  In this work, we compare the performance of the Quantum Approximate
Optimization Algorithm (QAOA) with state-of-the-art classical solvers such as
Gurobi and MQLib to solve the combinatorial optimization problem MaxCut on
3-regular graphs. The goal is to identify under which conditions QAOA can
achieve "quantum advantage" over classical algorithms, in terms of both
solution quality and time to solution. One might be able to achieve quantum
advantage on hundreds of qubits and moderate depth $p$ by sampling the QAOA
state at a frequency of order 10 kHz. We observe, however, that classical
heuristic solvers are capable of producing high-quality approximate solutions
in linear time complexity. In order to match this quality for $\textit{large}$
graph sizes $N$, a quantum device must support depth $p>11$. Otherwise, we
demonstrate that the number of required samples grows exponentially with $N$,
hindering the scalability of QAOA with $p\leq11$. These results put challenging
bounds on achieving quantum advantage for QAOA MaxCut on 3-regular graphs.
Other problems, such as different graphs, weighted MaxCut, maximum independent
set, and 3-SAT, may be better suited for achieving quantum advantage on
near-term quantum devices.
