Margin-Based Generalisation Bounds for Quantum Kernel Methods under Local Depolarising Noise

TL;DR

This work analyzes generalisation for Quantum Kernel Methods, focusing on QSVMs in the NISQ era under realistic local depolarising noise. It develops margin-based theory by deriving upper and lower bounds on QSVM margins and kernel values when local noise degrades quantum states, linking margin deterioration to generalisation performance. Empirical results on multiple datasets and real hardware demonstrate a strong correlation between geometric margins and test accuracy (e.g., Pearson $r>0.9$), and reveal that local noise offers a more faithful degradation model than global depolarising noise. A dataset-selection pipeline validates the bounds across diverse tasks, and the study highlights the importance of modeling per-qubit noise for robust QSVM generalisation in practice, with code made publicly available.

Abstract

Generalisation refers to the ability of a machine learning (ML) model to successfully apply patterns learned from training data to new, unseen data. Quantum devices in the current Noisy Intermediate-Scale Quantum (NISQ) era are inherently affected by noise, which degrades generalisation performance. In this work, we derive upper and lower margin-based generalisation bounds for Quantum Kernel-Assisted Support Vector Machines (QSVMs) under local depolarising noise. These theoretical bounds characterise noise-induced margin decay and are validated via numerical simulations across multiple datasets, as well as experiments on real quantum hardware. We further justify the focus on margin-based measures by empirically establishing margins as a reliable indicator of generalisation performance for QSVMs. Additionally, we motivate the study of local depolarising noise by presenting empirical evidence demonstrating that the commonly used global depolarising noise model is overly optimistic and fails to accurately capture the degradation of generalisation performance observed in the NISQ era.

Figures (13)

• Figure 1: Diagram depicting the optimal linear hyperplane returned by an SVM to separate two classes of data points (green and pink circles). The support vectors used to determine the hyperplane and the corresponding geometric margin are labelled. The equations of the separating and marginal hyperplanes are depicted in blue. A hard margin is depicted here since all samples are classified with a distance greater than the geometric margin away from the separating hyperplane.
• Figure 2: Diagram depicting the optimal linear hyperplane returned by the SVM to separate the two classes of data points (green and pink circles). The equations of the separating and marginal hyperplanes are depicted in blue. A soft margin is depicted here since some points are classified on the correct side of the linear hyperplane but with distances $1-\xi_i$ and $1-\xi_j$ from the separating hyperplane. Misclassified points which are placed on the wrong side of the hyperplane are also labelled.
• Figure 3: Quantum circuit layer implementing IQP encoding with nearest-neighbour entanglement using Hadamard gates (in purple), $R_z$ rotations for each input feature $\boldsymbol{x}_i$ (in red) and ZZ entangling gates (in green).
• Figure 4: Box-plots depicting the geometric margin distribution calculated using datasets with increasing fractions of corrupted training labels for various datasets. Each dataset is labelled at the top-right corner of each plot. These plots were generated without the outliers to maintain the emphasis on the median geometric margin.
• Figure 5: Plots overlaying the decreasing test accuracy and median geometric margin graphs for increasing fractions of corrupted training labels. The test accuracy (blue) is depicted with the error obtained from 5-fold CV, and the geometric margins (red) are taken from the box-plots in Figure \ref{['fig:boxplots']} above. Each dataset is labelled at the bottom-left corner of each plot.