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arXiv:2610.04190

Reduced Cost Quantum Kernel Training

Daniel R. Germain, Benjamin Goldweber, Kayla J. Rodriguez, Kristen Rhinehardt

quant-phSubmitted Oct 3, 2026• Announced Oct 6, 2026• 0 citations
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Abstract

Quantum kernel methods are promising for near-term quantum machine learning, but their training costs are driven by the quadratic scaling of Gram-matrix construction and the measurement cost of estimating individual kernel values. In this work, we investigate one-class anomaly detection as a resource-efficient operating regime for training non-variational quantum kernel models. Using the ULB Credit Card Fraud Detection dataset, we evaluate three quantum feature maps across training-set sizes and qubit counts. Power-law saturation models reveal strong diminishing returns with increasing training data, with 500-sample models retaining approximately 90--93\% of their fitted asymptotic average precision (AP). We further observe no severe exponential concentration among the evaluated feature maps over the investigated qubit range, thereby avoiding the exponentially increasing measurement requirements associated with that regime. Finally, Nyström approximation reduces kernel construction from $\mathcal{O}(N^2)$ to $\mathcal{O}(Nm)$ evaluations, with QSVR retaining 94--97\% of its exact-kernel AP using only five landmarks. For ZZ-QSVR, downsampling from 4,000 to 500 training samples and applying a five-landmark approximation decreases the number of required kernel evaluations from approximately $8.0$ million to $2,485$, a reduction of more than 3,200-fold, while AP decreases from $0.812$ to $0.737$. Under the minimum IonQ pricing assumption considered here, this corresponds to an illustrative reduction in quantum kernel training cost from approximately \$1.34 billion to \$417{,}480. These results demonstrate that training-set reduction and low-rank approximation can substantially reduce the resources required to train non-variational quantum kernel anomaly detectors while retaining meaningful anomaly-discrimination performance.

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arXiv:2610.17027•Oct 12, 2026•0 citations

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Quantum machine learning on near-term noisy quantum devices has generated substantial theoretical interest, but rigorous empirical comparisons under realistic noise on practically relevant data remain scarce. This paper compares two paradigmatic quantum learning models, a Quantum Support Vector Machine (QSVM) built on the ZZFeatureMap quantum kernel and a Variational Quantum Classifier (VQC) with an EfficientSU2/RealAmplitudes ansatz, against tuned classical baselines (SVM with four kernels, Random Forest, XGBoost, LightGBM and CatBoost) on the ULB Credit Card Fraud dataset (284,807 transactions, 0.17% fraud). All models share an identical 4-qubit PCA-reduced feature space, evaluated on the full unbalanced test fold over 15 fits (3 folds × 5 seeds) and reported as mean ± standard deviation with bootstrap confidence intervals, AUPRC as the primary metric. Noise robustness is assessed under depolarizing noise p∈{0,0.001,0.01,0.05}, with ranking preservation measured directly through Spearman ρ and Kendall τ between the noisy and noiseless decision scores rather than read off AUPRC, alongside the per-paradigm computational cost. At four qubits the classical baselines lead (AUPRC 0.60 to 0.74, CatBoost best), above the VQC (0.494) and the QSVM (0.240); the controlled QSVM-versus-RBF–SVM comparison puts the cost of the quantum kernel at about 0.45 AUPRC. Under noise the QSVM keeps its score ranking (ρ=0.998 at p=0.001, 0.906 at p=0.01) and an operational decision threshold (recall 0.87 to 0.89, stable calibration), while the VQC AUPRC peaks non-monotonically at p=0.01 (0.494 rising to 0.654, then easing to 0.569 at p=0.05) even as its ranking decays monotonically (ρ from 0.72 to near zero), so average precision on its own misrepresents how noise affects it. The quantum models do not surpass the tuned classical reference at four qubits; the contribution is methodological: under noise, AUPRC has to be read together with a genuine rank statistic, because the two can move in opposite directions.

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Quantum kernel methods are the most concrete near-term proposal for quantum machine learning on classical data, yet published benchmarks disagree on whether they outperform tuned classical kernels, and the disagreement cannot be resolved from the literature because the comparisons are not protocol-controlled. We argue that the obstacle is one of measurement, not theory: the usual setup fixes the quantum feature map's data scaling while grid-searching the classical kernel's hyperparameters, recomputes nothing per fold, and reports cost in simulator time. We therefore treat the comparison itself as the object of study and impose three controls. The bandwidth is tuned on the same grid budget as the classical $γ$, the kernel is recomputed inside every fold from fold-fitted preprocessing, and cost is counted in hardware circuit evaluations. On top of these we run a factorial design of 324 cells in which dimensionality, training size, class separability and data structure are explicit covariates, together with four standard datasets, a shots-versus-gate-noise ablation, and a cost sweep. The dominant effect we measure is not quantum: tuning the feature map's bandwidth is worth 0.120 accuracy on synthetic data and 0.339 on real data, exceeding every quantum-classical difference we observe. Tuned quantum kernels are then statistically indistinguishable from a tuned RBF on all four real datasets. The advantage on synthetic data is confined almost entirely to data generated by the feature map itself, and it shrinks with dimensionality (-0.0045 per qubit) while growing with sample size. Finite sampling costs an order of magnitude more accuracy than gate and readout noise, and a fidelity kernel needs $5\times10^{7}$ circuits at n = 10,000. Protocol variance is thus sufficient to explain the field's contradictory conclusions.

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A Hybrid Quantum Machine Learning Framework Based on Nyström Approximation for Scalability Enhancement of Quantum Kernels

Mohammad Hossein Rezvannia

Abstract Quantum Machine Learning (QML) is an emerging field at the intersection of quantum computing and artificial intelligence, offering high potential for extracting complex nonlinear patterns from high-dimensional data. However, many quantum kernel-based methods suffer from high computational costs, the requirement for numerous quantum circuit executions, and the limitations of current Noisy Intermediate-Scale Quantum (NISQ) hardware. These challenges, particularly when dealing with large datasets, represent a major barrier to the widespread adoption of these methods in practical applications. In this paper, a hybrid quantum machine learning framework based on Nyström approximation is proposed to effectively reduce the computational complexity of quantum kernels and enhance their scalability. In the proposed framework, classical data is first mapped to the quantum Hilbert space via parameterized quantum circuits, and the quantum kernel is defined based on the overlap of quantum states. Then, utilizing the Nyström approximation technique, the full kernel calculation is restricted to a representative subset of data, and the final learning process is conducted in the classical space. This approach reduces computational complexity from \(\:O\left({N}^{2}\right)\) to \(\:O\left(Nm\right)\), where N is the number of samples and m is the size of the selected subset. Analytical results and simulations on the Breast Cancer Wisconsin dataset demonstrate that the proposed framework reduces computational costs by over 60% while maintaining the geometric structure of the quantum feature space, with a classification accuracy loss of less than 3%. These results suggest that the proposed method can serve as a practical, scalable, and noise-resilient solution for applying quantum machine learning to real-world data-driven problems.

•Feb 18, 2026•0 citations
Quantum kernel methods are promising for near-term quantum machine learning, but their training costs are driven by the quadratic scaling of Gram-matrix construction and the measurement cost of estimating individual kernel values. In this work, we investigate one-class anomaly detection as a resource-efficient operating regime for training non-variational quantum kernel models. Using the ULB Credit Card Fraud Detection dataset, we evaluate three quantum feature maps across training-set sizes and qubit counts. Power-law saturation models reveal strong diminishing returns with increasing training data, with 500-sample models retaining approximately 90--93% of their fitted asymptotic average precision (AP). We further observe no severe exponential concentration among the evaluated feature maps over the investigated qubit range, thereby avoiding the exponentially increasing measurement requirements associated with that regime. Finally, Nyström approximation reduces kernel construction from O(N2)\mathcal{O}(N^2)O(N2) to O(Nm)\mathcal{O}(Nm)O(Nm) evaluations, with QSVR retaining 94--97% of its exact-kernel AP using only five landmarks. For ZZ-QSVR, downsampling from 4,000 to 500 training samples and applying a five-landmark approximation decreases the number of required kernel evaluations from approximately 8.08.08.0 million to 2,4852,4852,485, a reduction of more than 3,200-fold, while AP decreases from 0.8120.8120.812 to 0.7370.7370.737. Under the minimum IonQ pricing assumption considered here, this corresponds to an illustrative reduction in quantum kernel training cost from approximately $1.34 billion to $417{,}480. These results demonstrate that training-set reduction and low-rank approximation can substantially reduce the resources required to train non-variational quantum kernel anomaly detectors while retaining meaningful anomaly-discrimination performance.
Detecting credit card fraud is notoriously difficult because fraudulent charges make up less than 0.2% of total transactions. Quantum Machine Learning (QML), particularly Quantum Kernel Support Vector Machines (QSVM), promises richer feature representations by embedding classical data into high-dimensional Hilbert spaces. However, whether this theoretical advantage holds up when managing real-world financial data bottlenecks remains unclear. We designed a controlled benchmark comparing three quantum encodings, Angle, ZZFeatureMap, and IQP against classical models while keeping sample sizes, feature counts, and scaling strictly matched. To handle class skew without corrupting quantum state overlaps, we embedded synthetic minority oversampling (SMOTE) directly into the quantum Gram matrix workflow, raising PR-AUC scores by roughly +0.04 across feature maps. When trained on the full European credit card dataset with 30 features, classical baselines reached a top PR-AUC of 0.983. However, restricting classical models to match QSVM hardware constraints (500 samples, 4 PCA features) dropped their PR-AUC to 0.750. This puts our best quantum result 0.730 PR-AUC using IQP encoding on 150 samples into realistic context. Testing on physical quantum hardware reduced performance further to 0.62 due to CNOT gate errors and thermal noise. Overall, our findings pinpoint two main operational bottlenecks for near-term quantum kernels: the scaling of Gram matrix evaluation and heavy information loss from squeezing features onto small qubit registers.
Abstract Support vector machines (SVMs) are widely used for classification, but become computationally expensive as datasets grow and data become high-dimensional or nonlinearly separable. Quantum support vector machines (QSVMs) enhance the traditional SVMs by replacing conventional feature mappings with quantum-implemented embeddings that project data into high-dimensional Hilbert spaces, potentially improving expressive power and efficiency. However, training QSVMs on current noisy intermediate-scale quantum (NISQ) devices remains challenging, as constructing the quantum kernel matrix requires repeated circuit executions and scales quadratically with dataset size. We propose NQ-SVM , a hybrid quantum-classical method that incorporates the Nystr\"om approximation to enhance the scalability of QSVMs. By selecting a small set of landmark points, the Nystr\"om method yields a low-rank kernel approximation, reducing complexity from (O(n2))(O(n^2))(O(n2)) to (O(nm))(O(nm))(O(nm)), while preserving the quantum advantages in handling high-dimensional, non-linear data. Extensive simulations demonstrate that NQ-SVM achieves comparable classification accuracy to classical SVMs and full QSVMs, while substantially reducing training time and kernel evaluation costs. These results underscore the potential of NQ-SVM as a scalable and practical hybrid framework for near-term quantum machine learning (QML).
Quantum machine learning on near-term noisy quantum devices has generated substantial theoretical interest, but rigorous empirical comparisons under realistic noise on practically relevant data remain scarce. This paper compares two paradigmatic quantum learning models, a Quantum Support Vector Machine (QSVM) built on the ZZFeatureMap quantum kernel and a Variational Quantum Classifier (VQC) with an EfficientSU2/RealAmplitudes ansatz, against tuned classical baselines (SVM with four kernels, Random Forest, XGBoost, LightGBM and CatBoost) on the ULB Credit Card Fraud dataset (284,807 transactions, 0.17% fraud). All models share an identical 4-qubit PCA-reduced feature space, evaluated on the full unbalanced test fold over 15 fits (3 folds × 5 seeds) and reported as mean ± standard deviation with bootstrap confidence intervals, AUPRC as the primary metric. Noise robustness is assessed under depolarizing noise p∈{0,0.001,0.01,0.05}, with ranking preservation measured directly through Spearman ρ and Kendall τ between the noisy and noiseless decision scores rather than read off AUPRC, alongside the per-paradigm computational cost. At four qubits the classical baselines lead (AUPRC 0.60 to 0.74, CatBoost best), above the VQC (0.494) and the QSVM (0.240); the controlled QSVM-versus-RBF–SVM comparison puts the cost of the quantum kernel at about 0.45 AUPRC. Under noise the QSVM keeps its score ranking (ρ=0.998 at p=0.001, 0.906 at p=0.01) and an operational decision threshold (recall 0.87 to 0.89, stable calibration), while the VQC AUPRC peaks non-monotonically at p=0.01 (0.494 rising to 0.654, then easing to 0.569 at p=0.05) even as its ranking decays monotonically (ρ from 0.72 to near zero), so average precision on its own misrepresents how noise affects it. The quantum models do not surpass the tuned classical reference at four qubits; the contribution is methodological: under noise, AUPRC has to be read together with a genuine rank statistic, because the two can move in opposite directions.
Quantum kernel methods are the most concrete near-term proposal for quantum machine learning on classical data, yet published benchmarks disagree on whether they outperform tuned classical kernels, and the disagreement cannot be resolved from the literature because the comparisons are not protocol-controlled. We argue that the obstacle is one of measurement, not theory: the usual setup fixes the quantum feature map's data scaling while grid-searching the classical kernel's hyperparameters, recomputes nothing per fold, and reports cost in simulator time. We therefore treat the comparison itself as the object of study and impose three controls. The bandwidth is tuned on the same grid budget as the classical γγγ, the kernel is recomputed inside every fold from fold-fitted preprocessing, and cost is counted in hardware circuit evaluations. On top of these we run a factorial design of 324 cells in which dimensionality, training size, class separability and data structure are explicit covariates, together with four standard datasets, a shots-versus-gate-noise ablation, and a cost sweep. The dominant effect we measure is not quantum: tuning the feature map's bandwidth is worth 0.120 accuracy on synthetic data and 0.339 on real data, exceeding every quantum-classical difference we observe. Tuned quantum kernels are then statistically indistinguishable from a tuned RBF on all four real datasets. The advantage on synthetic data is confined almost entirely to data generated by the feature map itself, and it shrinks with dimensionality (-0.0045 per qubit) while growing with sample size. Finite sampling costs an order of magnitude more accuracy than gate and readout noise, and a fidelity kernel needs 5×1075\times10^{7}5×107 circuits at n = 10,000. Protocol variance is thus sufficient to explain the field's contradictory conclusions.
We develop quantum protocols for anomaly detection and apply them to the task of credit card fraud detection (FD). First, we establish classical benchmarks based on supervised and unsupervised machine learning methods, where average precision is chosen as a robust metric for detecting anomalous data. We focus on kernel-based approaches for ease of direct comparison, basing our unsupervised modelling on one-class support vector machines (OC-SVM). Next, we employ quantum kernels of different type for performing anomaly detection, and observe that quantum FD can challenge equivalent classical protocols at increasing number of features (equal to the number of qubits for data embedding). Performing simulations with registers up to 20 qubits, we find that quantum kernels with re-uploading demonstrate better average precision, with the advantage increasing with system size. Specifically, at 20 qubits we reach the quantum-classical separation of average precision being equal to 15%. We discuss the prospects of fraud detection with near- and mid-term quantum hardware, and describe possible future improvements.
Abstract Quantum Machine Learning (QML) is an emerging field at the intersection of quantum computing and artificial intelligence, offering high potential for extracting complex nonlinear patterns from high-dimensional data. However, many quantum kernel-based methods suffer from high computational costs, the requirement for numerous quantum circuit executions, and the limitations of current Noisy Intermediate-Scale Quantum (NISQ) hardware. These challenges, particularly when dealing with large datasets, represent a major barrier to the widespread adoption of these methods in practical applications. In this paper, a hybrid quantum machine learning framework based on Nyström approximation is proposed to effectively reduce the computational complexity of quantum kernels and enhance their scalability. In the proposed framework, classical data is first mapped to the quantum Hilbert space via parameterized quantum circuits, and the quantum kernel is defined based on the overlap of quantum states. Then, utilizing the Nyström approximation technique, the full kernel calculation is restricted to a representative subset of data, and the final learning process is conducted in the classical space. This approach reduces computational complexity from  O(N2)\:O\left({N}^{2}\right)O(N2) to  O(Nm)\:O\left(Nm\right)O(Nm), where N is the number of samples and m is the size of the selected subset. Analytical results and simulations on the Breast Cancer Wisconsin dataset demonstrate that the proposed framework reduces computational costs by over 60% while maintaining the geometric structure of the quantum feature space, with a classification accuracy loss of less than 3%. These results suggest that the proposed method can serve as a practical, scalable, and noise-resilient solution for applying quantum machine learning to real-world data-driven problems.