量子机器学习中,低频信号比高频信号更容易被模型学习。
Spectral Bias in Variational Quantum Machine Learning
- 用傅里叶级数分析量子电路,发现频谱偏差源于系数冗余。
- 训练时梯度大小与系数冗余正相关,影响学习速度。
- 高冗余电路对参数扰动更鲁棒,适合处理低频任务。
本文研究了量子机器学习中的频谱偏差现象,即在经典设置下,模型通常优先拟合目标函数的低频成分。针对参数化量子电路(PQC),我们利用其作为傅里叶级数的已有形式,证明该偏差源于傅里叶系数的‘冗余’——即多个项贡献同一频率分量。数据编码方式决定了系数冗余程度,训练中梯度幅度与冗余强相关。通过三种编码方案的实证验证了这一规律。此外,冗余更高的电路在对应频率上对参数随机扰动表现出更强鲁棒性。我们还考察了初始化尺度和纠缠结构的影响,发现大初始化和低纠缠结构会减缓收敛速度。
原文摘要 · Abstract (English)
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.
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