Solving excited states for long-range interacting trapped ions with neural networks

The computation of excited states in strongly interacting quantum many-body systems is of fundamental importance. Yet, it is notoriously challenging due to the exponential scaling of the Hilbert space dimension with the system size. Here, building on the natural-excited-states (NES) framework, we present a scalable implementation for large spin systems with long-range interactions, enabling the simultaneous computation of multiple low-lying excited states in an accurate and efficient fashion. We demonstrate our algorithm through concrete examples including the Haldane-Shastry model with all-to-all interactions, illustrating its applicability for efficiently computing multiple excited states and their associated observable expectation values. In addition, we apply this framework to two classes of long-range interacting trapped-ion systems in a two-dimensional Wigner crystal. For non-decaying all-to-all interactions with alternating signs, our computed low-lying excited states exhibit spatial correlation patterns similar to those of the ground states, providing an eigenstate-resolved perspective on recent experimental observations that the quasi-adiabatically prepared state accurately reproduces analytical ground-state correlations. For a system of up to 300 ions with power-law decaying antiferromagnetic interactions, we successfully uncover its gap scaling and correlation features. Our results establish a scalable and efficient algorithm for computing excited states of interacting quantum many-body systems, with potential applications ranging from benchmarking quantum devices to photoisomerization.