Local and Multi-Scale Strategies to Mitigate Exponential Concentration in Quantum Kernels

TL;DR

This paper empirically investigates local and multi-scale strategies to reduce exponential concentration in quantum kernels, demonstrating their effectiveness in enriching kernel spectra and mitigating concentration effects across various datasets.

Contribution

It introduces and benchmarks local and multi-scale quantum kernel strategies to address exponential concentration, showing their benefits over global fidelity kernels.

Findings

Local and multi-scale kernels mitigate concentration effects.

These strategies produce richer kernel spectra.

Impact on classification accuracy varies by dataset.

Abstract

Fidelity-based quantum kernels provide a direct interface between quantum feature maps and classical kernel methods, but they can exhibit exponential concentration: with increasing system size or circuit expressivity, the Gram matrix approaches the identity and suppresses informative similarity structure. We present an empirical study of two mitigation strategies implemented in Qiskit: (i) local (patch-wise) kernels that aggregate subsystem similarities, and (ii) multi-scale kernels that mix local and global similarity across patch granularities. We benchmark baseline, local, and multi-scale kernels under matched preprocessing, splits, and SVM protocols on several tabular datasets, sweeping the feature dimension $d\in\{4,6,\dots,20\}$. We report concentration diagnostics based on off-diagonal kernel statistics, spectral richness via effective rank, and centered alignment with labels. Across datasets, local and multi-scale constructions consistently mitigate concentration and yield richer kernel spectra relative to the global fidelity baseline, while the impact on classification accuracy depends on the dataset and dimension.