Performance Upper Bound of the Grover-Mixer Quantum Alternating Operator Ansatz

The Quantum Alternating Operator Ansatz (QAOA) represents a branch of quantum algorithms designed 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. From this, we derive the upper bounds for the probability of sampling an optimal solution and for the approximation ratio of maximum optimization problems, based on the objective value distribution. Using 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.