O-Sensing: Operator Sensing for Interaction Geometry and Symmetries

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 Erdős--Rényi 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.

Figures (6)

• Figure 1: Learning spatial geometry and symmetries from low-energy states of a quantum system. (a) The original spatial geometry, depicted as a random, connected Erdős--Rényi graph. Circles denote physical sites, while lines represent interactions. (b) Solving the eigenvalue problem yields low-energy states, revealing correlations between sites. Directly reading the original geometry from these correlations is infeasible since two spatially separated sites may be correlated. (c) The parent operator subspace derives a linear space of operators satisfying the eigenvalue equation. Randomly chosen basis (top half) obscure physical information, making interpretation difficult. Optimizing the basis (bottom half) identifies the sparsest basis, which optimally describes the system with clear physical meaning. This optimized basis includes the parent Hamiltonian, encoding the spatial geometry, and conserved quantities representing system symmetries. (d) Spatial geometry learned via sparse basis optimization of the parent operator subspace. The method reconstructs the original Hamiltonian with high precision, successfully retrieving the geometry.
• Figure 2: Parent operator subspace dimension $D_{\mathcal{K}}$ for an ensemble of 100 connected Erdős--Rényi graphs ($N_v=14, N_e=17$) with uniform antiferromagnetic coupling. Individual samples are ordered along the horizontal axis by their respective kernel dimensions. The results exhibit a hierarchy of discrete plateaus, starting from the universal baseline of $D_{\mathcal{K}} = N_v^2 + 3 = 199$.
• Figure 3: Phase diagram of Hamiltonian learnability. The success rate of reconstruction is plotted as a function of the number of edges $N_e$. We compare two interaction classes: the antiferromagnetic (AFM) model (blue circles), where all non-zero coupling constants are fixed at $J_{ij}=1$; and the random-sign model (orange triangles), where interactions $J_{ij}$ are chosen from $\{+1, -1\}$ with equal probability. The dip in the intermediate region indicates the regime where geometric duality confuses the sparsity criterion.
• Figure 4: Geometric mechanism of sparse operator discovery.(a) Ambient space view: The valid solution manifold $\mathcal{M}$ (blue ring) is determined by the intersection of the physical kernel subspace $\mathcal{K}$ (semi-transparent blue plane) and the $\ell_2$-normalization sphere (light gray). The teal arrows define the intrinsic projection coordinates within $\mathcal{K}$. (b) Subspace projection: Sparse optimization visualized within the $\mathcal{K}$-plane. The $\ell_1$-norm level sets form anisotropic cross-polytopes (red diamonds). Minimizing the $\ell_1$-norm is geometrically equivalent to identifying the minimal $\ell_1$-polytope that makes contact with $\mathcal{M}$. Because of the sharp vertices of the $\ell_1$-ball, this contact strictly occurs along a coordinate axis, naturally collapsing redundant degrees of freedom to isolate the optimal sparse solution $\mathbf{k}^*$ (black star).
• Figure 5: Geometry of the selected instance with $D_{\mathcal{K}}=205$.