Iqp quantum kernels show distinct expressivity regimes with data match
Expressivity of IQP Feature-Map Quantum Kernels via Reproducing Kernel Hilbert Spaces
Machine Learning
Summary
Quantum computers can help find patterns in complex data by representing it in special spaces, but understanding which patterns they can catch is tricky. The authors studied a particular quantum method called IQP kernels and found that their ability to capture patterns depends on settings that control how the data is transformed. They identified three types of pattern strengths—localized, repeating, and mixed—and showed that these only help when they fit the data's own structure. This work helps people know how to set parameters for these quantum tools when certain data patterns are expected.
What this means in practice
- •For quantum machine learning engineers: Tune IQP kernel parameters to improve classification and anomaly detection when data has known periodic or multimodal patterns.
- •For signal processing teams: Select bandwidth settings of quantum-inspired kernels to better capture frequency structures in complex signals using polynomial transformations.
Authors
Valeria Repetto, Stefano Melchionna, Romina D'Aurizio
Abstract
Quantum kernels are among the most promising methods in quantum machine learning, providing a natural way to embed data into high-dimensional Hilbert spaces via quantum circuits and capturing non-linear relationships in data. Their learning behaviour is governed by the function class they represent, which is formally characterized by the structure of the associated Reproducing Kernel Hilbert Space (RKHS). However, a theoretical characterization of this function class, and of how it depends on the choice of feature map and its hyperparameters, is still lacking. In this work, we focus on the Instantaneous Quantum Polynomial (IQP) feature map and derive a Fourier representation of the associated RKHS generators as finite sums of trigonometric functions evaluated on polynomially transformed input variables, with a finite set of accessible frequencies determined by the circuit construction and scaled by the bandwidth hyperparameter. This representation identifies three bandwidth regimes with qualitatively distinct expressive properties from localised, through multimodal, to periodic. Numerical experiments on synthetic datasets with controlled periodic and multimodal structure, across data dimensions and feature-map configurations, reproduce the three regimes in both supervised classification and unsupervised anomaly detection, benchmarked against a classical Gaussian kernel. These results make the relationship between expressivity and data structure explicit for IQP kernels: expressive power translates into a practical benefit only when the accessible frequencies match with the structure of the data, which provides a criterion for selecting the bandwidth when the periodic or multimodal structure of data is known.