Stochastic Schrödinger Diffusion Models for Pure-State Ensemble Generation

• URL: http://arxiv.org/abs/2605.03573v2
• Date: Mon, 11 May 2026 14:06:21 GMT
• Title: Stochastic Schrödinger Diffusion Models for Pure-State Ensemble Generation
• Abstract summary: In quantum machine learning (QML), classical data are often encoded as quantum pure states and processed directly as quantum representations.<n>We propose an intrinsic score-based generative framework on $mathbbCPd-1$ endowed with the Fubini-Study (FS) metric.<n>We show that SSDMs faithfully capture target pure-state ensemble statistics, including observable moments, overlap, and entanglement measures.
• Score: 29.736351530340674
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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