Hierarchy of discriminative power and complexity in learning quantum ensembles

2026-01-29

AI Summary

This work addresses the lack of a distance measure between quantum state ensembles that simultaneously offers strong discriminative power and statistical efficiency. The authors propose the MMD-k hierarchy, extending Maximum Mean Discrepancy to quantum ensembles, and establish—for the first time—a rigorous trade-off between discriminative capability and sample complexity as a function of the moment order \(k\). Theoretical analysis shows that for \(N\) pure states, MMD-\(k\) requires \(\Theta(N^{2-2/k})\) samples for fixed \(k\), while achieving full discriminative power (at \(k = N\)) demands \(\Theta(N^3)\) samples. In contrast, the quantum Wasserstein distance attains full discriminative power with only \(\Theta(N^2 \log N)\) samples. An MMD-k estimator based on the SWAP test is also presented, offering practical guidance for designing loss functions in quantum machine learning.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Intelligent Robots: State Estimation

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs

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.

Problem

• quantum ensembles
• distance metrics
• discriminative power
• statistical efficiency
• quantum machine learning

Innovation

• quantum ensembles
• maximum mean discrepancy
• MMD-k
• sample complexity
• quantum Wasserstein distance