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

2025-08-14

AI Summary

In quantum machine learning, angle encoding induces hybrid frequencies that grow doubly exponentially, causing parameter explosion; even when target functions possess critical spectral components, models fail to train due to insufficient parameters. This work identifies and alleviates this parameter trainability bottleneck via a novel frequency selection and dimension separation strategy. Specifically, we perform frequency-domain analysis to identify an effective subset of spectral components and enforce dimension-wise decoupling to curb parameter growth. Our method builds upon angle-encoded feature maps and integrates white-box function verification with parameter-constrained optimization. It enables end-to-end training and inference on noisy quantum simulators and real hardware (IBM Quantum). Experiments demonstrate successful fitting of two classes of high-dimensional functions that conventional QML approaches fail to converge on. While preserving accuracy, our approach reduces parameter counts by one to two orders of magnitude, substantially expanding the problem scale and dimensionality feasible on current NISQ devices.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• User Modeling, Personalization and Recommendation: ML for personalized search and recommendations
• Security and Privacy: Security and privacy of machine learning and AI applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Mitigate exponential mixed frequency growth in QML
• Improve trainability by reducing parameter requirements
• Enable tractable problems on current quantum hardware

Innovation

• Frequency selection to constrain parameters
• Dimensional separation for mixed frequencies
• Improved trainability on quantum hardware