Quantum machine learning increasingly relies on pure-state representations, motivating generative models that sample directly in quantum representation space rather than perturbing classical inputs and re-encoding. We introduce Stochastic Schrödinger Diffusion Models (SSDMs), a score-based generative framework that defines diffusion, scores, and reverse-time sampling intrinsically on the complex projective manifold $\mathbb{CP}^{d-1}$ under the Fubini--Study metric. SSDMs combine a Riemannian Ornstein--Uhlenbeck forward diffusion with a stochastic Schrödinger realization, and learn reverse-time dynamics driven by the Riemannian score. Our central technical contribution is a local-time learning objective that exploits the local Euclidean OU limit of intrinsic manifold diffusions in Fubini-Study normal coordinates to obtain an analytic teacher score, bypassing the intractable transition densities that limit existing Riemannian score-based models. Across synthetic, physics-inspired (TFIM, XXZ), and quantum feature-state benchmarks up to $14$ qubits, SSDMs match target pure-state ensembles by orders of magnitude on MMD and observable statistics over both ambient Euclidean and matched Riemannian score-based baselines, and improve representation-level diagnostics for downstream quantum kernel methods.

翻译：量子机器学习日益依赖纯态表示，这促使生成模型直接在量子表示空间中进行采样，而非扰动经典输入并重新编码。我们提出随机薛定谔扩散模型（SSDMs），这是一种基于分数的生成框架，在 Fubini-Study 度量下，将扩散、分数和反向时间采样内在定义于复投影流形 $\mathbb{CP}^{d-1}$ 上。SSDMs 结合了黎曼 Ornstein-Uhlenbeck 正向扩散与随机薛定谔实现，并学习由黎曼分数驱动的反向时间动力学。我们核心的技术贡献是一个局部时间学习目标，该目标利用 Fubini-Study 正则坐标下内在流形扩散的局部欧几里得 OU 极限来获得解析教师分数，从而绕过了限制现有黎曼基于分数的模型难处理的转移密度。在合成数据、物理启发模型（TFIM、XXZ）以及高达 $14$ 量子比特的量子特征态基准测试中，SSDMs 在 MMD 和可观测统计量上，无论是相对于环境欧几里得基线还是匹配的黎曼基于分数的基线，都能以数量级优势匹配目标纯态系综，并为下游量子核方法改进表示层面的诊断。