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arXiv:2606.22713

An interpretable closed form for entanglement entropy from bitstrings, guided by a graph neural network

Anas Saleh

quant-phphysics.comp-phSubmitted Jun 21, 2026• Announced Jun 23, 2026• 0 citations
v1 · Jun 21, 2026v2 · Aug 28, 2026
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Abstract

The empirical bitstring distribution is the most accessible observable on Rydberg-atom arrays, but the bipartite von~Neumann entropy it constrains is far costlier to obtain. We present a six-term linear closed form for the entropy, built on bitstring-derivable physics scalars, and characterize its accuracy, portability, scaling behaviour, and calibration cost. The feature set is selected with guidance from a trained graph neural network: probing the network localizes its entropy prediction to the two-point correlators on the bipartition boundary, and an exhaustive ground-truth search restricted to those boundary correlators isolates the form. It reaches $0.024$~nats mean absolute error in distribution: $6.4$ times the network's error, but in a form a human can read and apply without retraining. Fit once and applied unchanged, it has lower error than the base network on five of six out-of-distribution pools and ties the sixth. An independent density-matrix renormalization-group study to one hundred atoms -- five times the reach of exact diagonalization -- settles the size-extrapolation question: coefficients frozen at small size fail at scale, but the failure is structured. Refit per size the form holds to $25$--$50$~mnat (cross-validated); two of its six slopes follow clean inverse-size laws, one a downward curving growth, and the others are trendless; the fitted laws deploy the form label-free at roughly $40$--$80$~mnat. The result fixes a label-budget rule: at large sizes, a few dozen labels recalibrate the closed form to match a fine-tuned in-distribution ensemble on the same features, while nonlinear ML models pull ahead only given large labelled datasets.

Comments
22 pages, 7 figures

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Quantum simulators have the potential to solve quantum many-body problems that are beyond the reach of classical computers, especially when they feature long-range entanglement. To fulfill their prospects, quantum simulators must be fully controllable, allowing for precise tuning of the microscopic physical parameters that define their implementation. We consider Rydberg-atom arrays, a promising platform for quantum simulations. Experimental control of such arrays is limited by the imprecision on the optical-tweezer positions when assembling the array, hence introducing uncertainties in the simulated Hamiltonian. In this work, we introduce a scalable approach to Hamiltonian learning using graph neural networks (GNNs). We employ the density-matrix renormalization group to generate ground-state snapshots of the transverse-field Ising model realized by the array, for many realizations of the Hamiltonian parameters. Correlation functions reconstructed from these snapshots serve as input data to carry out the training. We demonstrate that our GNN model has a remarkable capacity to extrapolate beyond its training domain, regarding both the size and the shape of the system, yielding an accurate determination of the Hamiltonian parameters with a minimal set of measurements. We prove a theorem establishing a bijective correspondence between the correlation functions and the interaction parameters in the Hamiltonian, which provides a theoretical foundation for our learning algorithm. Our work could open the road to feedback control of the positions of the optical tweezers, hence providing a decisive improvement of analog quantum simulators.