Mitigating Exponential Mixed Frequency Growth through Frequency Selection and Dimensional Separation in Quantum Machine Learning

Abstract: To leverage the potential computational speedup of quantum computing (QC), research in quantum machine learning (QML) has gained increasing prominence. Angle encoding techniques in QML models have been shown to generate truncated Fourier series, offering asymptotically universal function approximation capabilities. By selecting efficient feature maps (FMs) within quantum circuits, one can leverage the exponential growth of Fourier frequencies for improved approximation. In multi-dimensional settings, additional input dimensions induce further exponential scaling via mixed frequencies. In practice, however, quantum models frequently fail at regression tasks. Through two white-box experiments, we show that such failures can occur even when the relevant frequencies are present, due to an insufficient number of trainable parameters. In order to mitigate the double-exponential parameter growth resulting from double-exponentially growing frequencies, we propose frequency selection and dimensional separation as techniques to constrain the number of parameters, thereby improving trainability. By restricting the QML model to essential frequencies and permitting mixed frequencies only among feature dimensions with known interdependence, we expand the set of tractable problems on current hardware. We demonstrate the reduced parameter requirements by fitting two white-box functions with known frequency spectrum and dimensional interdependencies that could not be fitted with the default methods. The reduced parameter requirements permit us to perform training on a noisy quantum simulator and to demonstrate inference on real quantum hardware.