Training-Free Entangler Selection for Quantum Neural Networks via Hilbert–Schmidt Geometry

Choosing where to place entangling gates is a central design choice in quantum machine learning circuits, yet entanglers are still typically chosen from fixed templates or by expensive search procedures requiring training or pairwise kernel evaluations. We propose the \emph{HSD-Lipschitz principle}, a training-free geometric criterion built on a simple intuition: a useful entangler should keep encoded states stable across the dataset while separating different classes. We formalise this through a two-sided bound on local light-cone subsystems. Under a light-cone locality assumption, dataset-level gradient variance is upper-bounded by reduced-state spread; under random initialisation, the expected between-class gradient signal is lower-bounded by class-conditional reduced-state distance. The resulting algorithm, HSD-greedy, adaptively grows an entanglement structure using only single-state reduced moments, estimable via classical shadows, and avoids the $\mathcal{O}(N^2)$ pairwise kernel evaluations of kernel-target alignment proxies. Empirically, both bounds hold without violation across $110$ topologies and $440$ parameter rows, and extend to $180$ additional configurations. HSD-greedy attains the strongest average AUC among training-free entangler selectors on synthetic and real-world benchmarks, at roughly $60\times$ lower selection cost than the strongest baseline in 13-qubit simulation. End-to-end execution on a 20-qubit IBM Eagle subchain substantially outperforms a fixed 19-CNOT Linear chain and matches or exceeds a matched-budget Random baseline while using far fewer two-qubit gates than the Linear chain.