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.

In quantum many-body systems with local interactions, quantum information and entanglement cannot spread outside of a linear light cone, which expands at an emergent velocity analogous to the speed of light. Local operations at sufficiently separated spacetime points approximately commute-given a many-body state jψi, O x ðtÞO y jψi ≈ O y O x ðtÞjψi with arbitrarily small errors-so long as jx − yj ≳ vt, where v is finite. Yet, most nonrelativistic physical systems realized in nature have long-range interactions: Two degrees of freedom separated by a distance r interact with potential energy VðrÞ ∝ 1=r α. In systems with long-range interactions, we rigorously establish a hierarchy of linear light cones: At the same α, some quantum information processing tasks are constrained by a linear light cone, while others are not. In one spatial dimension, this linear light cone exists for every many-body state jψi when α > 3 (Lieb-Robinson light cone); for a typical state jψi chosen uniformly at random from the Hilbert space when α > 5 2 (Frobenius light cone); and for every state of a noninteracting system when α > 2 (free light cone). These bounds apply to time-dependent systems and are optimal up to subalgebraic improvements. Our theorems regarding the Lieb-Robinson and free light cones-and their tightness-also generalize to arbitrary dimensions. We discuss the implications of our bounds on the growth of connected correlators and of topological order, the clustering of correlations in gapped systems, and the digital simulation of systems with long-range interactions. In addition, we show that universal quantum state transfer, as well as manybody quantum chaos, is bounded by the Frobenius light cone and, therefore, is poorly constrained by all Lieb-Robinson bounds.