Geometry Matters: Kernel Concentration in Quantum Feature Maps

Quantum machine learning (QML) is motivated by the premise that quantum circuits can embed classical data into feature Hilbert spaces in which learning may become more effective. We evaluate this premise empirically by comparing two feature maps, the entangling ZZFeatureMap and simple angle encoding, on multiclass MNIST classification. Using four kernel geometry metrics (kernel target alignment, intra/inter similarity ratio, effective rank, and off-diagonal variance), we show that ZZFeatureMap suffers severe kernel concentration as task complexity increases: its similarity matrix becomes nearly uniform, its effective rank exceeds 300, and its alignment with the labels collapses. In contrast, angle encoding preserves a structured, low-rank kernel and achieves 83% accuracy on 8-class, 8-qubit MNIST, outperforming ZZFeatureMap by 42 percentage points. We find that kernel target alignment and off-diagonal variance are reliable predictors of classification performance, while effective rank reveals when the kernel becomes diffusely concentrated. These metrics offer practical diagnostics for selecting feature maps in near-term quantum kernels. Overall, the results indicate that practical QML performance depends less on generic circuit expressivity than on whether the chosen feature map induces a useful geometry for the data.