Neural quantum support vector data description for one-class classification

Changjae Im; Hyeondo Oh; Daniel K. Park

quant-phMar 3, 2026

Neural quantum support vector data description for one-class classification

Authors: Changjae Im, Hyeondo Oh, Daniel K. Park

Organizations: Department of Statistics and Data Science, Yonsei University, Seoul, Republic of Korea · Department of Applied Statistics, Yonsei University, Seoul, Republic of Korea · Department of Quantum Information, Yonsei University, Seoul, Republic of Korea

Abstract

One-class classification (OCC) is a fundamental problem in machine learning with numerous applications, such as anomaly detection and quality control. With the increasing complexity and dimensionality of modern datasets, there is a growing demand for advanced OCC techniques with better expressivity and efficiency. We introduce Neural Quantum Support Vector Data Description (NQSVDD), a classical-quantum hybrid framework for OCC that performs end-to-end optimized hierarchical representation learning. NQSVDD integrates a classical neural network with trainable quantum data encoding and a variational quantum circuit, enabling the model to learn nonlinear feature transformations tailored to the OCC objective. The hybrid architecture maps input data into an intermediate high-dimensional feature space and subsequently projects it into a compact latent space defined through quantum measurements. Importantly, both the feature embedding and the latent representation are jointly optimized such that normal data form a compact cluster, for which a minimum-volume enclosing hypersphere provides an effective decision boundary. Experimental evaluations on benchmark datasets demonstrate that NQSVDD achieves competitive or superior AUC performance compared to classical Deep SVDD and quantum baselines, while maintaining parameter efficiency and robustness under realistic noise conditions.

Figures & tables

Figure 1: Overview of NQSVDD. Classical neural network layers extract features from complex input data. The output of classical layers is encoded into quantum state using ZZ feature embedding. Alternating repetition of the embedding layers and parameterized quantum layers makes the expressive power of the embedding richer. After the state preparation, variational quantum circuit and a set of Pauli measurements map the features into the latent space. The combination of classical layers, denoted as $\phi_{c}(\cdot;\boldsymbol{\mathcal{W}})$ , and quantum layers, denoted as $\phi_{q}(\cdot;\boldsymbol{\theta})$ , construct the overall feature mapping for OCC, trained to find the optimal latent space and hypersphere.

Figure 2: The parameterized two-qubit gate that can generate an arbitrary two-qubit unitary transformation in $SU(4)$ , requiring $15$ real parameters. $R_{i}(\theta)$ is a rotation gate of $i$ axis of Bloch sphere, and $U_{3}(\theta,\phi,\lambda)=R_{z}(\phi)R_{x}(-\pi/2)R_{z}(\theta)R_{x}(\pi/2)R_{z}(\lambda)$ . This two-qubit ansatz is applied for parameterized unitary layers in data embedding, and convolutional layers in QCNN.

Panel of Figure 3 — Figure 3: Avg. AUC (%) and std. (error bar) on (a) MNIST and (b) Fashion-MNIST datasets. NQSVDD, QSVDD and DSVDD have $1105$ , $75$ and $2152$ number of parameters, respectively. Each bar represents the average score of $10$ repetitions.

			Method
Target Class	NQSVDD	QSVDD(amp)	QSVDD(zz)	DSVDD	DSVDD(reduced)
0	$\boldsymbol{97.45}\pm 2.07$	$95.50\pm 1.84$	$88.70\pm 3.49$	$94.49\pm 3.80$	$92.42\pm 5.03$
1	$99.27\pm 0.79$	$97.96\pm 0.95$	$97.36\pm 1.36$	$\boldsymbol{99.54}\pm 0.40$	$99.29\pm 0.32$
2	$\boldsymbol{92.60}\pm 4.89$	$84.53\pm 5.17$	$82.70\pm 2.87$	$77.48\pm 10.84$	$70.25\pm 8.16$
3	$\boldsymbol{92.59}\pm 2.58$	$85.27\pm 3.64$	$83.71\pm 4.48$	$80.28\pm 8.60$	$80.35\pm 5.52$
4	$93.81\pm 3.83$	$\boldsymbol{94.74}\pm 2.15$	$88.32\pm 4.01$	$90.54\pm 2.88$	$87.29\pm 5.24$

Table 1: Average AUC scores in % with StdDevs with 10 random seeds per method and one-class experiment on MNIST and Fashion-MNIST dataset. Boldface indicates the best performing result within each class.

Method	Params	Avg. AUC
NQSVDD	$210$	$\boldsymbol{94.67}\pm 0.84$
QSVDD(amp)	$75$	$87.98\pm 7.59$
DSVDD	$288$	$92.30\pm 2.24$

Table 2: Average AUC scores in % with StdDevs with 20 random seeds per method on Credit Card Transaction dataset.

Method	Params	Avg. AUC
NQSVDD	$225$	$\boldsymbol{75.32}\pm 10.58$
QSVDD(amp)	$75$	$56.04\pm 14.35$
DSVDD	$456$	$59.45\pm 23.46$

Table 3: Average AUC scores in % with StdDevs with 20 random seeds per method on Network Intrusion dataset.

Backend Params
2-qubit Depolarizing	$0.00332$
1-qubit Gate Length (ns)	$32$
2-qubit Gate Length (ns)	$68$
$T_{1}$ ( $\mu$ s)	$183.29$
$T_{2}$ ( $\mu$ s)	$141.73$

Table 4: Median error rates and gate lengths for the IBM quantum device ibm_kingston used in the noisy simulation.

T_{1}

and

T_{2}

represents energy relaxation time and dephasing time, respectively.

Dataset	Method	Params	Condition	Avg. AUC
MNIST (target: 0)	NQSVDD	$1090$	Noiseless	$97.92\pm 1.33$
			Noise	$94.00\pm 2.37$
	DSVDD	$2152$	-	$92.11\pm 5.11$
Fashion-MNIST (target: T-shirt)	NQSVDD	$1090$	Noiseless	$90.14\pm 2.75$
			Noise	$85.84\pm 1.83$
	DSVDD	$2152$	-	$81.12\pm 6.69$

Table 5: Average AUC scores in % with StdDevs with five random seeds per method on MNIST, Fashion MNIST, Credit Card Transaction, Network Intrusion datasets under both noiseless and noisy conditions. The noisy NQSVDD is trained for

500

iterations with a batch size of four. The noiseless NQSVDD and the classical DSVDD adopt the same suboptimal configuration.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 9 — Figure A.1 : Avg. AUC (%) and std. (error bar) by the dimension of latent space on (a) “0” of MNIST and (b) “T-shirt” of Fashion-MNIST datasets with $10$ random seeds per latent dimension.

Figure A.2 : Avg. AUC (%) and std. (error bar) for different numbers of embedding layers in NQSVDD. For MNIST and Fashion MNIST, the average and std. of AUC scores are computed over all target-class experiments, using 10 random seeds per target class (100 experiments per the number of embedding layers). For Credit Card Transaction and Network Intrusion datasets, results are averaged over 10 random seeds for each number of embedding layers.

Panel of Figure 11 — Figure A.3 : Visualization of NQSVDD outputs for MNIST test data in a two-dimensional space. To enable visualization, the output dimension of NQSVDD is set to two. The upper plot illustrates the output of the initial model, and the bottom plot shows the mapping after $1500$ training iterations.

Panel of Figure 12 — Figure A.4 : Visualization of NQSVDD outputs for Fashion-MNIST test data in a two-dimensional space. To enable visualization, the output dimension of NQSVDD is set to two. The upper plot illustrates the output of the initial model, and the bottom plot shows the mapping after $1500$ training iterations.

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