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arXiv:2312.01655

Metric Learning with Quantum Feature Spaces

Sharma, Vinayak, Sen, Lord, Padhy, Ashish, Behera, Sourav, De Luca, Gennaro, Mukherjee, Shyamapada, Aviral Shrivastava

Published Dec 4, 2023• 0 citations
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Abstract

The $2^N$ dimensions of Hilbert space give quantum metric learning exponentially many mutually orthogonal states with which to separate classes, but the same exponential dimension also degrades its optimization landscape and the tractability of evaluating its metric. By letting a classical network predict all gate parameters of a fixed ansatz, we isolate the reachable state manifold and its fidelity metric. We prove that this separability is already present on the unentangled manifold $\mathbb{M}_{\text{unentangled}} = (\mathbb{CP}^1)^N$, which saturates the $2^N$ maximal orthogonal capacity needed for classification, while entanglement adds states only at intermediate separations, which are costly to resolve under finite sampling. The same size carries costs. Global fidelity causes exponential gradient collapse across both unentangled ($Θ(c^{-N})$) and entangled ($Θ(4^{-N})$) manifolds; as fidelities concentrate, the fidelity signal-to-noise ratio at a fixed shot budget also decays exponentially in $N$; and because $\mathbb{M}_{\text{unentangled}}$ already holds $2^N$ orthogonal states, expressive, unconstrained encoders under contrastive losses reach trivial minima by \emph{geometric shattering} even without entanglement. We introduce Quantum Projective Metric Learning (QPMeL), an exact $O(N)$ closed-form kernel on $\mathbb{M}_{\text{unentangled}}$ that circumvents circuit simulation. QPMeL resolves gradient collapse via a curriculum annealing from a $1$-local observable regime (restoring $Θ(1/N^2)$ polynomial variance) to the global product kernel. Empirically, QPMeL as a complete design exceeds eight and matches two of 11 published quantum few-shot designs.

DOI
10.48550/arxiv.2312.01655

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Quantum state tomography faces exponential scaling with system size, while recent neural network approaches achieve polynomial scaling at the cost of losing the geometric structure of quantum state space. We introduce geometric latent space tomography, combining classical neural encoders with parameterized quantum circuit decoders trained via a metric-preservation loss that enforces proportionality between latent Euclidean distances and quantum Bures geodesics. On two-qubit mixed states with purity 0.85--0.95 representing NISQ-era decoherence, we achieve high-fidelity reconstruction (mean fidelity F=0.942±0.03F = 0.942 \pm 0.03F=0.942±0.03) with an interpretable 20-dimensional latent structure. Critically, latent geodesics exhibit strong linear correlation with Bures distances (Pearson r=0.88r = 0.88r=0.88, R2=0.78R^2 = 0.78R2=0.78), preserving 78\% of quantum metric structure. Geometric analysis reveals intrinsic manifold dimension 6.35 versus 20 ambient dimensions and measurable local curvature (κ=0.011±0.006κ= 0.011 \pm 0.006κ=0.011±0.006), confirming non-trivial Riemannian geometry with O(d2)O(d^2)O(d2) computational advantage over O(4n)O(4^n)O(4n) density matrix operations. Unlike prior neural tomography, our geometry-aware latent space enables direct state discrimination, fidelity estimation from Euclidean distances, and interpretable error manifolds for quantum error mitigation without repeated full tomography, providing critical capabilities for NISQ devices with limited coherence times.
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Quantum kernel methods (QKMs) have emerged as a prominent framework for supervised quantum machine learning. Unlike variational quantum algorithms, which rely on gradient-based optimisation and may suffer from issues such as barren plateaus, non-variational QKMs employ fixed quantum feature maps, with model selection performed classically via convex optimisation and cross-validation. This separation of quantum feature embedding from classical training ensures stable optimisation while leveraging quantum circuits to encode data in high-dimensional Hilbert spaces. In this review, we provide a thorough analysis of non-variational supervised QKMs, covering their foundations in classical kernel theory, constructions of fidelity and projected quantum kernels, and methods for their estimation in practice. We examine frameworks for assessing quantum advantage, including generalisation bounds and necessary conditions for separation from classical models, and analyse key challenges such as exponential concentration, dequantisation via tensor-network methods, and the spectral properties of kernel integral operators. We further discuss structured problem classes that may enable advantage, and synthesise insights from comparative and hardware studies. Overall, this review aims to clarify the regimes in which QKMs may offer genuine advantages, and to delineate the conceptual, methodological, and technical obstacles that must be overcome for practical quantum-enhanced learning.