Geometric Flow Matching for Learning Quantum Many-Body States with Classical Shadows

Classical shadows provide an efficient representation of quantum many-body states, enabling machine learning approaches to predict properties of unseen states. The distribution of shadows can be viewed as a tomographic representation of a quantum state, from which many observables can be estimated. Learning its Hamiltonian-conditional distribution therefore provides a generative surrogate for estimating multiple properties of quantum states unseen during training. Existing approaches, however, either generate qubits autoregressively or diffuse continuous token embeddings; neither explicitly exploits the discrete and physically meaningful geometry of shadow outcomes. We ask whether incorporating this structure improves conditional shadow generation. We introduce ShadowFlow, a non-autoregressive flow-matching framework that explicitly incorporates the geometry of shadow outcomes through two complementary constructions. We evaluate ShadowFlow on one-dimensional quantum systems up to 100 qubits and two-dimensional systems up to 6x6, comparing against direct regression, autoregressive generation, diffusion, and continuous and discrete flow baselines. Across these benchmarks, ShadowFlow improves observable estimation over existing generative approaches in several settings. We also further demonstrate its extension beyond Pauli measurements to other IC-POVMs.