EFFECTIVE DIMENSION GOVERNS GENERALIZATION IN QUANTUM KERNEL VISION MODELS

Quantum machine learning models, such as quantum vision transformers (QViTs) and quantum convolutional networks (QCNNs), have shown promising results on small-scale vision tasks. However, two empirical observations have remained largely mysterious: why models with more entanglement often generalize better, and why injecting quantum noise can sometimes improve, rather than degrade, test accuracy. This paper investigates whether these phenomena are linked by a single underlying principle.

The authors propose that both entanglement and quantum noise act as "knobs" that manipulate the eigenspectrum of the quantum feature kernel. They focus on the effective dimension ($d_{\text{eff}}$), a measure of the participation ratio of the kernel's eigenvalues. By analyzing the spectral properties of the kernel, the authors derive an exact decomposition for depolarizing noise and demonstrate that amplitude damping also contracts the spectrum. They test these theoretical predictions across various entangling topologies and noise levels, using both simulated models and real hardware (IBM Heron).

The study establishes that $d_{\text{eff}}$ is a primary driver of generalization. In regimes where the model is overfitting, contracting $d_{\text{eff}}$ acts as a form of ridge-like regularization, which can improve test accuracy up to an optimal point. Conversely, in underfitting regimes, this contraction can hurt performance. The authors show that test accuracy collapses onto a single, stable function of $d_{\text{eff}}$ across different entangling ansatze, provided the models have comparable feature-space alignment. They also identify entanglement as a necessary "precondition" that ensures the model is in a regime where the $d_{\text{eff}}$ law holds; unentangled models fail to align well with the target labels and thus do not follow the same accuracy curve.