Optimal Classification of Three-Qubit Entanglement with Cascaded Support Vector Machine

TL;DR

A key contribution of this research is an optimization protocol based on systematic feature importance analysis that yields a tunable framework that significantly reduces the number of required features, while maintaining reliable model accuracy.

Abstract

We introduce a systematic framework for three-qubit entanglement classification using a cascaded architecture of Support Vector Machine (SVM) classifiers. Leveraging the well defined three-qubit structure with the four nested entanglement classes (S, B, W, and GHZ), we construct three distinct witness models ($\mathcal{M}_{B}$, $\mathcal{M}_{W}$, and $\mathcal{M}_{GHZ}$) that sequentially discriminate between these classes. The proposed Cascaded model achieves an overall classification accuracy of $95\%$ on a comprehensive dataset of mixed states. The framework's robustness and generalization capabilities are confirmed through rigorous testing against out-of-distribution (OOD) entangled states and various quantum noise channels, where the model maintains high performance. A key contribution of this research is an optimization protocol based on systematic feature importance analysis. This approach yields a tunable framework that significantly reduces the number of required features, while maintaining reliable model accuracy.

Figures (9)

• Figure 1: The schematic structure of three-qubit entanglement classes. $S$, $B$, $W$ and $GHZ$ refer to fully separable, biseparable, $W$ and $GHZ$ classes respectively. The four sets form a hierarchy structure $S \subseteq B \subseteq W \subseteq GHZ$.
• Figure 2: The schematics of three distinct witness models, $\mathcal{M}_{1}$, $\mathcal{M}_{2}$, and $\mathcal{M}_{3}$, all constructed to discriminate between the sets $\mathcal{S}_1$ and $\mathcal{S}_2 \setminus \mathcal{S}_1$. Among these, $\mathcal{M}_{3}$ demonstrates a higher degree of optimality, attaining improved accuracy by positioning its decision plane in closer proximity to the boundary of $\mathcal{S}_1$.
• Figure 3: Cascaded classification of an unlabeled quantum state $\rho$ using a sequence of three models. The process begins with $\mathcal{M}_{GHZ}$: the state $\rho$ may be classified as belonging to $GHZ \setminus W$; otherwise, it is assigned to the $W$ category. In this latter case, the subsequent application of $\mathcal{M}_{W}$ distinguishes between $W \setminus B$ and $B$. Finally, for states identified within $B$, the model $\mathcal{M}_{B}$ determines whether $\rho \in B \setminus S$ or $\rho \in S$. This cascaded framework achieves precise and interpretable classification through sequential hyperplane-based decision boundaries.
• Figure 4: ROC curves for the three witness models. Each panel displays the performance of soft-margin SVM classifiers employing either a RBF kernel or polynomial kernels of degrees 2–9. Across all models, the RBF kernel (red curves) consistently achieves superior discrimination, exhibiting higher true positive rates over the entire range of false positive rates when compared with the best-performing polynomial kernels.
• Figure 5: Confusion matrices for the three witness models. Each matrix compares predicted and true labels. Diagonal entries correspond to correct classifications, while off-diagonal entries indicate misclassifications between the entanglement classes.