Stochastic Schrödinger Diffusion Models for Pure-State Ensemble Generation

2026-05-05

AI Summary

This work addresses the challenges posed by the non-Euclidean geometry of complex projective space and the intractability of transition densities in generating ensembles of quantum pure states. To this end, we propose the Stochastic Schrödinger Diffusion Model (SSDM), which constructs an intrinsic diffusion process under the Fubini–Study metric. Perturbations are implemented via a forward stochastic Schrödinger equation, while generation is driven by a Riemannian score function in the reverse process. Our approach intrinsically extends score-based diffusion models to the manifold of quantum pure states and introduces a training objective based on a local Euclidean Ornstein–Uhlenbeck approximation, enabling efficient training without requiring explicit transition densities. Experiments demonstrate that SSDM accurately reproduces key statistical properties of target ensembles—including observable moments, overlap kernel maximum mean discrepancy (MMD), and entanglement measures—and significantly enhances generalization performance in downstream quantum machine learning tasks.

Abstract

In quantum machine learning (QML), classical data are often encoded as quantum pure states and processed directly as quantum representations, motivating representation-level generative modeling that samples new quantum states from an underlying pure-state ensemble rather than re-preparing them from perturbed classical inputs. However, extending \emph{score-based} diffusion models with well-defined reverse-time samplers to quantum pure-state ensembles remains challenging, due to the non-Euclidean geometry of the complex projective space $\mathbb{CP}^{d-1}$ and the intractability of transition densities. We propose \emph{Stochastic Schrödinger Diffusion Models} (SSDMs), an intrinsic score-based generative framework on $\mathbb{CP}^{d-1}$ endowed with the Fubini--Study (FS) metric. SSDMs formulate a forward Riemannian diffusion with a stochastic Schrödinger equation (SSE) realization, and derive reverse-time dynamics driven by the Riemannian score $\nabla_{\mathrm{FS}} \log p_t$. To enable training without analytic transition densities, we introduce a local-time objective based on a local Euclidean Ornstein--Uhlenbeck approximation in FS normal coordinates, yielding an analytic teacher score mapped back to the manifold. Experiments show that SSDMs faithfully capture target pure-state ensemble statistics, including observable moments, overlap-kernel MMD, and entanglement measures, and that SSDM-generated quantum representations improve downstream QML generalization via representation-level data augmentation.

Problem

• quantum pure-state ensemble
• score-based diffusion models
• complex projective space
• Riemannian geometry
• transition densities

Innovation

• Stochastic Schrödinger Diffusion Models
• score-based generative modeling
• complex projective space
• Fubini–Study metric
• quantum pure-state ensemble