Stochastic Schr\"odinger Diffusion Models for Pure-State Ensemble Generation

In quantum machine learning, classical data is often encoded as quantum pure states, but generative modeling of those states has been limited by the curved geometry of the underlying state space. A team of researchers (Jian Xu, Wei Chen, Chao Li, Jingyuan Zheng, Delu Zeng, John Paisley, and Qibin Zhao) has introduced Stochastic Schrödinger Diffusion Models (SSDMs) to solve this. The model defines a forward diffusion process using a stochastic Schrödinger equation on the complex projective manifold with the Fubini-Study metric, then derives reverse-time dynamics via the Riemannian score. To train without known transition densities, they propose a local-time objective that approximates the manifold locally with an Euclidean Ornstein-Uhlenbeck process, mapping back a learned teacher score.

The results show SSDMs can generate pure-state ensembles that faithfully reproduce key statistics, including observable moments, overlap-kernel maximum mean discrepancy, and entanglement measures. Moreover, using SSDM-generated quantum representations for data augmentation improves generalization in downstream quantum classification and regression tasks. This work bridges generative diffusion models with quantum state geometry, offering a scalable way to create synthetic quantum data for training quantum machine learning models without needing hardware or perturbed classical inputs.