Metric Learning with Quantum Feature Spaces

Metric learning trains an encoder so that distance in an embedding space reflects semantic similarity, underpinning few-shot classification and multimodal retrieval. Quantum metric learning (QMeL) replaces the classical embedding space with the state space of $N$ qubits, motivated by its $2^{N}$ dimensions and by quantum fidelity $F(x,x’)=\lvert\langle\psi(x)\mid\psi(x’)\rangle\rvert^{2}$ as a natural positive-semidefinite kernel. Metric learning, however, depends on metric geometry: geometric capacity at required margins, optimization landscapes, and evaluation tractability. The exponential dimension of Hilbert space serves the first of these – $N$ qubits hold $2^{N}$ mutually orthogonal states with which to separate classes – but the same size also concentrates kernels, flattens the gradients of global fidelity objectives, and raises the shot cost of resolving concentrated fidelities. Quantum spaces are nonetheless typically judged by dimension alone. This work evaluates them as metric-learning targets, weighing the separability their size provides against these costs.

Evaluating quantum state spaces is obstructed in prior models because their circuits act simultaneously as kernel and function approximator, preventing failures from being attributed to the geometry or the approximator. Moreover, terminal trainable unitaries cancel identically from fidelity, and single-axis encodings confine qubits to 1D great circles. We decouple these roles completely: a classical network $f_{w}$ predicts every gate parameter of a fixed quantum ansatz $U$ with zero trainable quantum gates, $\lvert\psi_{w}(x)\rangle=U(f_{w}(x))\lvert 0\rangle^{\otimes N}$. All representation learning is classical; the circuit contributes solely its reachable state manifold and its fidelity metric, isolating the target quantum geometry for rigorous analysis.

We prove that the separability motivating QMeL is already present on the unentangled manifold $\mathbb{M}_{\mathrm{unentangled}}=(\mathbb{CP}^{1})^{N}$, which saturates the $2^{N}$ maximal orthogonal capacity needed for classification, while entanglement adds states only at intermediate separations that are costly to resolve under finite sampling. The same size carries costs. Global fidelity causes exponential gradient collapse across both unentangled ($\Theta(c^{-N})$) and entangled ($\Theta(4^{-N})$) manifolds; as fidelities concentrate, the fidelity signal-to-noise ratio (FSNR) at a fixed shot budget also decays exponentially in $N$; and because $\mathbb{M}_{\mathrm{unentangled}}$ already holds $2^{N}$ orthogonal states, expressive, unconstrained encoders under contrastive losses reach trivial minima by geometric shattering even without entanglement.

We introduce Quantum Projective Metric Learning (QPMeL), an exact $O(N)$ closed-form kernel on $\mathbb{M}_{\mathrm{unentangled}}$ that circumvents circuit simulation. QPMeL instantiates the classically parameterized setup with the smallest ansatz that reaches all of $\mathbb{M}_{\mathrm{unentangled}}$: $U=\bigotimes_{q} R_{Z}(\gamma_{q})R_{Y}(2\theta_{q})$. A domain encoder maps inputs to a latent vector, and an angle head projects it through bounded activations to $N$ coordinate pairs $(\theta_{q},\gamma_{q})$. Training never runs the circuit. Instead, the Projective Metric Function (PMeF) maps each qubit’s angles to a real unit vector in $\mathbb{R}^{3}$ and computes fidelity via a Complex Kernel Function (CKF), yielding the exact product kernel $k_{w}(x,x’)=\prod_{q} F_{q}$ in $O(N)$ real arithmetic. To resolve the global fidelity barren plateau, QPMeL trains via a curriculum annealing from a $1$-local observable regime (restoring $\Theta(1/N^{2})$ polynomial gradient variance) to the global product kernel. The key contributions are:

1. Formalization of geometric capacity bounds and trainability trade-offs: $\mathbb{M}_{\mathrm{unentangled}}$ saturates the $2^{N}$ orthogonal capacity needed for classification, while global fidelity induces exponential gradient collapse on both unentangled and entangled manifolds.
2. Analysis of metric losses under finite-shot fidelity estimation, including exponential FSNR decay and geometric shattering under contrastive objectives.
3. QPMeL as a geometric base case: an exact $O(N)$ closed-form kernel on $\mathbb{M}_{\mathrm{unentangled}}$ with a local-to-global training curriculum that restores polynomial gradient variance without circuit simulation or shot noise.

Empirically, QPMeL as a complete design (encoder, angle head, kernel, and curriculum) exceeds eight and matches two of 11 published quantum few-shot baselines, with fewer or equal qubits in ten comparisons. It reaches nearly 99–100% on binary and 3-way MNIST/Fashion-MNIST tasks where prior methods degrade, and scales to 10-way 1-shot classification at $96.7\%$ (MNIST, 5 qubits) and $85.0\%$ (Fashion-MNIST, 6 qubits). On Omniglot, it matches or trails a 64-qubit re-uploading baseline while using only 11 qubits.

On THINGS-EEG2 retrieval, where published EEG encoders are held fixed and only the quantum head changes, QPMeL outperforms explicit variational and re-uploading heads on 200-way zero-shot retrieval ($19.9\%$ held-out on the convolutional encoder against at most $12.3\%$ for re-uploading). Circuit depth primarily induces kernel concentration rather than useful capacity, and expressive unconstrained encoders shatter under the global product kernel unless regularized. Execution on the 156-qubit IBM Marrakesh processor illustrates the estimation cost that the closed form on $\mathbb{M}_{\mathrm{unentangled}}$ avoids.

Ultimately, the failures observed in prior designs trace to kernel concentration, global fidelity objectives, and encoder regularization rather than to Hilbert-space dimension. Metric learning in quantum feature spaces depends on keeping the separability that dimension provides while containing its costs – by restricting the reachable manifold to one that retains all $2^{N}$ orthogonal states, localizing the objective, and regularizing the encoder – rather than on reaching more of the state space.