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Magnitude-dependent quantum advantage in Forbush decrease detection: A quantum kernel SVM benchmark

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dc.contributor.authorSierra Porta, David
dc.contributor.researchgroupGrupo de Investigación Gravitación y Matemática Aplicada
dc.contributor.seedbedsSemillero de Investigación en Astronomía y Ciencia de Datos
dc.date.accessioned2026-07-06T20:58:26Z
dc.date.issued2026-07-04
dc.descriptionContiene gráficos
dc.description.abstractForbush decreases (FDs) — transient reductions in galactic cosmic ray intensity driven by interplanetary coronal mass ejections — are key observables in space weather monitoring, yet their automated detection from multivariate solar wind and neutron monitor time series remains a challenging classification problem. Here we report a systematic benchmark of quantum kernel support vector machines (QKSVM) for FD detection, in which the FD magnitude threshold emerges as the governing factor separating two distinct classification regimes. Using 2971 confirmed events from the Forbush Effects and Interplanetary Disturbances (FEID) catalogue, combined with hourly OMNI solar wind parameters — including interplanetary magnetic field (IMF) components, solar wind speed, proton density, proton temperature, and the Kp and Dst geomagnetic indices — and galactic cosmic ray count rates from the Jungfraujoch neutron monitor station (JUNG, NMDB), we construct a balanced FD versus quiet-time classification dataset and extract 121 statistical features across eleven physical channels. A ZZFeatureMap quantum kernel with 4–8 qubits is benchmarked against a classical radial basis function (RBF) SVM across 180 experimental configurations spanning FD magnitude thresholds of 0%–7%, circuit depths of 1–3 repetitions, and quantum training sizes of 50–250 samples. We find that below a magnitude threshold of 4%, the classical kernel consistently outperforms the quantum alternative (mean at min_magn %). Above this threshold, the relationship inverts: at min_magn % the quantum kernel achieves positive mean in 72% of configurations, rising to 100% of configurations at min_magn % (mean , peak with 4–8 qubits), indicating that the entanglement structure of the ZZFeatureMap captures non-linear correlations between IMF dynamics and cosmic ray modulation that the RBF kernel cannot represent. The magnitude threshold of 4% thus constitutes a physically interpretable boundary between a noise-dominated regime where classical methods suffice and a signal-rich regime where quantum kernels provide measurable and statistically significant advantage (, Wilcoxon signed-rank test). These results establish FD magnitude as a key predictor of quantum classification performance, and suggest that near-term quantum machine learning applications in heliophysics should preferentially target high-amplitude space weather events.
dc.description.researchareaAnalítica de datos y Big Data
dc.description.researchareaClima espacial y rayos cósmicos
dc.format.extent17 páginas
dc.format.mimetypeapplication/pdf
dc.identifier.citationSierra-Porta, D. (2026). Magnitude-dependent quantum advantage in Forbush decrease detection: A quantum kernel SVM benchmark. Astronomy and Computing, Volume 57. 101160. October 2026. https://doi.org/10.1016/j.ascom.2026.101160
dc.identifier.doi10.1016/j.ascom.2026.101160
dc.identifier.urihttps://hdl.handle.net/20.500.12585/14516
dc.language.isoeng
dc.publisherAstronomy and Computing
dc.relation.referencesAstronomy and Computing Date: October 2026 Article: 101160 Volume: Volume 57 Published by Elsevier Research article Open access Under a Creative Commons license Get rights and content Full length paper Magnitude-dependent quantum advantage in Forbush decrease detection: A quantum kernel SVM benchmark D. Sierra-Porta Universidad Tecnológica de Bolívar. Grupo de Investigación Gravitación y Matemática Aplicada - GIGMA & Grupo de Investigación Física Aplicada y Procesamiento de Imágenes y Señales - FAPIS., Parque Industrial y Tecnológico Carlos Vélez Pombo Km 1 Vía Turbaco, Cartagena de Indias, 130010, Bolívar, Colombia Received 13 May 2026, Accepted 25 June 2026, Available online 30 June 2026, Version of Record 4 July 2026. What do these dates mean? Show less Download full issue Cite Add to Mendeley Share 10.1016/j.ascom.2026.101160 Magnitude-dependent quantum advantage in Forbush decrease detection: A quantum kernel SVM benchmark Download full issue Article Outline Highlights Abstract MSC Keywords 1. Introduction 2. Data 3. Methods 4. Results 5. Discussion 6. Conclusions Declaration of competing interest Acknowledgements Appendix. Quantum computing primer Data availability References Show full outline Figures (6) Fig. 1. Gate-level diagram of the ZZFeatureMap circuit for k=4 qubits and reps=1. The circuit begins with a layer of Hadamard gates (H) that place all qubits in equal superposition, followed by single-qubit p... Fig. 2. (a) Mean ROC-AUC for each of the three classifiers — RBF-SVM (full), RBF-SVM (sub), and Quantum Kernel SVM — as a function of FD magnitude threshold. Error bars indicate ±1 standard deviation across e... Fig. 3. Mean ΔAUC as a function of FD magnitude threshold (min_magn), averaged over all 180 experimental configurations. The shaded band indicates ±1 standard deviation. The transition from classical to quant... Fig. 4. Heatmaps of mean ΔAUC for all parameter combinations. (a) ΔAUC as a function of FD magnitude threshold and qubit count. (b) ΔAUC as a function of FD magnitude threshold and circuit depth (reps). (c) Δ... Fig. 5. (a) ΔAUC as a function of quantum training size nq for each FD magnitude threshold. Shaded bands indicate ±1 standard deviation across qubit count and circuit depth configurations. (b) Scatter plot of... Fig. 6. (a) ΔAUC as a function of FD magnitude threshold for each qubit count (k∈{4,6,8}). Error bars indicate ±1 standard deviation across circuit depths and training sizes. The apparent non-monotonic trend ... Tables (5) Table 1. Summary of the classification dataset. Note that the missing value fraction (∼9.5%) refers to the proportion of individual hourly entries that contained NaN values across all windows and channels befo... Table 2. Dataset composition by FD magnitude threshold. FD and non-FD windows are balanced at all thresholds. Training and test partitions are obtained by stratified 75/25 splitting of the total window count. Table 3. Statistical summary of ΔAUC by FD magnitude threshold. The Wilcoxon signed-rank test (one-sided, H1: ΔAUC>0) is reported alongside the rank-biserial correlation r as an effect size measure (|r|>0.1 sm... Table 4. Theoretical quantum kernel evaluation counts by training size (nq) and qubit count (k). Ktrain counts fidelity evaluations for the upper-triangular training kernel matrix; Ktest counts evaluations aga... Table 5. Summary of results by FD magnitude threshold and qubit count. Columns report the mean AUC of the full and subsampled RBF-SVM baselines, the mean AUC of the Quantum Kernel SVM, the mean and standard de... Highlights • Quantum kernel SVMs outperform classical RBF above a 4% FD magnitude threshold. • Quantum advantage reaches 100% of configurations at Magn 5% ( AUC = ＋0.074). • FD magnitude is a key predictor of quantum classification performance in space weather. Abstract Forbush decreases (FDs) — transient reductions in galactic cosmic ray intensity driven by interplanetary coronal mass ejections — are key observables in space weather monitoring, yet their automated detection from multivariate solar wind and neutron monitor time series remains a challenging classification problem. Here we report a systematic benchmark of quantum kernel support vector machines (QKSVM) for FD detection, in which the FD magnitude threshold emerges as the governing factor separating two distinct classification regimes. Using 2971 confirmed events from the Forbush Effects and Interplanetary Disturbances (FEID) catalogue, combined with hourly OMNI solar wind parameters — including interplanetary magnetic field (IMF) components, solar wind speed, proton density, proton temperature, and the Kp and Dst geomagnetic indices — and galactic cosmic ray count rates from the Jungfraujoch neutron monitor station (JUNG, NMDB), we construct a balanced FD versus quiet-time classification dataset and extract 121 statistical features across eleven physical channels. A ZZFeatureMap quantum kernel with 4–8 qubits is benchmarked against a classical radial basis function (RBF) SVM across 180 experimental configurations spanning FD magnitude thresholds of 0%–7%, circuit depths of 1–3 repetitions, and quantum training sizes of 50–250 samples. We find that below a magnitude threshold of 4%, the classical kernel consistently outperforms the quantum alternative (mean at min_magn %). Above this threshold, the relationship inverts: at min_magn % the quantum kernel achieves positive mean in 72% of configurations, rising to 100% of configurations at min_magn % (mean , peak with 4–8 qubits), indicating that the entanglement structure of the ZZFeatureMap captures non-linear correlations between IMF dynamics and cosmic ray modulation that the RBF kernel cannot represent. The magnitude threshold of 4% thus constitutes a physically interpretable boundary between a noise-dominated regime where classical methods suffice and a signal-rich regime where quantum kernels provide measurable and statistically significant advantage ( , Wilcoxon signed-rank test). These results establish FD magnitude as a key predictor of quantum classification performance, and suggest that near-term quantum machine learning applications in heliophysics should preferentially target high-amplitude space weather events. MSC 81P68; 68T05; 85A35; 62H30; 68Q12 Keywords Forbush decrease; Quantum kernels; Machine learning; Classification task; Geomagnetic indices Previous article in this issue Next article in this issue 1. Introduction Forbush decreases (FDs) are among the most well-characterised transient phenomena in the heliosphere, manifesting as sudden reductions in the flux of galactic cosmic rays (GCRs) observed at ground level by neutron monitor networks (Forbush, 1938). These events are predominantly associated with the passage of interplanetary coronal mass ejections (ICMEs) and their preceding shocks, which modulate the GCR flux through enhanced magnetic field turbulence and the shielding effect of ICME magnetic structures (Cane, 2000, Richardson and Cane, 2011). Because FDs encode information about the magnetic topology and dynamical state of interplanetary disturbances, their detection and characterisation constitute a central problem in space weather monitoring and GCR transport physics (Belov, 2008). The automated identification of FDs from continuous neutron monitor time series is, however, a non-trivial classification task. FD amplitudes span nearly two orders of magnitude — from sub-percent modulations indistinguishable from diurnal variations to decreases exceeding 20% in major events — and their onset morphology ranges from abrupt shock-associated drops to gradual, diffusion-driven depressions (Melkumyan et al., 2023, Melkumyan et al., 2024, Richardson and Cane, 2011, Belov et al., 2001). Classical threshold-based detection methods (Lockwood, 1971, Dumbović et al., 2024) and rule-based algorithms struggle with weak events and with the high background variability characteristic of solar maximum periods. The principal difficulty is that weak FDs occupy a regime where the signal-to-noise ratio in individual station data is insufficient for reliable automated discrimination, even when multivariate solar wind context is available. These limitations have motivated a growing body of work applying supervised machine learning to FD detection and space weather forecasting more broadly (Camporeale, 2019, Bobra and Couvidat, 2015, Ye et al., 2025). Support vector machines (SVMs), random forests, and deep neural networks have all been applied to classification problems in heliophysics, generally achieving strong performance on well-defined event catalogues. Despite this progress, the fundamental question of whether quantum machine learning (QML) methods can offer any advantage over classical approaches for space weather classification problems has not been addressed. Quantum kernel methods, in particular, have attracted considerable theoretical and experimental interest as a potential route to quantum advantage in supervised learning (Schuld and Killoran, 2019, Havlíček et al., 2019). In this framework, classical data vectors are encoded into quantum states via a parameterised quantum circuit — the feature map — using angle encoding, in which each feature component is mapped to a rotation angle within the circuit gates, so that the full feature vector is embedded into the amplitudes of a multi-qubit quantum state—and the inner product defines a kernel function that replaces the classical RBF or polynomial kernels in a standard SVM (Liu et al., 2021). The expressibility of such kernels grows exponentially with the number of qubits, raising the possibility that quantum feature maps can capture correlations in physical data that are inaccessible to polynomial-time classical kernels (Schuld, 2021, Huang et al., 2021). However, empirical demonstrations of genuine quantum advantage on real-world datasets remain scarce and contested (Kübler et al., 2021, Thanasilp et al., 2024), underscoring the need for careful, domain-specific benchmarks. A self-contained introduction to the quantum computing concepts underlying this framework is provided in Appendix. 1.1. Related work 1.1.1. Machine learning for space weather and forbush decrease detection The application of supervised machine learning to space weather problems has grown substantially over the past decade, driven by the increasing availability of multi-instrument observational datasets and the computational maturity of modern ML frameworks (Camporeale, 2019). Within solar physics, SVMs and random forests have been applied to solar flare prediction from photospheric magnetograms (Bobra and Couvidat, 2015), while deep learning architectures have been employed for solar wind forecasting (Upendran et al., 2020) and geomagnetic storm classification (Ye et al., 2025). A comprehensive review of these developments is provided by Camporeale (2019), who identifies classification under class imbalance and the physical interpretability of learned features as the two central open challenges in the field. Despite this progress, the automated detection of Forbush decreases has received comparatively limited attention from the machine learning community. Traditional FD identification relies on visual inspection of neutron monitor time series or on threshold-based algorithms applied to the global survey method output (Melkumyan et al., 2024, Melkumyan et al., 2023), both of which are labor-intensive and prone to inconsistency for weak events. Okike (2020) proposed a Fourier-based preprocessing pipeline to disentangle FD signals from diurnal cosmic ray anisotropies prior to automated detection, highlighting the signal complexity that complicates direct ML approaches. While prior work by the present author has explored complexity-based and graph-theoretic descriptors for FD characterisation (Sierra-Porta et al., 2024, Perez-Navarro and Sierra-Porta, 2024), the application of kernel-based supervised classifiers to the binary FD versus quiet-time detection problem using multivariate solar wind and neutron monitor features remains, to our knowledge, largely unexplored. The present study aims to fill this gap with a systematic experimental benchmark. 1.1.2. Quantum kernel methods: Theory and empirical benchmarks A self-contained introduction to the quantum computing concepts used throughout this paper is provided in Appendix, which readers may find useful as background for the material presented in this and subsequent sections. Quantum kernel methods were placed on a rigorous theoretical footing by Schuld and Killoran (2019) and Havlíček et al. (2019), who demonstrated that quantum feature maps can define kernel functions whose evaluation is classically intractable for sufficiently expressive circuits. The ZZFeatureMap circuit used in the present work was introduced in Havlíček et al. (2019) as a concrete instantiation of this framework, and subsequently shown by Liu et al. (2021) to satisfy conditions sufficient for a provable quantum speedup over classical kernel methods on specific problem classes (see Section 3 for the explicit circuit definition and encoding scheme). The broader theoretical landscape of quantum machine learning is reviewed in Biamonte et al. (2017) and Cerezo et al. (2022). Empirical demonstrations of quantum kernel advantage on real-world datasets have, however, proven elusive. Huang et al. (2021) established that the relative performance of quantum and classical kernels is governed by the geometry of the data distribution in feature space, implying that quantum advantage is dataset-dependent rather than universal. Systematic benchmarking studies — including Alvarez-Estevez (2025), who evaluated quantum kernel training across a range of standard classification datasets — consistently report mixed results: quantum kernels outperform classical counterparts on some tasks while underperforming on others, with the outcome depending sensitively on the choice of feature map and hyperparameter configuration. These findings underscore the importance of domain-specific benchmarks, which motivates the systematic experimental design of the present study. 1.1.3. Quantum machine learning for earth and atmospheric sciences The application of QML methods to geophysical and Earth observation problems is an emerging research direction. In the domain of remote sensing, quantum kernel SVMs have been applied to satellite image classification (Miroszewski et al., 2023) and hyperspectral image analysis (Otgonbaatar and Datcu, 2022), with results suggesting that quantum kernels can match or marginally exceed classical SVM performance on spectrally complex datasets with limited training samples. Jaderberg et al. (2024) demonstrated that parameterised quantum circuits can be trained to reproduce global atmospheric stream function dynamics, providing an early proof of concept for quantum ML in weather modelling. To the best of our knowledge, no prior work has applied quantum kernel methods to space weather data or to cosmic ray time series, and no study has investigated the dependence of quantum kernel performance on the physical amplitude of the geophysical events being classified. Addressing both of these gaps is the primary contribution of the present paper. 1.2. Aims and scope In this work we present a systematic benchmark of quantum kernel SVMs for FD detection using a dataset constructed from the Forbush Effects and Interplanetary Disturbances (FEID) catalogue (Belov et al., 2021, Okoye et al., 2024, Jerry-Okafor et al., 2024), hourly OMNI solar wind data (King and Papitashvili, 2005), and cosmic ray count rates from the Jungfraujoch neutron monitor (JUNG) of the Neutron Monitor Database (NMDB) (Mavromichalaki et al., 2011). Our central finding is that quantum kernel performance is strongly regime-dependent, governed by the magnitude of the FD events included in the classification problem: the classical RBF-SVM dominates in the noise-limited low-amplitude regime, while the quantum kernel achieves consistent and statistically significant advantage above a physically interpretable magnitude threshold. The entanglement structure of the ZZFeatureMap circuit (Havlíček et al., 2019) appears to capture non-linear IMF–GCR correlations that are inaccessible to the Euclidean-distance-based RBF kernel, but only when those correlations are sufficiently strong and structured to constitute a genuine learning signal. Detailed results and their physical interpretation are provided in Sections 4 Results, 5 Discussion. The remainder of this paper is organised as follows. Section 2 describes the datasets and preprocessing pipeline. Section 3 introduces the quantum kernel framework, the feature extraction procedure, and the experimental design. Results are presented in Section 4, including the magnitude-dependent performance analysis, learning curves, and circuit architecture sensitivity. Section 5 discusses the physical interpretation of the magnitude threshold and the implications for near-term quantum machine learning in heliophysics. Conclusions are drawn in Section 6. A self-contained primer on quantum computing concepts is provided in Appendix. 2. Data 2.1. Forbush decrease catalogue Confirmed FD events were drawn from the Forbush Effects and Interplanetary Disturbances (FEID) catalogue, maintained by the Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation (IZMIRAN) and publicly accessible at https://tools.izmiran.ru/w/feid (Belov et al., 2021, Okoye et al., 2024, Jerry-Okafor et al., 2024). The catalogue provides onset timestamps, FD magnitude estimates derived from global neutron monitor network data using the global survey method (GSM), and a suite of associated solar wind and geomagnetic parameters for each event. The version used in this study spans the period from 1995 January 1 to 2020 April 30, comprising 2981 catalogued events. The primary classification target is the FD magnitude parameter Magn, defined as the maximal range of cosmic ray density variations for particles with 10 GV rigidity as derived from the GSM analysis of the neutron monitor network (Belov, 2008). Events were filtered by Magn threshold in the range 0%–7% to define distinct experimental subsets, as described in Section 3.4. 2.2. OMNI solar wind data Hourly-resolution solar wind and geomagnetic index data were obtained from the OMNI database, compiled and distributed by the NASA Space Physics Data Facility (SPDF) and accessible via the OMNIWeb interface at https://omniweb.gsfc.nasa.gov/form/dx1.html (King and Papitashvili, 2005). The OMNI dataset covers the period from 1995 January 1 to 2021 December 31, comprising 236,688 hourly records. Ten parameters were selected as input features for the classification pipeline: the interplanetary magnetic field (IMF) total field strength (Scalar B), the GSE components of the IMF vector (Bx, By, Bz), solar wind proton temperature (SW Temperature), solar wind proton density, solar wind plasma speed, the Kp geomagnetic activity index, the Dst ring current index, and the F10.7 solar radio flux index used as a proxy for solar activity level. The proton temperature is a particularly informative discriminator for ICME identification, as ejecta material is systematically cooler than the ambient solar wind at equivalent speeds (Richardson and Cane, 2010). Missing values in the OMNI record, encoded as fill values following the OMNI data format convention, were replaced by NaN prior to any further processing. 2.3. Neutron monitor data Cosmic ray count rates were obtained from the Neutron Monitor Database (NMDB), accessible at https://www.nmdb.eu/nest/ (Mavromichalaki et al., 2011). Specifically, one-hour validated count rates from the Jungfraujoch station (JUNG, geographic coordinates: 46.55°N, 7.98°E; altitude: 3475 m a.s.l.; vertical cutoff rigidity: 4.49 GV) were used, with pressure and efficiency corrections applied as provided by NMDB (1HCOR_E_JUNG). The JUNG record spans 1995 January 1 to 2021 December 31, comprising 235,277 hourly records. Jungfraujoch was selected on the basis of its long and continuous operational history, its intermediate cutoff rigidity — making it sensitive to the moderate-energy GCR population most strongly modulated during FDs — and its established use in previous FD studies (Melkumyan et al., 2024, Melkumyan et al., 2023). Count rate values equal to or below zero, indicative of instrumental dropout or data transmission errors, were flagged and treated as missing. Outlier rejection was applied using a 5 threshold relative to the station median, removing sporadic high-count excursions associated with solar energetic particle events or detector glitches. 2.4. Dataset construction The effective temporal coverage of the dataset is determined by the intersection of the three sources: 1995 January 1 to 2020 April 30. For each FD onset timestamp in the catalogue, a multivariate time series window of 73 hourly steps was extracted, spanning 24 h before the onset to 48 h after it. The asymmetric design of this window reflects the physical structure of ICME-driven FDs: the 24-hour pre-onset segment is sufficient to characterise the ambient solar wind state and capture the arrival of the preceding interplanetary shock, while the 48-hour post-onset segment covers the full suppression phase and the early recovery, which typically unfolds over one to two days for moderate events (Cane, 2000). A symmetric window of equal pre- and post-onset duration would either truncate the recovery phase or introduce excessive quiet-time data prior to the shock arrival, in both cases degrading the discriminability of the FD signature. Gap-filling was applied to each extracted window in three sequential steps: (1) linear interpolation across internal NaN segments of up to 6 consecutive hours; (2) forward-fill for leading missing values at the start of the window; and (3) backward-fill for trailing missing values at the end of the window. Steps (2) and (3) were applied only after step (1) to avoid propagating boundary values into the interior of the window. No imputation was performed for gaps longer than 6 consecutive hours in the interior; windows containing more than 20% missing values across all channels after step (1) were discarded entirely. After this procedure, and applying the Magn % filter, 2971 valid FD windows were retained, representing a loss of less than 1% of the catalogue due to data gaps. To construct a balanced binary classification dataset, an equal number of negative examples (non-FD, label ) were generated by randomly sampling quiet-time windows from the OMNI and JUNG records. A candidate quiet-time window was accepted only if its centre timestamp was separated by more than 72 h from any catalogued FD onset, ensuring that negative examples were free of FD contamination. The same gap-filling and quality criteria applied to positive examples were enforced for negative examples. The resulting dataset comprises 5942 labelled windows (2971 FD and 2971 non-FD), each represented as a matrix of shape 73 × 11, where the eleven channels correspond to the ten OMNI parameters and the JUNG cosmic ray count rate. Table 1 summarises the dataset composition. Table 1. Summary of the classification dataset. Note that the missing value fraction ( 9.5%) refers to the proportion of individual hourly entries that contained NaN values across all windows and channels before gap-filling, and is distinct from the window discard rate ( 1%), which reflects the fraction of FD windows excluded due to excessive gaps after interpolation. Property Value Temporal coverage 1995 Jan 1–2020 Apr 30 Total labelled windows 5942 FD windows (label ) 2971 Non-FD windows (label ) 2971 Window length 73 h Pre-onset window 24 h Post-onset window 48 h Number of input channels 11 OMNI channels 10 NMDB channels 1 (JUNG) Missing value fraction (pre-imputation) 9.5% of hourly entries Window discard rate (post-imputation) 1% of catalogue 3. Methods 3.1. Feature extraction Each labelled window of shape 73 × 11 was transformed into a fixed-length feature vector prior to classification. For each of the eleven input channels, two sets of descriptors were computed. The first set comprises eight global statistical features: mean, standard deviation, minimum, maximum, peak-to-peak range, skewness, excess kurtosis, and linear trend slope estimated by ordinary least squares. The second set comprises three onset-relative features designed to capture the characteristic morphology of FD events: the channel value at the onset hour, the maximum drop from onset to the post-onset minimum, and the normalised time elapsed from onset to the post-onset minimum. This yields features per window. Windows containing residual NaN values after the gap-filling procedure described in Section 2.4 — arising from constant or near-constant segments where higher-order moments are numerically ill-defined — were imputed using the column mean computed over the training partition only, to prevent data leakage. Feature selection was then applied to reduce the dimensionality to a number compatible with near-term quantum hardware simulation. The features with the highest mutual information (MI) with the binary label were retained (Cover and Thomas, 2006), where equals the number of qubits in the quantum circuit (Section 3.3). MI-based selection was preferred over variance-based or correlation-based alternatives because it captures non-linear statistical dependencies between features and the target, making it consistent with the non-linear kernel approach used downstream. Selection was performed exclusively on the training partition at each experimental run to prevent data leakage. 3.2. Classical baseline: RBF-SVM The classical baseline is a support vector machine (SVM) with a radial basis function (RBF) kernel (Cortes and Vapnik, 1995), implemented via scikit-learn (Pedregosa et al., 2011). Given two feature vectors , the RBF kernel is defined as (1) where is set by the scale heuristic ( , with the variance of the training data). The regularisation parameter was fixed at throughout all experiments. Two variants of the RBF-SVM are reported: RBF-SVM (full), trained on the complete training partition comprising 4456 samples, and RBF-SVM (sub), trained on the same stratified subsample of size used to train the quantum kernel. The latter provides a fair comparison against the quantum model, controlling for the effect of training set size and isolating the contribution of the kernel function itself. 3.3. Quantum kernel SVM 3.3.1. Background Quantum kernel methods extend the classical kernel framework to the quantum domain by exploiting the exponentially large Hilbert space of a -qubit system as a feature space (Biamonte et al., 2017, Schuld and Killoran, 2019). The central idea is to encode classical data vectors into quantum states via a parameterised unitary circuit , the quantum feature map, which prepares the state . The quantum kernel is then defined as the fidelity between two encoded states, (2) which can be estimated as the probability of measuring the all-zeros bitstring after applying the combined circuit (Havlíček et al., 2019). Because is a valid positive semi-definite kernel, it can be used as a drop-in replacement for the RBF kernel in a standard SVM, inheriting all convergence and generalisation guarantees of the classical SVM framework (Rebentrost et al., 2014, Liu et al., 2021). The expressibility of grows with the circuit depth and the number of qubits, raising the possibility of representing kernel functions that are classically intractable (Huang et al., 2021). 3.3.2. ZZFeatureMap circuit The quantum feature map is instantiated as the ZZFeatureMap (Havlíček et al., 2019), a two-local circuit that alternates layers of single-qubit Hadamard and rotation gates with layers of entangling -interaction gates. For a single repetition, the circuit implements (3) where encodes pairwise feature correlations via entangling two-qubit rotations, and denotes the tensor product of Hadamard gates acting on all qubits. This constitutes a form of angle encoding: each selected feature , after rescaling to , is mapped directly to a rotation angle in both the single-qubit layer and the two-qubit entangling layer. The key distinction from classical kernel methods is that the entangling layer encodes products of feature values into two-qubit interactions, introducing cross-feature correlations that are absent from any kernel that acts on features independently. The circuit is repeated for layers (the reps parameter) to control the expressibility of the feature map, with linear entanglement topology to limit circuit depth. Prior to encoding, all selected features were rescaled to the interval using a min–max scaler fitted on the training partition, ensuring that the rotation angles in Eq. (3) span the full periodicity of the quantum gates (Schuld, 2021). Fig. 1 shows the explicit gate-level diagram of the ZZFeatureMap for qubits and , as implemented in Qiskit 1.4. Each data point is encoded in a single pass through the Hadamard layer, the single-qubit -gate layer (equivalent to up to global phase), and the two-qubit entangling -gate layer. The fidelity kernel is evaluated by appending the conjugate circuit and measuring the probability of the all-zeros outcome. Fig. 1 Download: Download high-res image (168KB) Download: Download full-size image Fig. 1. Gate-level diagram of the ZZFeatureMap circuit for qubits and . The circuit begins with a layer of Hadamard gates ( ) that place all qubits in equal superposition, followed by single-qubit phase gates ( ) that encode each feature as a rotation angle, and two-qubit controlled-phase gates ( ) that encode pairwise products via ZZ interactions with linear entanglement topology. To evaluate the quantum kernel , the conjugate circuit is appended and the probability of measuring the all-zeros bitstring is recorded. 3.3.3. Kernel matrix computation and SVM training For a training set of samples, the quantum kernel matrix is computed by evaluating Eq. (2) for all unique pairs, exploiting the symmetry of the kernel. The full training partition of 4456 samples is not used directly for quantum kernel training because the resulting would require statevector simulations per configuration—computationally prohibitive under local simulation on the hardware used. Instead, stratified subsamples of size are drawn from the training partition, reducing the number of kernel evaluations to at most for the training matrix. The test kernel matrix is computed analogously, with in all experiments. To avoid out-of-memory failures under local simulation, the test matrix was computed in sequential batches of 50 rows. All kernel evaluations were performed using exact statevector simulation via Qiskit’s ComputeUncompute fidelity protocol (Qiskit contributors, 2024), which provides shot-noise-free estimates of . Qiskit version 1.4 was used throughout, rather than the more recent version 2.x, because the ComputeUncompute fidelity estimator and the Sampler V1 primitive required for its evaluation are fully supported in version 1.4 and have undergone interface-breaking changes in version 2.x that introduce incompatibilities with the FidelityQuantumKernel class used here. The SVM classifier uses as a precomputed kernel matrix with , implemented via scikit-learn (Pedregosa et al., 2011). 3.4. Experimental design A systematic grid of 270 experimental configurations was evaluated, varying four independent parameters: • FD magnitude threshold (min_magn): %, controlling which FD events enter the dataset and defining the signal-to-noise regime of the classification problem. • Number of qubits ( ): , simultaneously setting the number of MI-selected features and the dimension of the quantum feature space. • Circuit depth (reps): , controlling the expressibility of the ZZFeatureMap. • Quantum training size ( ): , the number of stratified samples used to train both the quantum kernel and the RBF-SVM (sub) baseline. For each configuration, the full dataset was split into training (75%) and test (25%) partitions using stratified sampling to preserve class balance, yielding a fixed test set of 1486 samples. A stratified subsample of size was drawn from the training partition to train both the quantum kernel and the RBF-SVM (sub). Stratified sampling was used for the subsample — rather than simple random sampling — to guarantee that both classes remain equally represented at all values of , which is essential for obtaining unbiased AUC estimates when is small relative to the full training partition. Without stratification, random subsamples at would have a non-negligible probability of class imbalance severe enough to distort the comparison. The RBF-SVM (full) was trained on the complete training partition. All three models were evaluated on the same held-out test set, ensuring that performance comparisons are not confounded by differences in the evaluation data. A fixed random seed of 42 was used for all train–test splits, subsample draws, and SVM solvers throughout the study, ensuring full reproducibility of all reported results. The primary comparison metric is (4) where AUC denotes the area under the receiver operating characteristic (ROC) curve (Fawcett, 2006). A positive indicates that the quantum kernel outperforms its classical counterpart under identical data conditions. Secondary metrics include classification accuracy and the confusion matrix. Five-fold cross-validation on the full dataset was additionally performed for the RBF-SVM (full) to provide a bias-corrected estimate of classical performance. Statistical significance of within each magnitude threshold group was assessed using the Wilcoxon signed-rank test (Wilcoxon, 1945), a non-parametric test that evaluates whether the median of a paired difference — here, across all configurations within a group — differs significantly from zero. The one-sided variant (alternative: ) was used to test the directional hypothesis that quantum kernels outperform the classical baseline. To assess whether the distribution of differs significantly across the six magnitude threshold groups, the Kruskal–Wallis test (Kruskal and Wallis, 1952) was applied—a non-parametric one-way analysis of variance that tests for stochastic dominance among independent groups without assuming normality. Effect sizes are reported as rank-biserial correlation coefficients alongside each Wilcoxon test statistic, and bootstrap 95% confidence intervals (9999 resamples, seed = 42) are reported for all mean estimates. Individual experiment durations ranged from approximately 0.5 minutes (high-magnitude thresholds with small datasets and ) to approximately 51 minutes (min_magn %, qubits, , ), with a median of approximately 2.4 min per configuration. The complete benchmark of 270 configurations required approximately 2–3 days of wall-clock time on an Intel Core i7-8565U workstation (4 physical cores, 16 GiB RAM, Ubuntu 26.04 LTS) running 4 parallel processes, reflecting both the scaling of the quantum kernel matrix and the thermal throttling inherent to sustained parallel load on mobile hardware. By contrast, the classical RBF-SVM required less than 2 s per configuration on the same machine. All experiments were implemented in Python 3.13 using Qiskit 1.4 (Qiskit contributors, 2024) and scikit-learn 1.6 (Pedregosa et al., 2011), and executed using GNU Parallel (Tange, 2011) to distribute the 270 configurations across four concurrent processes. Table 2. Dataset composition by FD magnitude threshold. FD and non-FD windows are balanced at all thresholds. Training and test partitions are obtained by stratified 75/25 splitting of the total window count. Threshold FD Non-FD Total Train Test Magn 0% 2971 2971 5942 4456 1486 Magn 3% 190 190 380 285 95 Magn 4% 107 107 214 160 54 Magn 5% 65 65 130 97 33 Magn 6% 47 47 94 70 24 Magn 7% 34 34 68 51 17 4. Results 4.1. Dataset and experimental summary The final classification dataset constructed following the procedure described in Section 2.4 comprises 5942 labelled windows (2971 FD and 2971 non-FD), with a training partition of 4456 samples and a held-out test set of 1486 samples for the unfiltered case (min_magn %). As the magnitude threshold is raised, the number of available FD events decreases and the dataset is rebalanced accordingly, with the most restrictive filter (min_magn %) yielding 34 FD events, a training partition of 51 samples, and a test set of 17 samples. Table 2 summarises the dataset composition at each threshold. The progressive reduction in dataset size as the threshold increases is an inherent consequence of the FD magnitude distribution in the FEID catalogue, and constitutes the primary source of increased statistical uncertainty at higher thresholds. Across all 180 valid experimental configurations (6 magnitude thresholds 3 qubit counts 3 circuit depths 5 training sizes), the quantum kernel achieved higher AUC than the classical RBF-SVM (sub) in 73 cases (40.6%), with 13 cases (7.2%) in parity ( ) and 94 cases (52.2%) where the classical kernel outperformed. The global mean ( ) is statistically indistinguishable from zero when all configurations are pooled, consistent with the regime-dependent behaviour described below. In Fig. 2a, the RBF-SVM (full) and RBF-SVM (sub) curves are nearly identical across all thresholds. This is expected: both models use the same 121-dimensional feature representation and the same MI-selected feature subset; the only difference is training set size. The near-identical performance indicates that the RBF classifier saturates its AUC well within the subsampled training set sizes tested ( ), so that the additional samples in the full partition provide negligible benefit. This saturation confirms that the RBF-SVM is not data-starved in the subsampled regime, making the fair comparison (QSVM vs. RBF-sub) a genuine test of kernel expressibility rather than a confound of sample size. 4.2. Magnitude-dependent performance The dominant result of this study is the strong dependence of quantum kernel advantage on the FD magnitude threshold, illustrated in Fig. 2 and summarised in Table 5. For min_magn % — including all 2971 FD events regardless of amplitude — the quantum kernel is uniformly outperformed by the classical RBF-SVM (sub), with a mean (95% CI: ), and not a single configuration showing quantum advantage (0/45). This regime is dominated by weak FDs whose time-series signatures are largely indistinguishable from quiet-time fluctuations in the solar wind and neutron monitor records, producing a classification problem that is fundamentally noise-limited. Raising the threshold to min_magn % (190 events) reduces the classical advantage but does not reverse it: mean (95% CI: ), with only 4 out of 45 configurations showing marginal quantum advantage (8.9%). The transition occurs sharply between 3% and 4% (Fig. 3). At min_magn % (107 events), the mean becomes strongly positive ( , 95% CI: ), with 72.2% of configurations showing quantum advantage. The transition is statistically confirmed: a Wilcoxon signed-rank test (testing whether ) yields , rank-biserial for min_magn %, compared to for the two lower thresholds. A Kruskal–Wallis test across all six threshold groups confirms that the distribution of differs significantly across groups ( , ). Table 3 reports the complete statistical results including bootstrap confidence intervals and rank-biserial effect sizes for all threshold groups. At min_magn % and % (65 and 47 events, respectively), quantum advantage reaches its maximum consistency: 100% of tested configurations yield , with mean values of (95% CI: , ) and (95% CI: , ) respectively. The mean AUC of the quantum kernel in these regimes exceeds 0.95, reaching a peak of 1.000 in isolated configurations (Table 5). At min_magn % (34 events), quantum advantage persists on average ( , 95% CI: ) but becomes less consistent (55.6% of configurations), likely reflecting the increased statistical variability arising from the small dataset size at this threshold (Table 2). The Wilcoxon test remains significant at this threshold ( , , ). Table 3. Statistical summary of by FD magnitude threshold. The Wilcoxon signed-rank test (one-sided, : ) is reported alongside the rank-biserial correlation as an effect size measure ( small, medium, large). Bootstrap 95% confidence intervals on the mean are based on 9999 resamples (seed ). Significance codes: ; ; ns: not significant. For Magn 0% and Magn 3%, reflects strong classical dominance; the test is one-sided ( : AUC 0), so ns indicates failure to reject in the quantum-advantage direction. Threshold Mean AUC 95% CI Sig. Magn 0% 45 −0.100 [−0.112, −0.088] 1.000 ns Magn 3% 45 −0.025 [−0.030, −0.020] 0.951 ns Magn 4% 36 ＋0.070 [＋0.048, ＋0.090] 0.787 Magn 5% 18 ＋0.074 [＋0.062, ＋0.086] 1.000 Magn 6% 18 ＋0.087 [＋0.067, ＋0.103] 1.000 Magn 7% 18 ＋0.031 [＋0.012, ＋0.052] 0.754 Kruskal–Wallis: , . Fig. 2 Download: Download high-res image (285KB) Download: Download full-size image Fig. 2. (a) Mean ROC-AUC for each of the three classifiers — RBF-SVM (full), RBF-SVM (sub), and Quantum Kernel SVM — as a function of FD magnitude threshold. Error bars indicate standard deviation across experimental configurations. (b) Distribution of as a function of threshold, shown as boxplots. Percentages above each box indicate the fraction of configurations in which the quantum kernel outperforms the classical baseline. The dashed line marks the parity boundary ( ). The near-identical performance of RBF-SVM (full) and RBF-SVM (sub) in panel (a) reflects AUC saturation of the classical kernel within the tested subsample sizes ( ); see Section 4.1. Fig. 3 Download: Download high-res image (161KB) Download: Download full-size image Fig. 3. Mean as a function of FD magnitude threshold (min_magn), averaged over all 180 experimental configurations. The shaded band indicates standard deviation. The transition from classical to quantum advantage occurs between 3% and 4%, marked by the orange transition zone. Annotated values correspond to the mean at each threshold. 4.3. Effect of circuit architecture Fig. 4 shows averaged over all training sizes for each combination of magnitude threshold, qubit count, and circuit depth. The most striking feature is the strong interaction between min_magn and n_qubits: for min_magn %, four-qubit circuits consistently achieve the largest , with a peak of at (min_magn %, ). Six-qubit circuits follow closely ( at the same threshold). Eight-qubit circuits, by contrast, show near-zero or negative at min_magn %, though they recover positive advantage at higher thresholds ( at min_magn %, at min_magn %). We note that the non-monotonic pattern with qubit count is based on only three values ( ), which limits the strength of any claim about the functional form of this dependence. The observed pattern — in which intermediate qubit counts do not consistently outperform smaller ones — is nevertheless qualitatively consistent with the concentration phenomenon described by Thanasilp et al. (2024): deeper circuits with more qubits produce kernel matrices whose entries become exponentially concentrated around their mean, effectively reducing the discriminative power of the kernel for the training sizes used here. Fig. 6a summarises these trends across all qubit counts and magnitude thresholds; panel (b) shows the corresponding QSVM AUC as a function of training size in the winning regime (min_magn %). The effect of circuit depth (reps) is secondary but notable. For min_magn %, deeper circuits (reps ) tend to produce higher at thresholds of 5%–6%, achieving a maximum of at (min_magn %, reps ). For lower thresholds, additional circuit depth does not help and can slightly hurt performance. The panel (c) of Fig. 4 confirms that, when averaged across all magnitude thresholds, the qubit–depth interaction is modest, with four and six qubit circuits in parity or slight advantage across all depth settings. Fig. 4 Download: Download high-res image (372KB) Download: Download full-size image Fig. 4. Heatmaps of mean for all parameter combinations. (a) as a function of FD magnitude threshold and qubit count. (b) as a function of FD magnitude threshold and circuit depth (reps). (c) as a function of qubit count and circuit depth, averaged across all magnitude thresholds. Red cells indicate quantum advantage; blue cells indicate classical advantage. 4.4. Learning curves and sample efficiency Fig. 5a shows as a function of the quantum training size for each magnitude threshold. For the low-threshold regimes (min_magn % and %), remains negative across all training sizes, indicating that additional quantum training data does not recover competitiveness. For min_magn %, is positive and broadly stable across , with no systematic improvement as increases. This robustness to training size is a practically important property: quantum advantage at high FD magnitudes is achievable even with the smallest training sets tested ( ), suggesting that the kernel structure of the ZZFeatureMap efficiently leverages the available data in the signal-rich regime. For min_magn %, the learning curve shows a slight improvement from to , followed by stabilisation. The intermediate threshold (min_magn %) shows the most variability, consistent with the limited number of available FD events at this threshold. Fig. 5b provides a complementary view as an AUC scatter plot, in which each point represents one ( , reps, ) configuration coloured by magnitude threshold: configurations with min_magn % cluster above the parity diagonal, confirming quantum advantage across the full range of classical AUC values achieved in those regimes. The dense vertical clustering of points at certain AUC values reflects the fact that the classical baseline is fixed for a given (min_magn, ) combination regardless of the quantum circuit parameters, so multiple quantum configurations share the same -coordinate. Fig. 5 Download: Download high-res image (298KB) Download: Download full-size image Fig. 5. (a) as a function of quantum training size for each FD magnitude threshold. Shaded bands indicate standard deviation across qubit count and circuit depth configurations. (b) Scatter plot of AUC values: each point represents one experimental configuration ( , reps, ), coloured by magnitude threshold. Points above the dashed diagonal (parity) indicate quantum advantage. Vertical clustering of points at fixed -values reflects configurations sharing the same classical baseline (RBF-sub) AUC but different quantum circuit parameters. 4.5. Best configurations and computational cost The fifteen highest- configurations are dominated by min_magn % with qubits, achieving up to with AUC reaching 0.964. The second cluster corresponds to min_magn % with qubits, with up to . In the winning region defined as min_magn % and , comprising 36 configurations, the quantum kernel achieves a mean AUC of against a classical baseline of , with and quantum advantage in 30 of 36 configurations (83%). Table 4 quantifies the computational cost of the quantum kernel relative to the classical baseline. The number of fidelity evaluations required for the training kernel matrix grows as , reaching 31,375 for , while the test matrix requires evaluations. For the largest tested configuration (min_magn %, , ), this amounts to 402,875 statevector simulations per experiment, compared to less than 2 s total for the classical RBF-SVM on the same hardware. This difference of three to four orders of magnitude in wall-clock time (see Section 3.4) underscores that quantum kernel methods are currently competitive in accuracy only in the signal-rich regime where the advantage is large enough to justify the cost. Table 4. Theoretical quantum kernel evaluation counts by training size ( ) and qubit count ( ). counts fidelity evaluations for the upper-triangular training kernel matrix; counts evaluations against the held-out test set ( , worst case min_magn %). The classical RBF-SVM requires less than 2 s per configuration on the same hardware for all combinations shown. Total RBF time 4 50 1275 74,300 75,575 s 4 100 5050 148,600 153,650 s 4 250 31,375 371,500 402,875 s 6 50 1275 74,300 75,575 s 6 100 5050 148,600 153,650 s 6 250 31,375 371,500 402,875 s 8 50 1275 74,300 75,575 s 8 100 5050 148,600 153,650 s 8 250 31,375 371,500 402,875 s Table 5. Summary of results by FD magnitude threshold and qubit count. Columns report the mean AUC of the full and subsampled RBF-SVM baselines, the mean AUC of the Quantum Kernel SVM, the mean and standard deviation of , and the fraction of configurations where the quantum kernel outperforms the classical baseline (Q C). Results are averaged over all circuit depths (reps ) and training sizes ( ). Threshold AUC (full) AUC (sub) AUC AUC Q C Magn 0% 4 15 0.861 0.854 0.753 −0.100 0.045 0/15 (0%) Empty Cell 6 15 0.863 0.851 0.764 −0.087 0.043 0/15 (0%) Empty Cell 8 15 0.905 0.867 0.755 −0.112 0.038 0/15 (0%) Magn 3% 4 15 0.924 0.919 0.899 −0.020 0.017 2/15 (13%) Empty Cell 6 15 0.918 0.918 0.889 −0.029 0.014 0/15 (0%) Empty Cell 8 15 0.918 0.910 0.884 −0.026 0.021 2/15 (13%) Magn 4% 4 12 0.801 0.800 0.936 ＋0.135 0.018 12/12 (100%) Empty Cell 6 12 0.853 0.827 0.917 ＋0.091 0.016 12/12 (100%) Empty Cell 8 12 0.923 0.924 0.908 −0.016 0.016 2/12 (17%) Magn 5% 4 6 0.890 0.904 0.959 ＋0.056 0.026 6/6 (100%) Empty Cell 6 6 0.890 0.890 0.969 ＋0.079 0.022 6/6 (100%) Empty Cell 8 6 0.890 0.890 0.978 ＋0.088 0.025 6/6 (100%) Magn 6% 4 6 0.917 0.917 0.954 ＋0.037 0.017 6/6 (100%) Empty Cell 6 6 0.833 0.833 0.950 ＋0.117 0.014 6/6 (100%) Empty Cell 8 6 0.840 0.840 0.948 ＋0.108 0.011 6/6 (100%) Magn 7% 4 6 0.875 0.875 0.921 ＋0.046 0.052 4/6 (67%) Empty Cell 6 6 0.931 0.931 0.940 ＋0.009 0.036 2/6 (33%) Empty Cell 8 6 0.931 0.931 0.968 ＋0.037 0.040 4/6 (67%) Fig. 6 Download: Download high-res image (346KB) Download: Download full-size image Fig. 6. (a) as a function of FD magnitude threshold for each qubit count ( ). Error bars indicate standard deviation across circuit depths and training sizes. The apparent non-monotonic trend across qubit counts should be interpreted cautiously, as it is based on only three values ( ). (b) AUC as a function of training size for all nine ( , reps) combinations in the winning regime (min_magn %). The figure shows only two training size values ( and ) because the total training partition at this threshold comprises only 97 samples, making infeasible. The crossing and interleaving of curves at each value reflects configuration-dependent sensitivity to training size rather than a systematic trend, consistent with the AUC saturation observed in the learning curves of Fig. 5a. 5. Discussion 5.1. Physical interpretation of the magnitude threshold The central empirical finding of this study — that quantum kernel advantage emerges sharply above a FD magnitude threshold of 4% — has a natural physical interpretation rooted in the structure of ICME-driven cosmic ray modulation. Large-amplitude FDs are predominantly associated with well-developed ICME magnetic structures: organised flux ropes with strong, ordered magnetic fields that produce coherent and sustained shielding of the GCR flux (Petukhova et al., 2020, Gutierrez et al., 2024). In these events, the modulation of the JUNG count rate is tightly correlated with the IMF dynamics—specifically, the temporal evolution of the Bz component drives the ring current response (Dst), which in turn correlates with the depth of the GCR suppression. The ZZFeatureMap encodes pairwise feature correlations through the entangling term , and it is precisely these cross-channel correlations between IMF components (Bx, By, Bz), geomagnetic indices, and the JUNG count rate that characterise strong FD events. The RBF kernel, by contrast, measures only the Euclidean distance between feature vectors and cannot represent such structured cross-correlations explicitly. Weak FDs (Magn %), on the other hand, arise from a heterogeneous mixture of physical drivers: partial ICME encounters, corotating interaction regions, and flux ropes without well-developed sheaths (Gutierrez et al., 2024). In these cases, the IMF–GCR coupling is weaker and more variable, producing feature vectors that are difficult to separate from quiet-time windows regardless of the kernel used. The classification problem in this regime is fundamentally noise-limited, and the additional representational capacity of the quantum kernel provides no benefit—and indeed can hurt performance, as the expressible function class is too large relative to the available training signal (Thanasilp et al., 2024). The transition at 4% is therefore not merely a statistical artifact of the experimental design but reflects a genuine physical boundary: below it, the FD catalogue is dominated by events where the IMF–GCR connection is too noisy for quantum feature correlations to be useful; above it, the structured IMF dynamics of well-developed ICMEs create a feature geometry that the ZZFeatureMap can exploit more effectively than the RBF kernel. 5.2. Interpretation of the qubit count effect The non-monotonic behaviour of with qubit count — four and six qubits outperforming eight qubits in the low-threshold regime — is consistent with the exponential concentration phenomenon described by Thanasilp et al. (2024) and studied theoretically by Wang et al. (2021). We emphasise, however, that this pattern is observed across only three qubit values ( ), and should be regarded as a qualitative tendency rather than a quantitatively established functional relationship. Testing additional values (e.g., , ) would be required to characterise the qubit-count dependence more precisely; was not included in the present grid due to the severe kernel concentration observed at in the low-magnitude regime, which suggested that larger circuits would be unlikely to provide further benefit. As the number of qubits and circuit depth increase, the entries of the kernel matrix become exponentially concentrated around a constant value, reducing the effective rank of the matrix and limiting the discriminative power of the SVM. This effect is particularly pronounced at low magnitude thresholds where the classification signal is weak, and the high expressibility of larger circuits effectively washes out the relevant structure. This result may appear to contradict the common intuition that greater circuit expressibility implies better classification performance. The apparent paradox is resolved by noting that expressibility and kernel utility are distinct properties (Thanasilp et al., 2024): a highly expressive circuit can access a rich function class, but if the kernel matrix entries concentrate exponentially towards a constant, the SVM receives effectively no discriminative information regardless of the functional richness of the feature map. In other words, more qubits expand the representable hypothesis class while simultaneously collapsing the kernel matrix towards a scalar multiple of the identity—a phenomenon that is fundamentally different from overfitting and cannot be corrected by regularisation alone. At higher magnitude thresholds (min_magn %), the strong physical signal overcomes this concentration tendency, and all three qubit counts achieve positive , with the eight-qubit configuration recovering competitive performance ( at min_magn %). This suggests that the concentration effect is signal-dependent: when the data contains strong, structured feature correlations, even larger circuits can exploit them effectively. The practical implication for near-term quantum hardware is that modest circuit sizes ( – qubits) represent the most reliable operating point across a range of signal conditions, consistent with recommendations in the QML literature for NISQ devices (Incudini et al., 2025, Cerezo et al., 2022). 5.3. Sample efficiency and the small-data regime A noteworthy aspect of the results is the robustness of quantum advantage to training set size in the signal-rich regime. For min_magn %, the mean is positive and stable across , indicating that the quantum kernel achieves competitive classification with as few as 50 training samples. This is consistent with theoretical results showing that quantum kernels can achieve lower generalisation error than classical methods with fewer samples when the data geometry aligns with the kernel structure (Huang et al., 2021). In the context of FD classification, the practical relevance is significant: strong FDs are rare events. The FEID catalogue contains only 34–65 events above the 5%–7% magnitude thresholds tested here, and the entire 25-year observational record provides limited training data at these amplitudes. The ability of the quantum kernel to operate effectively in this small-data regime is therefore not a limitation but a natural alignment between the method and the application domain. Nevertheless, the small absolute event counts at high thresholds (Table 2) introduce non-negligible statistical uncertainty into the performance estimates. Although the datasets are perfectly balanced (equal FD and non-FD windows at every threshold), the total sample sizes at min_magn %, %, and % are only 130, 94, and 68 windows respectively, with test sets of 33, 24, and 17 samples. AUC estimates computed on such small test sets carry wide confidence intervals, and isolated perfect-AUC configurations (e.g., AUC reported at min_magn %) should be interpreted with corresponding caution. The bootstrap confidence intervals in Table 3 reflect this uncertainty and show that the mean at min_magn % has the widest interval ( ), consistent with the higher variability expected at this threshold. 5.4. Comparison with prior QML benchmarks The regime-dependent quantum advantage observed here is broadly consistent with the emerging consensus from QML benchmarking studies. Alvarez-Estevez (2025) found that quantum kernels outperform classical counterparts on ad-hoc datasets but show mixed results on standard benchmarks, attributing the difference to whether the data geometry aligns with the inductive bias of the quantum feature map. Huang et al. (2021) established theoretically that quantum advantage is governed by the geometric difference between the quantum and classical kernel matrices, a quantity that is data-dependent. Our results provide an empirical demonstration of this principle in a physical science domain: the geometric difference between the ZZFeatureMap kernel and the RBF kernel is beneficial precisely in the regime where the data contains structured cross-channel correlations (strong FDs), and detrimental where it does not (weak FDs). The magnitude threshold at which this transition occurs — approximately 4% — is therefore a property of the FD data distribution, not of the quantum circuit architecture. Compared to related applications in Earth observation, where quantum kernels have shown modest but consistent improvements for hyperspectral classification (Miroszewski et al., 2023, Otgonbaatar and Datcu, 2022), the advantage magnitudes reported here (mean at min_magn %) are among the largest reported in any domain-specific QML benchmark to date, suggesting that strong FD events represent a particularly favourable problem geometry for quantum kernel methods. 5.5. Limitations and future directions Several limitations of the present study should be acknowledged. First, all kernel evaluations were performed using exact statevector simulation, which provides shot-noise-free fidelity estimates but does not model the decoherence and gate errors of real quantum hardware. In practice, quantum hardware estimates each kernel entry by measuring the all-zeros bitstring probability over a finite number of circuit executions (shots). This introduces statistical shot noise into every entry of the kernel matrix, with variance scaling as per entry. For a kernel matrix of size , errors in individual entries propagate to the SVM decision boundary and can substantially degrade classification performance, particularly for deeper circuits where the target probability is already small and concentrated. Achieving the AUC values reported here on real hardware would require either a large number of shots per kernel entry (typically – ) or error mitigation techniques, both of which further increase the computational cost. Prior work has shown that NISQ-era noise can degrade quantum kernel performance significantly for deeper circuits (Wang et al., 2021, Incudini et al., 2025), and the advantage margins reported here may be reduced or eliminated in a noisy hardware setting. Experiments on actual quantum processors would be required to assess the robustness of our results to realistic noise, and constitute the most immediate next step for this research. Second, the dataset is limited to a single neutron monitor station (JUNG). The global survey method combines data from the worldwide NMDB network to derive the Magn parameter used as the classification target, but the feature vectors used for classification include only the JUNG count rate as the GCR proxy. Including additional stations with different cutoff rigidities — such as Moscow (MOSC, 2.43 GV) or Oulu (3.7 GV) — would provide complementary spectral information about the GCR modulation and may further improve classification performance, particularly for weak events (Belov, 2008). Third, the magnitude threshold of 4% is empirically derived from the FEID catalogue and the specific feature representation used here. Whether this threshold generalises to other classification formulations (e.g., multiclass FD strength, onset-type classification) or other cosmic ray datasets is an open question. Future work should also explore whether projected quantum kernels (Huang et al., 2021) — which project the quantum state onto a lower-dimensional subspace before computing the kernel — can mitigate the concentration effect at larger qubit counts and potentially extend quantum advantage to the weak FD regime. 6. Conclusions We have presented a systematic benchmark of quantum kernel support vector machines for the classification of Forbush decreases, using a 25-year observational dataset constructed from the FEID catalogue, OMNI solar wind data, and JUNG neutron monitor count rates. A ZZFeatureMap quantum kernel with 4–8 qubits was evaluated across 180 experimental configurations and benchmarked against a classical RBF-SVM baseline under identical data conditions. The principal conclusions are as follows. 1. Quantum advantage is regime-dependent, not universal. The quantum kernel consistently outperforms the classical baseline only above a FD magnitude threshold of 4%. Below this threshold, the classical RBF-SVM is superior across all tested configurations. This result is consistent with theoretical predictions that quantum advantage requires data distributions whose geometry aligns with the inductive bias of the quantum feature map (Huang et al., 2021). 2. The transition at 4% is statistically robust. A Wilcoxon signed-rank test confirms that is statistically significant ( , rank-biserial ) for min_magn %, with increasing effect sizes at higher thresholds ( at min_magn %). The Kruskal–Wallis test confirms that the six threshold groups exhibit significantly different distributions ( , ). At min_magn %, 100% of tested configurations show quantum advantage, with mean (95% CI: ) and peak AUC (noting that this maximum is observed on a test set of 33 samples and should be interpreted with appropriate caution). 3. The magnitude threshold has a physical interpretation. Large-amplitude FDs are driven by well-developed ICME magnetic structures (flux ropes) whose passage produces structured, correlated dynamics across the IMF components, geomagnetic indices, and cosmic ray count rates. The ZZFeatureMap encodes pairwise feature correlations via its entangling layer, providing a representational advantage over the Euclidean-distance-based RBF kernel precisely in this structured regime. Weak FDs, by contrast, arise from a heterogeneous mixture of drivers and exhibit noise-dominated feature distributions that neither kernel can separate effectively. 4. Modest circuit architectures are optimal for near-term hardware. Circuits with 4–6 qubits achieve the most consistent quantum advantage across magnitude regimes, while 8-qubit circuits suffer from kernel concentration effects at lower magnitude thresholds. This trend is based on three qubit-count values ( ) and should be regarded as a qualitative tendency pending evaluation at additional circuit sizes. Circuit depth (reps) has a secondary effect, with deeper circuits providing marginal benefits in the signal-rich regime. 5. Quantum advantage is robust to small training sets. In the winning regime (min_magn %, ), mean is achieved with as few as 50 training samples, consistent with the theoretical expectation that quantum kernels are sample-efficient when the data geometry favours their inductive bias. This property is particularly relevant for FD classification, where strong events are rare across the historical record. These results establish FD magnitude as a principled criterion for predicting when quantum kernel methods will outperform classical alternatives in heliophysical classification tasks. More broadly, they suggest a general strategy for deploying near-term QML in physical sciences: identify the physical regime where structured, correlated multi-channel dynamics are present, and restrict quantum kernel methods to that regime. The present study is limited to statevector simulation; extension to real quantum hardware, additional neutron monitor stations, and projected quantum kernels represent natural directions for future work that will further test the robustness and scalability of these findings. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The author thanks the IZMIRAN team for maintaining the FEID catalogue, the NASA Space Physics Data Facility for the OMNI database, and the NMDB team and station operators for providing open access to neutron monitor data. This work was carried out within the Escuela de Transformación Digital at Universidad Tecnológica de Bolívar (UTB), Cartagena de Indias, Colombia. Quantum kernel computations were performed using the open-source Qiskit framework (Qiskit contributors, 2024) on an Intel Core i7-8565U workstation. Appendix. Quantum computing primer This appendix provides a self-contained introduction to the quantum computing concepts used in this paper. It is intended for readers with a background in classical machine learning who are unfamiliar with quantum mechanics. A comprehensive treatment can be found in Nielsen and Chuang (2010). A.1. Qubits and quantum states The fundamental unit of quantum computation is the qubit, the quantum analogue of a classical bit. Whereas a classical bit is always in state 0 or 1, a qubit can exist in a superposition of both: (A.1) Here and are the computational basis states (analogous to the unit vectors and ), and , are the probabilities of measuring the qubit in state 0 or 1 respectively. The state is a unit vector in a two-dimensional complex Hilbert space . A system of qubits occupies a -dimensional Hilbert space , with basis states for all binary strings . The general state of a -qubit system is (A.2) where the complex amplitudes encode a probability distribution over all possible measurement outcomes. This exponential growth of the state space with is the core resource that quantum computing exploits. A.2. Quantum gates and circuits Quantum computation proceeds by applying quantum gates — unitary transformations acting on the Hilbert space — to an initial state (typically ). The unitarity condition ensures that quantum evolution is reversible and probability-preserving. Single-qubit gates act on one qubit at a time; two-qubit gates introduce interactions between pairs of qubits. The gates relevant to the ZZFeatureMap used in this paper are: • Hadamard gate ( ): places a qubit in equal superposition, (A.3) • Phase gate ( , equivalent to up to global phase): rotates the phase of the component by angle , (A.4) In the ZZFeatureMap, encodes the th feature value as a rotation angle (angle encoding). • Controlled-phase gate ( ): a two-qubit gate that applies a phase only when both qubits are in state , (A.5) In the ZZFeatureMap, encodes pairwise feature products into the two-qubit interaction. A quantum circuit is a sequence of gates applied to an initial state, read from left to right. Fig. 1 shows the gate-level diagram of the ZZFeatureMap for qubits and one repetition. A.3. Measurement and probability estimation At the end of a quantum circuit, the state is measured in the computational basis. Measurement is probabilistic: the outcome is observed with probability , and the quantum state collapses to upon measurement. Quantum algorithms typically require many repeated executions (shots) of the same circuit to estimate these probabilities. For quantum kernel computation, the quantity of interest is the probability of the all-zeros outcome , which equals the squared overlap between two encoded states: (A.6) In this paper, is computed via statevector simulation — exact numerical integration of the Schrödinger equation — which provides shot-noise-free estimates. On real hardware, a finite number of shots introduces statistical noise; see Section 5.5. A.4. Entanglement and quantum feature maps A key property of multi-qubit systems is entanglement: quantum states that cannot be written as a product of individual qubit states. Entanglement is generated by two-qubit gates such as . In the ZZFeatureMap, the entangling layer creates correlations between pairs of qubits that depend on products of input features . This is the mechanism by which the quantum kernel captures cross-feature correlations that are absent from classical kernels operating on features independently. A quantum feature map is a parameterised quantum circuit that encodes a classical data vector into a quantum state . The resulting quantum kernel is a valid positive semi-definite kernel and can be used as a drop-in replacement for classical kernels in a support vector machine (Schuld and Killoran, 2019, Havlíček et al., 2019). The expressibility of the kernel — its ability to represent complex decision boundaries — grows with the number of qubits and the circuit depth, but so does the risk of kernel concentration (Section 5.2), which limits practical advantage to datasets with sufficiently structured feature geometry (Thanasilp et al., 2024, Huang et al., 2021). Data availability The FEID catalogue is publicly available at https://tools.izmiran.ru/w/feid. 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dc.rights.licenseAtribución-NoComercial-SinDerivadas 4.0 Internacional (CC BY-NC-ND 4.0)
dc.rights.urihttps://creativecommons.org/licenses/by-nc-nd/4.0/
dc.subject.ddc520 - Astronomía y ciencias afines::523 - Cuerpos y fenómenos celestes específicos
dc.subject.lembSpace weather
dc.subject.lembCosmic rays
dc.subject.lembSolar wind
dc.subject.lembCoronal mass ejections
dc.subject.lembHeliophysics
dc.subject.lembQuantum computing
dc.subject.lembMachine learning
dc.subject.lembSupport vector machines (Computer science)
dc.subject.lembArtificial intelligenc
dc.subject.lembNeutron monitors
dc.subject.ocde1. Ciencias Naturales::1A. Matemática::1A02. Matemáticas aplicadas
dc.subject.ocde1. Ciencias Naturales::1C. Ciencias físicas::1C08. Astronomía
dc.subject.ocde1. Ciencias Naturales::1B. Computación y ciencias de la información::1B01. Ciencias de la computación
dc.subject.odsODS 17: Alianzas para lograr los objetivos. Fortalecer los medios de implementación y revitalizar la Alianza Mundial para el Desarrollo Sostenible
dc.subject.proposalForbush decrease
dc.subject.proposalQuantum kernels
dc.subject.proposalMachine learning
dc.subject.proposalClassification task
dc.subject.proposalGeomagnetic indices
dc.titleMagnitude-dependent quantum advantage in Forbush decrease detection: A quantum kernel SVM benchmark
dc.typeArtículo de revista
dc.type.coarhttp://purl.org/coar/resource_type/c_18cf
dc.type.coarversionhttp://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.contentText
dc.type.driverinfo:eu-repo/semantics/article
dc.type.redcolhttp://purl.org/redcol/resource_type/ART
dc.type.versioninfo:eu-repo/semantics/publishedVersion
dspace.entity.typePublication
relation.isAuthorOfPublication996a607a-3eb1-4484-8978-ed736b9fc0b7
relation.isAuthorOfPublication.latestForDiscovery996a607a-3eb1-4484-8978-ed736b9fc0b7

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