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

2026-06-19

AI Summary

This work addresses the geometric mismatch between complex projective space and Euclidean manifolds that hinders quantum pure state generative modeling. The authors propose an Intrinsic Flow Matching (IFM) framework that constructs horizontally parameterized conditional paths via Pancharatnam phase alignment, enabling direct learning of tangent-space velocity fields on the complex projective manifold. This approach achieves deterministic probability transport without relying on local score estimation or stochastic reverse sampling, as required by conventional methods. IFM provides theoretical guarantees for endpoint accuracy and training stability. Empirical results demonstrate that IFM significantly outperforms Euclidean flow matching across multiple tasks—including multi-qubit states, spin coherent states, and amplitude-encoded MNIST—particularly excelling in high-dimensional settings and scenarios sensitive to quantum coherence.

Abstract

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 $\mathbb{CP}^{d-1}$ 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.

Problem

• Quantum pure states
• Complex projective space
• Generative modeling
• Geometric mismatch
• Manifold learning

Innovation

• Intrinsic Flow Matching
• Quantum Pure-State Manifolds
• Phase-Aligned Transport
• Complex Projective Space
• Horizontal Parameterization