In this work, we investigate the phenomenon of spectral bias in quantum machine learning, where, in classical settings, models tend to fit low-frequency components of a target function earlier during training than high-frequency ones, demonstrating a frequency-dependent rate of convergence. We study this effect specifically in parameterised quantum circuits (PQCs). Leveraging the established formulation of PQCs as Fourier series, we prove that spectral bias in this setting arises from the ``redundancy'' of the Fourier coefficients, which denotes the number of terms in the analytical form of the model contributing to the same frequency component. The choice of data encoding scheme dictates the degree of redundancy for a Fourier coefficient. We find that the magnitude of the Fourier coefficients' gradients during training strongly correlates with the coefficients' redundancy. We then further demonstrate this empirically with three different encoding schemes. Additionally, we demonstrate that PQCs with greater redundancy exhibit increased robustness to random perturbations in their parameters at the corresponding frequencies. We investigate how design choices affect the ability of PQCs to learn Fourier sums, focusing on parameter initialization scale and entanglement structure, finding large initializations and low-entanglement schemes tend to slow convergence.

翻译：本研究探讨了量子机器学习中的谱偏置现象——该现象在经典场景中表现为模型在训练过程中倾向于先拟合目标函数的低频分量，后拟合高频分量，呈现出频率依赖的收敛速率。我们特别在参数化量子电路（PQCs）中研究了这一效应。基于将PQCs表述为傅里叶级数的既定理论框架，我们证明了该场景下的谱偏置源于傅里叶系数的“冗余度”，即模型解析形式中对同一频率分量有贡献的项数。数据编码方案的选择决定了傅里叶系数的冗余程度。我们发现训练过程中傅里叶系数梯度的大小与其冗余度呈强相关性，并通过三种不同编码方案进行了实证验证。此外，我们证明具有更高冗余度的PQCs在相应频率上对其参数的随机扰动表现出更强的鲁棒性。我们进一步探究了设计选择（重点关注参数初始化尺度与纠缠结构）如何影响PQCs学习傅里叶级数的能力，发现较大的初始化参数与低纠缠方案往往会减缓收敛速度。