证明了多体投影系综可逼近任意量子态分布，理论扎实且训练更高效。
Universality of Many-body Projected Ensemble for Learning Quantum Data Distribution
- 用单个多体波函数生成随机量子态，构建投影系综框架。
- 在1-Wasserstein距离下，可逼近任意纯态分布，误差可控。
- 提出分层训练的增量版MPE，提升复杂量子数据学习能力。
通过学习潜在量子分布来生成量子数据，在理论与实践上均面临挑战，却是理解量子系统的关键任务。量子机器学习中的核心问题之一是近似普适性：参数化模型能否逼近任意量子分布。本文针对多体投影系综（MPE）框架，证明了其普适性定理——该框架利用单一多体波函数生成随机态，能够以1-Wasserstein距离误差逼近任意纯态分布。这一结果为量子机器学习提供了严格的通用表达力保证，填补了关键理论空白。为提升实用性，本文提出一种分层训练的增量式MPE方法，改善模型可训练性。数值实验在聚类量子态和量子化学数据集上验证了MPE在学习复杂量子数据分布方面的有效性。
原文摘要 · Abstract (English)
Generating quantum data by learning the underlying quantum distribution poses challenges in both theoretical and practical scenarios, yet it is a critical task for understanding quantum systems. A fundamental question in quantum machine learning (QML) is the universality of approximation: whether a parameterized QML model can approximate any quantum distribution. We address this question by proving a universality theorem for the Many-body Projected Ensemble (MPE) framework, a method for quantum state design that uses a single many-body wave function to prepare random states. This demonstrates that MPE can approximate any distribution of pure states within a 1-Wasserstein distance error. This theorem provides a rigorous guarantee of universal expressivity, addressing key theoretical gaps in QML. For practicality, we propose an Incremental MPE variant with layer-wise training to improve the trainability. Numerical experiments on clustered quantum states and quantum chemistry datasets validate MPE's efficacy in learning complex quantum data distributions.
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