A Statistical Approach to Analyzing the Quantum Alternating Operator Ansatz with Grover Mixer

The Quantum Alternating Operator Ansatz (QAOA) is a promising quantum heuristic to combinatorial optimization, designed for the Noisy Intermediate-Scale Quantum (NISQ) era. A key quantum operator of the algorithm is the mixing, which introduces quantum interference. This work investigates analytically the performance of QAOA with the Grover mixer, a mixing operator inspired by Grover’s algorithm. A critical feature of QAOA with Grover mixer is that it depends only on the probability distribution of the solutions, ignoring the structure of the problem. Due to this, it was conjectured that QAOA with this mixer is bounded to the quadratic speed-up of Grover’s algorithm over the classical brute force on the unstructured search problem. On the other hand, this feature allows the development of a statistical approach, which we explore in this work. For a variant of QAOA called Grover mixing QAOA (GM-QAOA), the analysis leads to an expression for the expectation value whose number of terms depends exponentially on the number of layers of the QAOA. In contrast, for Grover mixer Threshold QAOA (GM-Th-QAOA), a simpler variant that has an intimate relationship with Grover’s algorithm, we obtain a constant-time expression, which allows us to get bounds for different performance metrics, such as the statistical quantities of standard score and quantile. Subsequently, we generalized QAOA with the Grover mixer in a framework we called Grover-based QAOA. By using a contradiction argument with the optimality of Grover’s algorithm, we extend all the bounds of GM-Th-QAOA to the general context of Grover-based QAOA. As a result, we get the main contribution of this work: the proof of the aforementioned conjecture on the performance of QAOA with Grover mixer. We apply our bounds to the class of complete bipartite graphs on the Max-Cut problem, demonstrating that we cannot obtain a polynomial-time algorithm with Grover-based QAOA for these instances, showing that the performance limitation of the Grover mixer can be very severe. Our results suggest that achieving competitive performance with QAOA requires a mixing operator capable of leveraging the structure of combinatorial optimization problems.