Title: ShadowFM: Geometric Approaches for Learning Quantum Many-Body States with Flow Matching on Classical Shadows
Primary area: generative models
Keywords: Discrete Flow Matching, Riemannian Flow Matching, Generative Modeling of Quantum Many-Body Physics, Classical Shadow Tomography

Abstract: We introduce \textbf{ShadowFM}, a generative framework for learning ground states of quantum many-body systems by learning distributions of classical shadows of a target quantum state with geometric flow matchings. 
Specifically, we conditionally generate shadows of the given Hamiltonian's ground state by incorporating geometric modifications into the existing flow matching frameworks.
We interpret the embedding space of discrete shadow measurements as a spherical manifold, inspired by the Bloch sphere representation, and motivate geometric approaches for modeling flow on the curved manifold.
This approach enables us to capture the intrinsic symmetries of quantum measurements and allows more accurate sampling of Hamiltonian-conditioned shadows, which is a direction that was not explored in the previous works assuming Euclidean geometry.
In this work, we propose two methods, Riemannian-based and probability path-based method, to learn a more accurate transport dynamics on an anisotropic probability simplex and a Riemannian manifold aligned with the Bloch sphere geometry, respectively.
We demonstrate that this geometric alignment leads to more faithful sampling of shadow sequences which leads to more accurate prediction of an unseen quantum state's observables, such as correlation functions and entanglement entropy.
We believe this work provides a novel perspective on leveraging geometric generative modeling for learning quantum states.

TL;DR: We propose ShadowFM, a geometric generative framework that learns quantum many-body ground states by modeling shadow distributions on curved manifolds aligned with Bloch sphere geometry.