Comparing Classical and Quantum Variational Classifiers on the XOR Problem

Miras Seilkhan; Adilbek Taizhanov

cs.LGFeb 27, 2026

Comparing Classical and Quantum Variational Classifiers on the XOR Problem

Authors: Miras Seilkhan, Adilbek Taizhanov

Abstract

Quantum machine learning applies principles such as superposition and entanglement to data processing and optimization. Variational quantum models operate on qubits in high-dimensional Hilbert spaces and provide an alternative approach to model expressivity. We compare classical models and a variational quantum classifier on the XOR problem. Logistic regression, a one-hidden-layer multilayer perceptron, and a two-qubit variational quantum classifier with circuit depths 1 and 2 are evaluated on synthetic XOR datasets with varying Gaussian noise and sample sizes using accuracy and binary cross-entropy. Performance is determined primarily by model expressivity. Logistic regression and the depth-1 quantum circuit fail to represent XOR reliably, whereas the multilayer perceptron and the depth-2 quantum circuit achieve perfect test accuracy under representative conditions. Robustness analyses across noise levels, dataset sizes, and random seeds confirm that circuit depth is decisive for quantum performance on this task. Despite matching accuracy, the multilayer perceptron achieves lower binary cross-entropy and substantially shorter training time. Hardware execution preserves the global XOR structure but introduces structured deviations in the decision function. Overall, deeper variational quantum classifiers can match classical neural networks in accuracy on low-dimensional XOR benchmarks, but no clear empirical advantage in robustness or efficiency is observed in the examined settings.

Figures & tables

Figure 1: Illustration of the XOR classification problem. The two classes (black and white points) cannot be separated by a single straight line, demonstrating the linear inseparability of XOR.

Panel of Figure 2 — Figure 2: Schematic illustration of a one-hidden-layer multilayer perceptron (MLP). Only a subset of hidden units is shown; the total number of hidden units $h$ is a tunable hyperparameter.

Figure 3: Bloch sphere representation of a single qubit. Any pure qubit state can be visualized as a point on the unit sphere, parameterized by the polar and azimuthal angles $(\theta,\phi)$ . Source: Wikipedia – Bloch sphere [ 12 ] .

Panel of Figure 4 — Figure 4: High-level structure of a Variational Quantum Classifier (VQC): feature map, trainable ansatz, and measurement.

Panel of Figure 5 — Figure 5: Illustration of the three XOR dataset variants used in this study: (a) Dataset A — clean, discrete XOR; (b) Dataset B — clustered XOR with additive Gaussian noise; (c) Dataset C — continuous XOR defined by a threshold-based rule.

Parameter	Value
Datasets
Types	Clean XOR (A), Noisy Clustered XOR (B), Continuous XOR (C)
Noise level (Dataset B)	$\sigma\in\{0.00,0.05,0.10,0.20,0.30\}$
Samples per cluster (Dataset B)	$\{25,50,100,250,500\}$
Threshold (Dataset C)	$t=0.5$
Training protocol

Table 1: Experimental settings: datasets, training protocol, and random seeds.

Parameter	Value
Classical models
Logistic regression	2 input features + bias (3 trainable parameters)
MLP hidden units	$h\in\{1,2,4,8\}$
Activation function	Sigmoid
Optimizer	Stochastic Gradient Descent (SGD)
Training epochs	3000–5000 (depending on setting)

Table 2: Model configurations used in this study (classical and quantum).

$L$	#params	shots	encoding	observable	device/interface
1	6	analytic	$RX(\pi x_{1}),RX(\pi x_{2})$	$\langle Z_{0}\rangle$	default.qubit / autograd
1	6	128	$RX(\pi x_{1}),RX(\pi x_{2})$	$\langle Z_{0}\rangle$	default.qubit / autograd
1	6	1024	$RX(\pi x_{1}),RX(\pi x_{2})$	$\langle Z_{0}\rangle$	default.qubit / autograd
2	12	analytic	$RX(\pi x_{1}),RX(\pi x_{2})$	$\langle Z_{0}\rangle$	default.qubit / autograd
2	12	1024	$RX(\pi x_{1}),RX(\pi x_{2})$	$\langle Z_{0}\rangle$	default.qubit / autograd

Table 3: Variational quantum classifier (VQC) configurations used in the experiments.

Panel of Figure 6 — Figure 6: Decision boundaries learned by the classical MLP ( $h=4$ ) on representative XOR settings.

Panel of Figure 7 — Figure 7: Decision boundaries learned by the VQC. The background color represents the prediction score $f(x)=2p(y=1\mid x)-1\in[-1,1]$ using a shared color scale. The dashed line indicates the decision boundary ( $p=0.5$ ).

Panel of Figure 8 — Figure 8: Learning curves for the MLP ( $h=4$ ) and the variational quantum classifier (VQC), including training and test loss, accuracy, and representative runs across random initializations.

Panel of Figure 9 — Figure 9: Robustness of classical and quantum models. Test accuracy as a function of data noise level, dataset size, and measurement shot noise (VQC).

Panel of Figure 10 — Figure 10: Sensitivity to random initialization across models: accuracy and BCE evaluated over multiple training seeds.

Model	Train acc.	Test acc.	Test loss (BCE)
LR	0.759 $\pm$ 0.000	0.713 $\pm$ 0.000	0.696 $\pm$ 0.000
MLP ( $h=4$ )	1.000 $\pm$ 0.000	1.000 $\pm$ 0.000	0.016 $\pm$ 0.005
VQC ( $L=1$ , analytic)	0.529 $\pm$ 0.004	0.508 $\pm$ 0.007	0.692 $\pm$ 0.000
VQC ( $L=1$ , 1024 shots)	0.531 $\pm$ 0.018	0.525 $\pm$ 0.058	0.692 $\pm$ 0.003
VQC ( $L=1$ , 128 shots)	0.496 $\pm$ 0.016	0.488 $\pm$ 0.056	0.696 $\pm$ 0.009
VQC ( $L=2$ , analytic)	1.000 $\pm$ 0.000	1.000 $\pm$ 0.000	0.0185 $\pm$ 0.0000

Table 4: Model performance on Dataset B (

\sigma=0.10

n=100

). Reported values are mean

\pm

std over five model seeds (

0

–

4

Model	#params	Train time (s)
LR	3	0.005 $\pm$ 0.000
MLP ( $h=4$ )	17	0.053 $\pm$ 0.001
VQC ( $L=1$ , analytic)	6	360.500 $\pm$ 4.500
VQC ( $L=1$ , 1024 shots)	6	442.500 $\pm$ 3.700
VQC ( $L=1$ , 128 shots)	6	410.880 $\pm$ 1.730
VQC ( $L=2$ , analytic)	12	943.300 $\pm$ 7.300

Table 5: Model complexity and training cost on Dataset B (

\sigma=0.10

n=100

). Reported values are mean

\pm

std over five model seeds.

Panel of Figure 11 — Figure 11: Decision surfaces of the variational quantum classifier for circuit depths $L=1$ and $L=2$ in the analytic regime. The color scale is consistent with Fig. 7 .

Panel of Figure 12 — Figure 12: Decision function comparison for a fixed trained VQC ( $L=2$ ) on clean XOR. The color encodes $f(x)=2p(y=1\mid x)-1$ evaluated on a dense grid. Hardware execution preserves the global XOR-like structure but introduces visible local fluctuations and reduced smoothness compared to the ideal simulator.

Panel of Figure 13 — Figure 13: Quantifying hardware-induced deviations for a fixed trained VQC. Left: absolute distortion map over the evaluation grid, revealing structured, spatially varying deviations (largest near the outer regions of the grid). Right: controlled comparison showing that finite-shot simulation remains close to the ideal baseline, whereas hardware execution increases the deviation magnitude. In our run, the mean absolute deviation between hardware and finite-shot simulation is $\approx 0.118$ , while the MSE summary indicates a clear gap between shot noise and device noise.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 14 — Figure 14: MLP width ablation on Dataset B ( $\sigma=0.10$ , $n_{\text{per\_cluster}}=100$ ). Performance improves with increasing width and saturates for moderate model sizes.

Panel of Figure 15 — Figure 15: Two-dimensional parameter-space loss landscape slices in parameter space for the variational quantum classifier at circuit depths $L=1$ and $L=2$ (analytic regime). The horizontal color scale indicates the binary cross-entropy loss value and is shared across both surfaces.

Figure 16: Distribution of decision scores $f(x)=2p-1$ over the evaluation grid: ideal simulation vs. IBM hardware. Hardware execution shows a visible compression toward zero, consistent with reduced contrast caused by device noise.

Panel of Figure 17 — Figure 17: Qualitative results for Dataset C: learning dynamics and threshold dependence across classical and variational models.

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Comparative Study of Quantum and Classical Machine Learning Models in Binary Classification

Anand Kumar Mishra, Ramanuj Awasthi

School of Engineering & Technology (UIET), CSJM University, Kanpur, Uttar Pradesh, India

Quantum Machine Learning Quantum Kernel Methods Binary Classification Variational Quantum Circuits

Open paper AskPDF arXiv

A potential path forward is Quantum Machine Learning (QML), which aims to leverage quantum computing in conjunction with classical machine learning to enhance computing efficiency and the expressiveness of models. In this paper, two different quantum classifiers - Variational Quantum Classifier (VQC) and Quantum Kernel Support Vector Machine (QSVM) - are compared with three classical classifiers as baseline classifiers - Logistic Regression, Support Vector Machine (SVM), and a Multi-Layer Perceptron (MLP) - on the Breast Cancer Wisconsin dataset. The quantum circuits were created in the PennyLane framework and simulated on a classical backend. However, in terms of accuracy, classical Logistic Regression performed better with an accuracy of 97.8%, classical SVM and QSVM with an accuracy of 95.6% each, although the Quantum VQC achieved a lower accuracy of 88.9% and had a recall of 100% for the benign class, though it correctly identified only 12 of the 17 malignant cases (a malignant-class recall of approximately 70.6%). The drawback of quantum models is the higher training time; however, since the quantum circuit needs to be classically simulated, the quantum SVM took 23.29 seconds compared to less than 0.01 seconds for the classical linear models. These results indicate that for small structured datasets, classifiers based on quantum computing have not yet surpassed well-tuned classical counterparts. In some respects (e.g., benign-class recall), they perform competitively, though not on malignant-class recall, where the VQC in particular performed worse than the classical baselines, which is worth further investigation on real quantum computers.

Abstract

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