Metric Learning with Quantum Feature Spaces

Metric learning depends on the geometry of its embedding space, not merely its dimension, yet quantum metric learning is usually motivated by the exponential dimension of Hilbert space. We evaluate quantum feature spaces against the properties classical metric learning requires of an embedding space: capacity for margins, bounded diameter, non-vanishing gradients, valid kernels, non-concentrating distances, cheap distance evaluation, and resistance to overfitting. To separate geometry from function approximation, we let a classical network predict every gate parameter, making the model a quantum kernel method with a learned feature map. We prove that data-independent entanglement is a metric isometry, while active entanglement enlarges the state manifold at the cost of fidelity concentration, vanishing gradients, and exponential shot requirements. The unentangled product manifold $({\mathbb{CP}}^1{)}^N$ avoids these failures: its fidelity is an exact positive-semidefinite kernel computable classically in $\mathcal{O}(N)$. It still suffers from geometric shattering: the encoder can scatter training samples into nearly orthogonal states, so generalization depends on controlling the encoder, not on entanglement.