Performance evaluation of Quantum Support Vector Machines under Realistic Noise and Mitigation

This work focuses on qSVMs which implement SVMs using quantum kernels while retaining the convex optimization properties of classical SVMs. The quantum kernel replaces the classical kernel, computing similarities between input data embedded in high-dimensional Hilbert spaces using quantum feature maps.

Quantum kernels are evaluated on locally simulated quantum devices using realistic hardware noise models that emulate the behavior of near-term quantum processors. The training kernel is obtained by estimating the fidelity between quantum states corresponding to each pair of data points. To account for readout errors inherent to NISQ devices, measurement outcomes are post-processed using M3 mitigation, which corrects observed probabilities through the estimation of marginal response matrices over subsets of qubits. In addition, we consider the use of a bandwidth scaling factor that rescales the input features prior to encoding, allowing control over the expressivity of the quantum feature space. This scaling is applied selectively, enabling a direct comparison between scaled and unscaled kernels, both with and without measurement error mitigation, and supporting a joint analysis of their effects on kernel quality and downstream classification performance.

After computing the quantum kernel matrices, the corresponding classification task is carried out using a classical SVM, where the optimization problem is solved within the standard convex quadratic formulation. The regularization hyperparameter C is determined through cross-validation, ensuring a consistent model selection procedure across all configurations. The resulting model is then evaluated on the test kernel to obtain the final classifier predictions. This integrated workflow enables a systematic assessment of how noise, measurement error mitigation, and bandwidth scaling influence qSVM performance under realistic near-term quantum conditions.

To ensure a statistically robust evaluation, the entire experimental procedure is repeated multiple times using different random train-test splits of the dataset. This repeated sampling enables a reliable characterization of variability in both classification accuracy and kernel matrix estimations across configurations. The statistical significance of the observed differences, such as those between noisy and noiseless settings or with and without mitigation, is assessed using non-parametric statistical analyses providing a consistent framework for determining whether the observed deviations are meaningful, supporting a rigorous comparison of noise effects, bandwidth scaling, and mitigation strategies.

The correct functioning of the proposed qSVM framework is demonstrated through a fully implemented quantum kernel pipeline built using established quantum machine learning libraries. The implementation leverages standard Qiskit-based tools for circuit construction, kernel evaluation, and simulation, ensuring a reproducible workflow based on established quantum machine learning libraries without requiring low-level implementation of all components.

The input data used in this study are taken from the MNIST-1D dataset [5], which has been proposed in the literature as a more representative benchmark for evaluating quantum machine learning models compared to simpler alternatives [6]. Before quantum encoding, the data are standardized and rescaled to the interval [0,2π] to match the input requirements of the quantum feature maps. To further reduce computational and qubit requirements, the dataset is compressed to 8 dimensions using Principal Component Analysis (PCA).

Quantum feature encoding is performed using four different feature maps, namely the Z feature map, the ZZ feature map, a variant of the Covariant feature map [7], and the 3D+CNOT feature map [8]. These feature maps encode classical data into quantum states, and the corresponding kernel entries are computed as fidelities between pairs of quantum states generated by these circuits. This process is executed using shot-based simulators that support both ideal and noisy executions.

To emulate realistic quantum hardware conditions, simulations are performed using noise models derived from IBM TorontoV2 and CESGA Qmio backend specifications. These models include device connectivity constraints, native gate sets, and calibrated error parameters, allowing the evaluation of quantum kernels under conditions that closely resemble real quantum processors.

The pipeline is evaluated under multiple configurations combining noise, quantum kernel bandwidth scaling, and M3 measurement error mitigation. Noiseless simulations are used as a reference baseline, while noisy simulations enable the assessment of hardware-induced effects. M3 mitigation is applied as a post-processing correction to measurement outcomes in noisy runs, and bandwidth scaling is applied at the data encoding stage by globally rescaling input features before quantum embedding.

For each configuration, kernel matrices are computed using repeated circuit executions with the sampling shots set to 1000, and a classical SVM is trained using standard library implementations of convex quadratic optimization. The experiments are repeated over 30 independent random train-test splits to ensure statistical reliability and robustness against sampling variability.

Overall, this setup provides a complete and reproducible demonstration of the qSVM workflow, combining library-based quantum circuit execution, realistic noise simulation, and classical post-processing into a unified experimental framework.

The results of this study provide evidence for the practical viability of quantum kernel methods on near-term noisy quantum hardware. In particular, qSVMs exhibit strong robustness to realistic noise, as classification performance remains largely stable across most configurations even when kernel matrices are significantly affected by hardware imperfections. Importantly, statistical comparisons between noisy and noiseless settings indicate that these differences are often not significant at the level of classification accuracy, despite observable deviations in the kernel matrix structure.

Measurement error mitigation (M3) primarily affects the structure of the kernel matrices by improving their agreement with the ideal, noiseless kernels. This results in a more faithful reconstruction of the underlying similarity structure, which is reflected in significant differences at the kernel level for several configurations, even when changes in classification accuracy are more moderate. In some cases, M3 helps recover kernel structures that are statistically closer to the ideal reference, particularly when combined with bandwidth scaling.

Within the same experimental framework, quantum kernel bandwidth scaling consistently leads to substantial improvements in classification accuracy across all feature maps, with scaled configurations systematically outperforming their unscaled counterparts. Moreover, its performance remains stable across different noise models, maintaining the robustness of the learning process. 

Overall, the results show that while noise significantly impacts kernel representations, its effect on final classification is more limited. At the same time, both bandwidth scaling and M3 mitigation play complementary roles, with scaling primarily improving performance and mitigation improving kernel fidelity, jointly contributing to more reliable quantum kernel-based learning under realistic hardware conditions.

Despite the encouraging results obtained in simulation, a key limitation of this study is the reliance on noise models rather than direct execution on physical quantum hardware. While hardware-informed simulators capture important device characteristics such as connectivity constraints and calibrated error rates, they may still fail to fully reproduce the complex, time-dependent, and context-specific nature of hardware noise. Therefore, validation on real quantum processors remains necessary to confirm whether the observed robustness of qSVMs and the relative benefits of bandwidth scaling and M3 mitigation persist under actual experimental conditions.

In addition, quantum kernel methods based on fidelity estimation face inherent scalability challenges that are not fully captured in small-scale simulations. In particular, the need to compute pairwise fidelities between quantum states leads to quadratic scaling in the number of data points, while the underlying quantum circuits may require a number of qubits that grows with the dimensionality of the encoded data. Moreover, issues such as concentration effects in high-dimensional Hilbert spaces and the practical cost of repeated circuit evaluations can limit the applicability of these methods to larger, real-world datasets. These factors highlight that, although promising in the NISQ regime, quantum kernel approaches must address both hardware limitations and algorithmic scaling constraints to become viable at industrially relevant scales.