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The Quantum Approximate Optimization Algorithm, QAOA, uses a shallow depth circuit to produce a parameter dependent state. For a given optimization problem instance, the quantum expectation of the cost function is the parameter dependent objective function of the. We demonstrate that if the parameters are fixed and the instance comes a reasonable distribution then the objective function value is in the sense that typical instances have (nearly) the same value the objective function. This applies not just for optimal parameters as the landscape is instance independent. We can prove this is true for low quantum circuits for instances of MaxCut on large 3-regular graphs. Our generalize beyond this example. We support the arguments with numerical that show remarkable concentration. For higher depth circuits the also show concentration and we argue for this using the Law of Large. We also observe by simulation that if we find parameters which result good performance at say 10 bits these same parameters result in good at say 24 bits. These findings suggest ways to run the QAOA that or eliminate the use of the outer loop optimization and may allow us to good solutions with fewer calls to the quantum computer.
Brandão et al. (Mon,) studied this question.