Universality of Many-body Projected Ensemble for Learning Quantum Data Distribution
This addresses a fundamental theoretical gap in quantum machine learning for researchers and practitioners, though it is incremental as it builds on existing frameworks.
The paper tackles the problem of whether quantum machine learning models can approximate any quantum distribution by proving a universality theorem for the Many-body Projected Ensemble framework, showing it can approximate any pure state distribution within a 1-Wasserstein distance error. Numerical experiments on clustered quantum states and quantum chemistry datasets validate its efficacy.
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.