Quantum kernel learning Model constructed with small data

We aim to use quantum machine learning to detect various anomalies in image inspection by using small size data. Assuming the possibility that the expressive power of the quantum kernel space is superior to that of the classical kernel space, we are studying a quantum machine learning model. Through trials of image inspection processes not only for factory products but also for products including agricultural products, the importance of trials on real data is recognized. In this study, training was carried out on SVMs embedded with various quantum kernels on a small number of agricultural product image data sets collected in the company. The quantum kernels prepared in this study consisted of a smaller number of rotating gates and control gates. The F1 scores for each quantum kernel showed a significant effect of using CNOT gates. After confirming the results with a quantum simulator, the usefulness of the quantum kernels was confirmed on a quantum computer. Learning with SVMs embedded with specific quantum kernels showed significantly higher values of the AUC compared to classical kernels. The reason for the lack of learning in quantum kernels is considered to be due to kernel concentration or exponential concentration similar to the Baren plateau. The reason why the F1 score does not increase as the number of features increases is suggested to be due to exponential concentration, while at the same time it is possible that only certain features have discriminative ability. Furthermore, it is suggested that controlled Toffoli gate may be a promising quantum kernel component.

💡 Research Summary

This paper investigates the feasibility of quantum kernel methods for anomaly detection in image inspection when only a very small training set is available. The authors focus on a practical use‑case: detecting invisible vine‑crack defects inside apples, which serve as a proxy for industrial products that may exhibit subtle anomalies. A dataset of 66 images (33 normal, 33 anomalous) was collected from commercially available apples; 24 normal and 24 anomalous images were randomly selected for training, and 9 of each class were reserved for testing. Images were captured under controlled LED illumination, cut in half to expose internal structure, binarized, and down‑sampled from 4032 × 3024 to 403 × 302 pixels to make the data tractable for machine‑learning pipelines.

Feature extraction was performed by principal component analysis (PCA). The first three principal components (PCs) captured the external shape, browning, and internal vine‑crack respectively, while the cumulative contribution ratio (CCR) reached 0.85 after ten PCs. These PCs were used as the input vector for both classical and quantum kernels.

Eleven quantum kernel circuits (QK0–QK10) were designed, varying in the number and type of rotation and controlled gates. QK0 and QK1 are simple circuits containing only Hadamard and Ry rotations. QK2–QK5 introduce staircase patterns of controlled‑Ry and controlled‑Rx gates. QK6–QK9 incorporate CNOT gates in different configurations, and QK10 replaces each CNOT with a controlled‑Toffoli gate, dramatically increasing circuit depth. All circuits were first evaluated on IBM’s qasm simulator and then on the real quantum processor ibmq_Osaka.

Performance was measured with two metrics: F1‑score (to assess classification balance) and ROC‑AUC (to capture overall discriminative ability). On the simulator, QK9 achieved an AUC of 0.90, substantially higher than the classical radial‑basis‑function (RBF) kernel’s 0.62. QK10 also reached 0.89 on the simulator but dropped to 0.59 on the real device, indicating severe noise accumulation at the large circuit depth (≈273 gates). QK9, with a modest depth of 32 gates, showed almost identical results on simulator and hardware, confirming its robustness.

Analysis of the F1‑score as a function of the number of PCs revealed that kernels containing CNOT gates (QK7–QK10) benefit from additional features, whereas simpler kernels (QK0–QK5) plateau early. The authors attribute the stagnation of performance with increasing feature dimension to kernel concentration or exponential concentration phenomena, where random high‑dimensional quantum states tend to produce nearly identical inner products, limiting the expressive power of the kernel.

The discussion highlights that while Toffoli‑based kernels theoretically provide richer non‑linear transformations, current noisy intermediate‑scale quantum (NISQ) hardware cannot reliably execute such deep circuits. Errors grow sharply between depths of 32 and 273 gates, as confirmed by a systematic depth‑analysis. Consequently, QK9 emerges as the most promising candidate for near‑term applications, balancing expressive power and hardware feasibility.

In conclusion, the study demonstrates that quantum kernels can outperform classical RBF kernels even with a tiny training set, provided that the quantum circuit is carefully designed to stay within the noise limits of existing devices. The work suggests future directions such as error‑mitigation techniques, circuit compression, hybrid quantum‑classical models, and validation on larger, more diverse industrial datasets.