Quantum computers and simulators may offer significant advantages over their classical counterparts, providing insights into quantum many-body systems and possibly improving performance for solving exponentially hard problems, such as optimization and satisfiability. Here, we report the implementation of a low-depth Quantum Approximate Optimization Algorithm (QAOA) using an analog quantum simulator. We estimate the ground-state energy of the Transverse Field Ising Model with long-range interactions with tunable range, and we optimize the corresponding combinatorial classical problem by sampling the QAOA output with high-fidelity, single-shot, individual qubit measurements. We execute the algorithm with both an exhaustive search and closed-loop optimization of the variational parameters, approximating the ground-state energy with up to 40 trapped-ion qubits. We benchmark the experiment with bootstrapping heuristic methods scaling polynomially with the system size. We observe, in agreement with numerics, that the QAOA performance does not degrade significantly as we scale up the system size and that the runtime is approximately independent from the number of qubits. We finally give a comprehensive analysis of the errors occurring in our system, a crucial step in the path forward toward the application of the QAOA to more general problem instances.

We study the quasiparticle excitation and quench dynamics of the one-dimensional transverse-field Ising model with power-law (1/r α ) interactions. We find that long-range interactions give rise to a confining potential, which couples pairs of domain walls (kinks) into bound quasiparticles, analogous to mesonic bound states in high-energy physics. We show that these quasiparticles have signatures in the dynamics of order parameters following a global quench and the Fourier spectrum of these order parameters can be expolited as a direct probe of the masses of the confined quasiparticles. We introduce a two-kink model to qualitatively explain the phenomenon of long-range-interaction-induced confinement, and to quantitatively predict the masses of the bound quasiparticles. Furthermore, we illustrate that these quasiparticle states can lead to slow thermalization of onepoint observables for certain initial states. Our work is readily applicable to current trapped-ion experiments.Long-range interacting quantum systems occur naturally in numerous quantum simulators [1][2][3][4][5][6][7][8][9][10]. A paradigmatic model considers interactions decaying with distance r as a power law 1/r α . This describes the interaction term in trappedion spin systems [3,[11][12][13][14][15], polar molecules [16][17][18][19], magnetic atoms [5,20,21], and Rydberg atoms [1,2,22,23]. One remarkable consequence of long-range interactions is the breakdown of locality, where quantum information, bounded by linear 'light cones' in short-range interacting systems [24], can propagate super-ballistically or even instantaneously [25][26][27][28][29][30]. Lieb-Robinson linear light cones have been generalized to logarithmic and polynomial light cones for long-range interacting systems [25,26,31], and non-local propagation of quantum correlations in one-dimensional (1D) spin chains has been observed in trapped-ion experiments [12,13]. Moreover, 1D long-range interacting quantum spin chains can host novel physics that is absent in their short-range counterparts, such as continuous symmetry breaking [32,33].More recently, it has been shown that confinement-which has origins in high-energy physics-has dramatic signatures in the quantum quench dynamics of short-range interacting spin chains [34]. Owing to confinement, quarks cannot be directly observed in nature as they form mesons and baryons due to strong interactions [35,36]. An archetypal model with analogous confinement effects in quantum many-body systems is the 1D short-range interacting Ising model with both transverse and longitudinal fields [37][38][39][40][41][42]. For a vanishing longitudinal field, domain-wall quasiparticles propagate freely and map out light-cone spreading of quantum information [41][42][43][44]. As first proposed by , a nonzero longitudinal field induces an attractive linear potential between two domain walls and confines them into mesonic bound quasiparticles. Recently, Kormos et al. investigated the effect of these bound states on quench dynamics and showed that the non...