# Intrinsic Flow Matching on Quantum Pure-State Manifolds with Phase-Aligned Transport

arXiv: https://arxiv.org/abs/2606.21256 · HTML: https://mlsanity.com/paper/2606.21256/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 31 of its 65 references that are in the map.
- First posted on arXiv 19 Jun 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 30 of 31)

- [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)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Discrete Flow Matching](https://mlsanity.com/paper/2407.15595/) (2024)
- [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [Equivariant flow matching](https://mlsanity.com/paper/2306.15030/) (2023)
- [Quantum generative adversarial networks](https://mlsanity.com/paper/1804.08641/) (2018)
- [Riemannian Diffusion Models](https://mlsanity.com/paper/2208.07949/) (2022)
- [FlowMM: Generating Materials with Riemannian Flow Matching](https://mlsanity.com/paper/2406.04713/) (2024)
- [Improving the Training of Rectified Flows](https://mlsanity.com/paper/2405.20320/) (2024)
- [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)
- [Functional Flow Matching](https://mlsanity.com/paper/2305.17209/) (2023)
- [Generative quantum machine learning via denoising diffusion probabilistic models](https://mlsanity.com/paper/2310.05866/) (2023)
- [Dynamic Conditional Optimal Transport through Simulation-Free Flows](https://mlsanity.com/paper/2404.04240/) (2024)
- [Fast and Smooth Interpolation on Wasserstein Space](https://mlsanity.com/paper/2010.12101/) (2020)
- [Scaling Riemannian Diffusion Models](https://mlsanity.com/paper/2310.20030/) (2023)
- [Quantum Denoising Diffusion Models](https://mlsanity.com/paper/2401.07049/) (2024)
- [Continuous normalizing flows on manifolds](https://mlsanity.com/paper/2104.14959/) (2021)
- [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)
- [Quantum Flow Matching](https://mlsanity.com/paper/2508.12413/) (2025)
- [Distributed Wasserstein Barycenters via Displacement Interpolation](https://mlsanity.com/paper/2012.08610/) (2020)
- [A Benamou-Brenier Proximal Splitting Method for Constrained Unbalanced Optimal Transport](https://mlsanity.com/paper/2512.09250/) (2025)

## Abstract (arXiv)

Quantum pure-state ensembles live on complex projective space, making flat Euclidean generative modeling geometrically mismatched. We introduce Intrinsic Flow Matching (IFM), a deterministic transport framework on mathbbCPd-1 that learns tangent velocity fields using Pancharatnam phase-aligned conditional paths. IFM replaces local score teachers and reverse-time stochastic sampling with manifold probability flow, while horizontal parameterization removes redundant ambient directions. We show that the IFM objective recovers the induced marginal transport field, represents deterministic projective ensemble flows, and yields endpoint and stability guarantees. Empirically, IFM often improves over ambient Euclidean flow matching across higher-qubit, multimodal, spin-coherent, physics-inspired, and amplitude-encoded MNIST image-vector benchmarks, with strongest gains on high-dimensional and coherence-sensitive tasks but not uniformly across every metric.