# Local-Time Riemannian Score Matching on the Quantum Pure-State Manifold

arXiv: https://arxiv.org/abs/2605.03573 · HTML: https://mlsanity.com/paper/2605.03573/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 57 references that are in the map.
- First posted on arXiv 5 May 2026.

## Builds on (most cited first, 26 of 26)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [WaveGrad: Estimating Gradients for Waveform Generation](https://mlsanity.com/paper/2009.00713/) (2020)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [Power of data in quantum machine learning](https://mlsanity.com/paper/2011.01938/) (2020)
- [Diffusion probabilistic modeling of protein backbones in 3D for the motif-scaffolding problem](https://mlsanity.com/paper/2206.04119/) (2022)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Quantum generative adversarial networks](https://mlsanity.com/paper/1804.08641/) (2018)
- [Riemannian Diffusion Models](https://mlsanity.com/paper/2208.07949/) (2022)
- [Protein structure generation via folding diffusion](https://mlsanity.com/paper/2209.15611/) (2022)
- [Differentiable Learning of Quantum Circuit Born Machine](https://mlsanity.com/paper/1804.04168/) (2018)
- [The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine](https://mlsanity.com/paper/1904.02214/) (2019)
- [Generative quantum machine learning via denoising diffusion probabilistic models](https://mlsanity.com/paper/2310.05866/) (2023)
- [Quantum-Noise-Driven Generative Diffusion Models](https://mlsanity.com/paper/2308.12013/) (2023)
- [Scaling Riemannian Diffusion Models](https://mlsanity.com/paper/2310.20030/) (2023)
- [Quantum Denoising Diffusion Models](https://mlsanity.com/paper/2401.07049/) (2024)
- [Dequantified Diffusion-Schr"odinger Bridge for Density Ratio Estimation](https://mlsanity.com/paper/2505.05034/) (2025)
- [EVODiff: Entropy-aware Variance Optimized Diffusion Inference](https://mlsanity.com/paper/2509.26096/) (2025)
- [Quantum Generative Diffusion Model: A Fully Quantum-Mechanical Model for Generating Quantum State Ensemble](https://mlsanity.com/paper/2401.07039/) (2024)
- [Mixed-State Quantum Denoising Diffusion Probabilistic Model](https://mlsanity.com/paper/2411.17608/) (2024)
- [Unraveling Quantum Environments: Transformer-Assisted Learning in Lindblad Dynamics](https://mlsanity.com/paper/2505.06928/) (2025)
- [Quantum Diffusion Model for Quark and Gluon Jet Generation](https://mlsanity.com/paper/2412.21082/) (2024)

## Built on by (most cited first, 1 of 1)

- [Intrinsic Flow Matching on Quantum Pure-State Manifolds with Phase-Aligned Transport](https://mlsanity.com/paper/2606.21256/) (2026)

## Abstract (arXiv)

Score-based diffusion can be defined intrinsically on the manifold of quantum pure states, mathbbCPd-1 with the Fubini--Study metric, but no closed-form transition density is available, so the score must be supervised by a local-time teacher taken from the Euclidean limit of the diffusion in normal coordinates. This paper is about what makes that teacher work, and where it stops working. Three training choices turn out not to be incidental: the increment must be divided by the diffusion clock rather than by the elapsed time, since the published expression assumes unit diffusion and a non-unit schedule introduces a time-change mismatch varying by a factor of 400 across the horizon; the logarithm and exponential maps should be the closed-form Fubini--Study ones, which is the largest single effect we measure; and the global phase must be randomised, because horizontal projection alone does not make a score network descend to the quotient. With these choices the model improves on the published Riemannian local-time baseline in every cell of an eight-benchmark, four-metric comparison over ten seeds, significantly on five of eight after Holm correction, and beats an ambient Euclidean baseline by an order of magnitude everywhere. We then bound what the approximation costs by replacing it with the exact heat kernel of mathbbCPd-1, computable up to complex dimension seven, where the local-time teacher loses a factor of 1.2 to 2.7.