  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 towards the
application of the QAOA to more general problem instances.
