Exhaustive Entanglement-Topology Enumeration and Edge-Vertex-Level Attribution in Quantum Feature Map Design for Classification

Abstract

Quantum kernel methods encode classical data into quantum states using specially designed feature-map circuits and then train a classical SVM on the resulting kernel matrix. Prior studies of entanglement structure typically compare only a few basic topologies, such as linear, circular, and full entanglement, leaving the broader space of entanglement graphs largely unexplored. This study exhaustively evaluates all possible entanglement topologies of a TwoLocal feature map and benchmarks them against classical methods and standard quantum feature maps on two synthetic datasets (Adhoc3_150 and Adhoc4_150) and two real-world datasets (Blood and Banknote). On Adhoc4_150, the best TwoLocal topology achieves an accuracy of 0.7556, matching Pauli Z; on Blood, it reaches 0.6889, exceeding RBF SVM and Pauli Z at 0.6667; and on Banknote, it achieves 1.0000, compared with 0.9778 for RBF SVM. In contrast, Random Forest achieves the highest accuracy on Adhoc3_150 at 0.8222, exceeding all evaluated quantum configurations. Across all four datasets, increasing the number of entangling pairs does not consistently improve accuracy, while the standard linear, circular, and full Pauli ZZ topologies never outperform Pauli Z. To further characterize this behavior, we propose a factorial linear modeling approach that quantifies the contribution of individual entanglement edges and qubit connectivity to classification accuracy. On Adhoc4_150, two edges exhibit significant negative effects, whereas Blood and Banknote show both significant positive and negative edge effects. The q0q1 edge is significant across all three four-feature datasets but reverses direction across datasets. These results indicate that the effect of entanglement depends jointly on the feature-map architecture, dataset, and specific qubit connections rather than on entanglement density alone. The proposed framework can also be applied to sampled topology sets, providing a basis for targeted entanglement analysis beyond exhaustively enumerable low-qubit systems.