Spectral Bias in Variational Quantum Machine Learning

TL;DR

This paper explores spectral bias in quantum machine learning, showing how Fourier coefficient redundancy influences training dynamics and robustness in parameterized quantum circuits, with implications for circuit design.

Contribution

It provides a theoretical proof linking spectral bias to Fourier coefficient redundancy in PQCs and empirically demonstrates how encoding schemes and circuit design affect learning.

Findings

Redundancy of Fourier coefficients causes spectral bias in PQCs.

Encoding scheme choice influences the degree of Fourier coefficient redundancy.

Higher redundancy correlates with increased robustness to parameter perturbations.

Abstract

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.