  The Quantum Alternating Operator Ansatz (QAOA) represents a branch of quantum
algorithms for solving combinatorial optimization problems. A specific variant,
the Grover-Mixer Quantum Alternating Operator Ansatz (GM-QAOA), ensures uniform
amplitude across states that share equivalent objective values. This property
makes the algorithm independent of the problem structure, focusing instead on
the distribution of objective values within the problem. In this work, we prove
the probability upper bound for measuring a computational basis state from a
GM-QAOA circuit with a given depth, which is a critical factor in QAOA cost.
Using this, we derive the upper bounds for the probability of sampling an
optimal solution, and for the approximation ratio of maximum optimization
problems, both dependent on the objective value distribution. Through numerical
analysis, we link the distribution to the problem size and build the regression
models that relate the problem size, QAOA depth, and performance upper bound.
Our results suggest that the GM-QAOA provides a quadratic enhancement in
sampling probability and requires circuit depth that scales exponentially with
problem size to maintain consistent performance.
