Hierarchy of discriminative power and complexity in learning quantum ensembles

Abstract:Distance metrics are central to machine learning, yet distances between ensembles of quantum states remain poorly understood due to fundamental quantum measurement constraints. We introduce a hierarchy of integral probability metrics, termed MMD-$k$, which generalizes the maximum mean discrepancy to quantum ensembles and exhibits a strict trade-off between discriminative power and statistical efficiency as the moment order $k$ increases. For pure-state ensembles of size $N$, estimating MMD-$k$ with arbitrary measurement schemes requires $\Theta(N^{1-1/k})$ samples for constant $k$. At the same time, we prove that any stable distance metric with full discriminative power admits an $O(N\log N)$ upper bound and an $\Omega(N)$ instance-gap lower bound. For quantum Wasserstein distance, with sufficiently large fixed state dimension, we establish a nearly linear lower bound in the ensemble size at constant additive accuracy, together with an $O(N\log N)$ upper bound. These results provide principled guidance for the design of loss functions in quantum machine learning, as we illustrate in training quantum denoising diffusion probabilistic models.

Quantum Physics (quant-ph); Statistics Theory (math.ST); Machine Learning (stat.ML)

arXiv:2601.22005 [quant-ph]

(or arXiv:2601.22005v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2601.22005 Focus to learn more arXiv-issued DOI via DataCite

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