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.

翻译：距离度量是机器学习的核心，但由于量子测量的基本限制，量子态系综之间的距离仍未被充分理解。我们引入了一类积分概率度量层次结构，称为MMD-$k$，它将最大均值差异推广到量子系综，并展现出随着矩阶数$k$增加，判别能力与统计效率之间的严格权衡。对于规模为$N$的纯态系综，使用基于SWAP测试的实验可行估计器估计MMD-$k$需要$Θ(N^{2-2/k})$样本（$k$为常数），而在$k = N$时实现完全判别能力需要$Θ(N^3)$样本。相比之下，量子Wasserstein距离仅需$Θ(N^2 \log N)$样本即可达到完全判别能力。这些结果为量子机器学习中损失函数的设计提供了原理性指导，我们通过在训练量子去噪扩散概率模型中的应用予以说明。