Hierarchy of discriminative power and complexity in learning quantum ensembles

• URL: http://arxiv.org/abs/2601.22005v1
• Date: Thu, 29 Jan 2026 17:13:48 GMT
• Title: Hierarchy of discriminative power and complexity in learning quantum ensembles
• Authors: Jian Yao, Pengtao Li, Xiaohui Chen, Quntao Zhuang,
• Abstract summary: We introduce a hierarchy of integral probability metrics, termed MMD-$k$, which generalizes the maximum mean discrepancy to quantum ensembles.<n>We show that for pure-state ensembles of size $N$, estimating MMD-$k$ using experimentally feasible SWAP-test-based estimators requires $(N2-2/k)$ samples for constant $k$, and $(N3)$ samples for constant $k$.
• Score: 15.100044984577915
• License: http://creativecommons.org/licenses/by/4.0/
• 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 exhibit 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-$k$ using experimentally feasible SWAP-test-based estimators requires $Θ(N^{2-2/k})$ samples for constant $k$, and $Θ(N^3)$ samples to achieve full discriminative power at $k = N$. In contrast, the quantum Wasserstein distance attains full discriminative power with $Θ(N^2 \log N)$ samples. These results provide principled guidance for the design of loss functions in quantum machine learning, which we illustrate in the training quantum denoising diffusion probabilistic models.

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