Spectral Bias in Variational Quantum Machine Learning

In this work, we investigate the phenomenon of spectral bias in quantum machine learning (QML), in classical machine learning this manifests as models fitting low-frequency components of a target function before high-frequency ones. We perform the first comprehensive study characterising how this phenomena manifests in parameterised quantum circuits (PQCs) both theoretically and empirically. Leveraging the established formulation of PQCs as Fourier series, we prove that spectral bias in this setting can arise 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 then further demonstrate this empirically with several different encoding schemes. We show this has direct consequences for quantum-assisted simulation of physical systems such as PDEs. Additionally, we demonstrate that PQCs with greater redundancy exhibit increased robustness to random perturbations in their parameters at the corresponding frequencies.