  The quantum approximate optimization algorithm (QAOA) is a quantum heuristic
for combinatorial optimization that has been demonstrated to scale better than
state-of-the-art classical solvers for some problems. For a given problem
instance, QAOA performance depends crucially on the choice of the parameters.
While average-case optimal parameters are available in many cases, meaningful
performance gains can be obtained by fine-tuning these parameters for a given
instance. This task is especially challenging, however, when the number of
circuit executions (shots) is limited. In this work, we develop an end-to-end
protocol that combines multiple parameter settings and fine-tuning techniques.
We use large-scale numerical experiments to optimize the protocol for the
shot-limited setting and observe that optimizers with the simplest internal
model (linear) perform best. We implement the optimized pipeline on a
trapped-ion processor using up to $32$ qubits and $5$ QAOA layers, and we
demonstrate that the pipeline is robust to small amounts of hardware noise. To
the best of our knowledge, these are the largest demonstrations of QAOA
parameter tuning on a trapped-ion processor.
