量子集合学习中的判别能力与复杂性层次
Hierarchy of discriminative power and complexity in learning quantum ensembles
AI总结：
本文提出MMD-k度量，用于量子集合学习，揭示判别能力与统计效率的权衡，并展示其在量子机器学习中的应用。
摘要：
距离度量是机器学习的核心，然而量子态集合之间的距离仍由于基本的量子测量限制而理解不足。我们引入了一种积分概率度量的层次结构，称为MMD-k，它将最大均值差异扩展到量子集合，并展示出随着时刻阶数k增加，判别能力与统计效率之间存在严格的权衡。对于大小为N的纯态集合，使用实验可行的SWAP测试基估计器估计MMD-k需要Θ(N^{2-2/k})个样本（对于常数k），而要达到完全判别能力需Θ(N^3)个样本（在k=N时）。相比之下，量子瓦瑟斯坦距离在Θ(N² log N)个样本下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指导，我们通过训练量子去噪扩散概率模型来说明这一点。
英文摘要：
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 $Θ(N^{1-1/k})$ 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 $Ω(N)$ 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.
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