequency selection approach achieved near-optimal performance (median $R^2 \approx 0.95$) with 78\% of the parameters needed by the best standard approach in 10 randomly chosen target functions.

翻译：量子机器学习研究因其相对于经典方法的潜在计算优势而迅速发展。角度编码作为一种特征映射（FM）因其简单性及自然生成截断傅里叶级数的特性，成为将经典数据嵌入量子模型的流行选择，提供了通用函数逼近能力。量子电路中的高效特征映射可利用傅里频率的指数缩放特性，而多维输入通过混合频率项引入额外的指数增长。尽管这种表达能力前景广阔，但实际实施面临重大挑战。通过对白盒目标函数进行受控实验，我们证明即使所有相关频率在理论上均可访问，训练仍可能失败。我们阐释了两种已知主要原因如何导致优化失败：相对于模型频率内容的可训练参数不足，以及由拟设动态李代数维度所施加的限制；同时我们还揭示了一个额外的参数负担：控制模型中非唯一频率的必要性。为解决此问题，我们提出近零权重初始化以抑制不必要的重复频率。对于具有先验频率知识的目标函数，我们引入频率选择作为一种实用解决方案，该方案能降低参数需求，并缓解原本会因参数不足而导致问题无法处理的指数增长。在10个随机选择的目标函数中，我们的频率选择方法以最佳标准方法所需参数的78%实现了接近最优的性能（中位$R^2 \approx 0.95$）。