🤖 AI Summary
This work addresses the spectral learning bias inherent in quantum machine learning models when approximating functions containing high- or multi-frequency components, which hinders their ability to capture complex spectral structures. To overcome this limitation, the study introduces, for the first time, a multi-stage residual learning framework into the quantum domain. By iteratively training additional parameterized quantum circuit modules to fit the residual error from the previous stage, the model progressively enhances its spectral representational capacity. Integrating quantum Fourier series approximation, diverse encoding strategies, and multi-qubit architectures, the proposed approach significantly reduces test mean squared error on synthetic multi-frequency benchmarks featuring Gaussian, Lorentzian, and triangular envelopes. This effectively mitigates the quantum Fourier parametrization bias and substantially improves learning performance on multi-frequency signals.
📝 Abstract
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.