Hierarchy of discriminative power and complexity in learning quantum ensembles

TL;DR

This paper introduces a hierarchy of quantum distance metrics called MMD-$k$, revealing a trade-off between their discriminative power and statistical efficiency, and compares them to quantum Wasserstein distance for quantum ensemble analysis.

Contribution

The paper develops a new hierarchy of quantum distance metrics, MMD-$k$, and analyzes their sample complexity and discriminative power, guiding quantum machine learning loss function design.

Findings

MMD-$k$ exhibits a trade-off between discriminative power and sample efficiency.

Estimating MMD-$k$ requires $ heta(N^{2-2/k})$ samples for pure-state ensembles.

Quantum Wasserstein distance achieves full discriminative power with $ heta(N^2 ext{log} N)$ samples.

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 $\Theta(N^{2-2/k})$ samples for constant $k$, and $\Theta(N^3)$ samples to achieve full discriminative power at $k = N$. In contrast, the quantum Wasserstein distance attains full discriminative power with $\Theta(N^2 \log N)$ 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.