Mitigating Frequency Learning Bias in Quantum Models via Multi-Stage Residual Learning
This work addresses a specific bottleneck in quantum machine learning for researchers and practitioners, offering a practical framework to enhance spectral expressivity, but it is incremental as it adapts an existing classical method to the quantum domain.
The paper tackled the problem of quantum machine learning models struggling to learn functions with multiple frequency components, particularly high-frequency or non-dominant ones, by adapting multi-stage residual learning from classical Fourier neural operators to the quantum domain. The result showed that residual learning alone significantly improved test MSE over a single-stage baseline, with systematic experiments highlighting the importance of qubit count, encoding scheme, and residual learning for resolving multiple frequencies.
Quantum machine learning models based on parameterized circuits can be viewed as Fourier series approximators. However, they often struggle to learn functions with multiple frequency components, particularly high-frequency or non-dominant ones; a phenomenon we term the quantum Fourier parameterization bias. Inspired by recent advances in classical Fourier neural operators (FNOs), we adapt the multi-stage residual learning idea to the quantum domain, iteratively training additional quantum modules on the residuals of previous stages. We evaluate our method on a synthetic benchmark composed of spatially localized frequency components with diverse envelope shapes (Gaussian, Lorentzian, triangular). Systematic experiments show that the number of qubits, the encoding scheme, and residual learning are all crucial for resolving multiple frequencies; residual learning alone can improve test MSE significantly over a single-stage baseline trained for the same total number of epochs. Our work provides a practical framework for enhancing the spectral expressivity of quantum models and offers new insights into their frequency-learning behavior.