Performance Upper Bound of Grover-Mixer Quantum Alternating Operator Ansatz

TL;DR

This paper establishes theoretical upper bounds on the performance of the Grover-Mixer QAOA, revealing its limitations and scaling behavior in solving combinatorial optimization problems.

Contribution

It provides the first probability upper bounds for GM-QAOA measurement outcomes and links these bounds to problem size and circuit depth through numerical analysis.

Findings

GM-QAOA offers quadratic enhancement in sampling probability.

Performance requires exponential circuit depth scaling with problem size.

Upper bounds depend on the distribution of objective values.

Abstract

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.