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

TL;DR

This paper introduces a neural network-based algorithm capable of efficiently computing multiple low-lying excited states in long-range interacting quantum systems, with applications demonstrated on trapped-ion models and the Haldane-Shastry model.

Contribution

The authors develop the neural quantum excited-state (NQES) algorithm that computes excited states without explicit orthogonalization, applicable to higher dimensions and large systems.

Findings

Successfully computed excited states for systems with up to 300 ions.

Reproduced experimental correlation patterns in trapped-ion systems.

Uncovered gap scaling and correlation features in long-range interacting systems.

Abstract

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, we introduce a neural network-based algorithm that can simultaneously output multiple low-lying excited states of a quantum many-body spin system in an accurate and efficient fashion. This algorithm, dubbed the neural quantum excited-state (NQES) algorithm, requires no explicit orthogonalization of the states and is generally applicable to higher dimensions. We demonstrate, through concrete examples including the Haldane-Shastry model with all-to-all interactions, that the NQES algorithm is capable of efficiently computing multiple excited states and their related observable expectations. In addition, we apply the NQES algorithm 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 bear spatial correlation patterns similar to those of the ground states, which closely match 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, which holds potential applications ranging from benchmarking quantum devices to photoisomerization.