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.
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∈{4,6,…,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.
Quantum kernel methods utilize high-dimensional quantum feature spaces as a theoretically motivated strategy to address data scarcity in supervised classification. However, theoretical analyses predict exponential concentration of quantum kernel similarity values as qubit count increases. This study presents the first statistically controlled empirical evaluation of fixed-feature-map quantum support vector machines (QSVMs) for few-shot keyword spotting. The ZZFeatureMap and PauliFeatureMap circuits (2 to 3 qubits, PCA-compressed MFCC features, ~51% variance retained) were evaluated across 24 configurations and four low-resource regimes (5–30 samples per class) on the Google Speech Commands dataset, using ten independent seeds with Wilcoxon signed-rank tests and Bonferroni correction. Across all configurations and sample sizes, QSVM accuracy remained close to the five-class random baseline (19.3% at 30 samples per class; random baseline 20%), while RBF SVM achieved 38.7% and a compact CNN achieved 65.3%. Direct kernel similarity measurement confirmed severe concentration, where the 3-qubit QSVM produced near-identical same-class and different-class similarity means (both approximately 0.192), with 94.1% overlap. Under matched MFCC+PCA features, all classical kernels outperformed QSVM, including linear SVM (Cliff’s δ=−0.98 , p=0.002 ) and RBF SVM ( δ=−0.82 , p=0.002 , Bonferroni-corrected α=0.003 ), with CNN achieving complete stochastic dominance ( δ=−1.00 ). These results empirically validate quantum kernel concentration in structured acoustic classification within evaluated NISQ constraints, and apply specifically to the fixed-feature-map paradigm, not quantum kernel methods in general. The identified failure modes, namely exponential concentration, geometric misalignment, and limited circuit expressivity, provide concrete directions for future research into trainable quantum feature maps and hybrid quantum-classical architectures.
Quantum kernel methods have been proposed as a promising approach for leveraging near-term quantum computers for supervised learning, yet rigorous benchmarks against strong classical baselines remain scarce. We present a comprehensive empirical study of quantum kernel support vector machines (QSVMs) across nine binary classification datasets, four quantum feature maps, three classical kernels, and multiple noise models, totalling 970 experiments with strict nested cross-validation.
Our analysis spans four phases: (i) statistical significance testing, revealing that none of 29 pairwise quantum-classical comparisons reach significance at α=0.05; (ii) learning curve analysis over six training fractions, showing steeper quantum slopes on six of eight datasets that nonetheless fail to close the gap to the best classical baseline; (iii) hardware validation on IBM ibm_fez (Heron r2), demonstrating kernel fidelity r≥0.976 across six experiments; and (iv) seed sensitivity analysis confirming reproducibility (mean CV 1.4%).
A Kruskal-Wallis factorial analysis reveals that dataset choice dominates performance variance (ε2=0.73), while kernel type accounts for only 9%. Spectral analysis offers a mechanistic explanation: current quantum feature maps produce eigenspectra that are either too flat or too concentrated, missing the intermediate profile of the best classical kernel, the radial basis function (RBF). Quantum kernel training (QKT) via kernel-target alignment yields the single competitive result -- balanced accuracy 0.968 on breast cancer -- but with ~2,000x computational overhead.
Our findings provide actionable guidelines for quantum kernel research. The complete benchmark suite is publicly available to facilitate reproduction and extension.
Quantum kernel methods have emerged as a promising approach for leveraging high-dimensional feature spaces in machine learning, particularly in domains where classical kernel methods face scalability limitations. In this work, we present the first large-scale study of fidelity-quantum-kernel support vector machines for hyperspectral data classification without requiring heavy prior feature selection or dimensionality reduction. By simulating quantum kernels using tensor network contraction techniques and GPU acceleration, we overcome the computational bottlenecks traditionally associated with quantum models, achieving quadratic scaling O(n^2) in the number of qubits. Our approach enables the evaluation of quantum kernels on hyperspectral data with hundreds of spectral bands, aligning quantum feature spaces with real-world remote sensing applications. We provide an in-depth analysis of kernel bandwidth optimization, demonstrating its crucial role in mitigating exponential concentration effects and ensuring the model's ability to generalize. Experimental results on binary classification (Indian Pines and Methane Detection) and multiclass classification (Indian Pines) demonstrate that quantum kernels achieve competitive performance compared to a broad range of state-of-the-art classical baselines. As illustrative cases, on four 50-band splits selected from Indian Pines, the quantum model achieved a 78.0 pm6.2% accuracy for a binary classification task compared to 72.0 pm5.0% for the standard radial basis function (RBF) kernel. For a four-class classification task, the quantum kernel reached 83.3 pm3.1% accuracy, outperforming several state-of-the-art baselines. On five 75-band splits selected from the Methane Detection dataset, the quantum approach yielded 58.5\pm5.0% accuracy versus 55.1\pm2.5% for the classical counterpart...
Abstract Fidelity quantum kernels provide a provable advantage on classification problems where a group structure in the data can be exploited. However, in practical applications, the group structure may be unknown or approximate, and scaling to the ‘utility’ regime is affected by exponential concentration. We prove that an ideal behavior of fidelity kernels is always associated with a (possibly unknown) group structure in the feature map. We also propose a mitigation strategy for fidelity kernels, called Bit Flip Tolerance (BFT), to alleviate exponential concentration. Applied to real-world data with unknown structure, related to the charge schedule of electric vehicles, BFT proves useful on 40 + qubits, where mitigated accuracies reach 80%, in line with classical, compared to 33% without BFT. Through a synthetic dataset with 156 qubits, we obtain an accuracy of 80%, compared to 83% of classical models, and 37% of unmitigated quantum. This constitutes the largest experiment of quantum machine learning on IBM devices to date.
Kernel methods compare inputs through feature maps. Quantum kernels follow the same principle: input data are encoded into quantum states, which define quantum feature representations in Hilbert spaces. Kernel values are then obtained by estimating inner products between these states using suitable quantum circuit measurements. As a result, quantum kernels may be intractable to compute classically while remaining efficiently computable on quantum hardware, potentially leading to a quantum advantage. However, designing effective quantum kernels remains a major challenge. Many quantum kernels, such as the fidelity kernel, suffer from exponential concentration. This results in near-identity kernel matrices that fail to capture meaningful data correlations and lead to overfitting and poor generalization. In this paper, we propose a novel strategy for constructing quantum kernels that achieve good generalization performance, drawing inspiration from benign overfitting in classical machine learning. We introduce the concept of Local-Global quantum kernels, which combine two components: a local quantum kernel based on measurements of small subsystems, and a global quantum kernel derived from full-system measurements. To support the effectiveness of the proposed construction, we show theoretically and empirically that Local-Global quantum kernels exhibit benign overfitting.
We present ThylerSpectral-Audit-Q, a three-layer automated diagnostic framework for quantum kernel pathologies, validated across 124 configurations spanning 7 data domains plus a systematic six-family quantum kernel survey: tabular (digits, diabetes, California housing, covtype), clinical (Iris, Wine, Breast Cancer), quantum-native (V0/V1 ZZFeatureMap, V2 IQPFeatureMap), molecular graph (MUTAG, PTC_FM), industrial (semiconductor proxy, Steel Plates Faults, satellite proxy), time series (AR2, ECG5000 via ZZFeatureMap, QHMM, and TSHK kernels). The kernel survey applies six encoding families (PauliFeatureMap, AngleEncoding, AmplitudeEncoding, EfficientSU2, RandomCircuitKernel, QFTFeatureMap) to three tabular benchmark datasets at n ∈ {4, 6, 8}. Contribution 1 — Diagnostic framework. Three sequential layers — exponential concentration (EC) detection (Shannon effective rank ratio r_eff; five-tier spectral taxonomy), dequantization filter (Tang-Chia eigenvalue truncation), and kernel-target alignment evaluation (centered KTA) — combine into a composite PROCEED/CAUTION/ABORT Spectral Health Card. All four primary datasets (breast cancer, make_moons, Iris, Wine) yield ABORT or CAUTION verdicts under ZZFeatureMap. The framework generalizes: IQPFeatureMap reproduces EC_COLLAPSE at n=8 (r_eff=0.943) without recalibration. A data-kernel structural sensitivity is also detected: the same kernel yields LIKELY_DEQUANTIZABLE on molecular graph data (r_eff=0.153) versus CONFIRMED_DEQUANTIZABLE on tabular data (r_eff=0.095) — a distinction invisible to kernel architecture inspection alone. Contribution 2 — Advantage regime, retired at n_train=150 (v19). At n_train=50, Hamming Quantum Kernel (HQK) on quantum-native data produced the paper's only non-ABORT signal: large effect sizes over SVM-Linear on V2 (IQPFeatureMap) data (d=+0.949–1.015; under the corrected resampled t-test for repeated-CV dependence, p_NB=0.0605 for λ=0.5 and p_NB=0.0485 for λ=0.1 — marginal). Because the pre-registered robustness criterion required corrected significance in both λ configurations, a recovery experiment at n_train=150 was pre-registered (exploracion_010) and has now been executed (S052). Result: every V2 effect size reverses sign — λ=0.5: d=−0.215 (p_NB=0.6375, ns); λ=0.1: d=−1.243 (p_NB=0.0189 — a significant disadvantage, HQK now performing worse than SVM-Linear/RBF) — and the pre-registered effect-size-floor control (d≥0.5 in both λ, designed to distinguish gradual collapse from sign reversal) fails in both configurations. The secondary same-architecture V0/V1 arm also reverses sign (d=+0.27→−0.374 vs Linear, ns). Per the pre-registered decision rule, the cross-architecture advantage claim is retired, not restored. This converges with exploracion_012's finding that V6 — a circuit verified diagonal-unitary by construction but structurally distinct from V0/V1/V2's ZZ/IQP coupling — also shows significant disadvantage at n_train=150 (d=−1.10 to −1.13, p_NB<0.05 in both λ): at the matched n_train=150 standard, no HQK configuration tested in this paper shows a robust advantage over classical baselines. The dual concentration bottleneck — global EC (fidelity collapse) and local EC (Pauli-Z observables ⟨Z_i⟩→0 as n increases) — disables both fidelity kernels and projected quantum kernels simultaneously, and this negative result is now the paper's consistent empirical picture rather than a partial exception to it. A complementary negative structural result: exhaustive hyperparameter optimization over ZZFeatureMap (63 grid configurations: γ × reps × entanglement, plus 100-step continuous variational search) on classical time series data (AR2_n4, r_eff=0.2975) finds zero configurations with Δ_kta > 0 in full audit, confirming that quantum-native data is a necessary but not sufficient structural condition for any advantage signal under the full three-layer audit within the ZZFeatureMap family. The D6 kernel survey further shows that K2 (AngleEncoding) achieves Δ_kta>0 on classical tabular data, but is blocked from PROCEED by the dequantization filter, confirming that Layers 1–2 are essential to prevent false PROCEED verdicts. Contribution 3 — Quantum kernel phase diagram. The 124 audited configurations form a quantum kernel phase diagram in (r_eff × Δ_acc × n_qubits) space with two robustly confirmed empirical phases (ABORT, CAUTION) and measurable phase boundaries, plus a third, provisional phase (PROCEED, 7/124 configs) that has not been re-verified at the n_train=150 standard that retired Contribution 2 above. Seven geometric patterns are observable, including a cross-domain discovery: the STRUCTURED_LOW_RANK regime (0.10 < r_eff ≤ 0.35) appears independently in molecular graph data (r_eff ∈ [0.036, 0.312]) and industrial quality-control data (r_eff=0.22), despite no domain relationship — a finding invisible to kernel architecture inspection alone. Contribution 4 — Domain-agnostic spectral classification. A k-nearest-neighbor classifier trained exclusively on spectral audit features achieves F1-macro=1.000 and MCC=1.000 on an external test set of 20 configurations from three entirely unseen domains: molecular graphs, industrial QC, and time series. Random Forest feature importance reveals dataset_type_code=0% — the framework's diagnostic accuracy is domain-invariant by construction. To the best of our knowledge, ThylerSpectral-Audit-Q is the only pre-execution diagnostic tool integrating EC detection, dequantization filtering, and structural alignment evaluation into a single automated pipeline with an empirical multi-domain phase diagram, validated across 124 configurations spanning 7 data domains and a systematic six-family quantum kernel survey.