O-Sensing: Operator Sensing for Interaction Geometry and Symmetries

TL;DR

O-Sensing is a novel method that infers the interaction geometry, Hamiltonian, and symmetries of quantum many-body systems from low-lying eigenstates using sparsity and spectral entropy optimization.

Contribution

It introduces O-Sensing, a protocol that extracts Hamiltonians and symmetries directly from eigenstates by leveraging sparsity and entropy maximization, revealing interaction structures without prior knowledge.

Findings

Successfully reconstructs interaction geometry in Heisenberg models on Erdős–Rényi graphs.

Uncovers additional long-range conserved operators beyond local interactions.

Identifies a phase diagram with a 'confusion' regime where dual descriptions emerge.

Abstract

We ask whether the Hamiltonian, interaction geometry, and symmetries of a quantum many-body system can be inferred from a few low-lying eigenstates without knowing which sites interact with each other. Directly solving the eigenvalue equations imposes constraints that yield a highly degenerate subspace of candidate operators, where the local Hamiltonian is hidden among an extensive family of conserved quantities, obscuring the interaction geometry. Here we introduce O-Sensing, a protocol designed to extract the Hamiltonian and symmetries directly from these states. Specifically, O-Sensing employs parsimony-driven optimization to extract a maximally sparse operator basis from the degenerate subspace. The Hamiltonian is then selected from this basis by maximizing spectral entropy (effectively minimizing degeneracy) within the sampled subspace. We validate O-Sensing on Heisenberg models on connected Erd\H{o}s--R\'enyi graphs, where it reconstructs the interaction geometry and uncovers additional long-range conserved operators. We establish a learnability phase diagram across graph densities, featuring a pronounced ``confusion'' regime where parsimony favors a dual description on the complement graph. These results show that sparsity optimization can reconstruct interaction geometry as an emergent output, enabling simultaneous recovery of the Hamiltonian and its symmetries from low-energy eigenstates.