Analytical results for the quantum alternating operator ansatz with Grover mixer
Guilherme Adamatti Bridi, Franklin de Lima Marquezino · Physical Review A · 2024
The Grover mixer operator is a variational version of Grover's diffusion operator, introduced as a mixing operator for the quantum alternating operator ansatz (QAOA) and used in a variant known as Grover mixer QAOA (GM-QAOA). An important property of the QAOA with Grover mixer is that its expectation value is invariant over any permutation of states. As a consequence, the algorithm is independent of the structure of the problem. If, on the one hand, this characteristic raises serious doubts about the capacity of the algorithm to overcome the bound of the unstructured search problem, on the other hand, it can pave the way to its analytical study. In this sense, a prior work introduced a statistical approach to analyze GM-QAOA that results in an analytical expression for the expectation value depending on the probability distribution associated with the problem Hamiltonian spectrum. Although the method provides surprising simplifications in calculations, the expression depends exponentially on the number of layers, which makes direct analytical treatment unfeasible. In this work, we extend the analysis to the more simple context of Grover mixer threshold QAOA, a variant that replaces the phase separation operator of GM-QAOA to encode a threshold function. As a result, we obtain an expression for the expectation value independent of the number of layers and, with it, we provide bounds for different performance metrics. Furthermore, we extend the analysis to a more general context of QAOA with Grover mixer, which we called Grover-based QAOA. In that framework, which allows the phase separation operator to encode any compilation of the cost function, we generalize all the bounds by using an argument by contradiction with the optimality of Grover's algorithm on the unstructured search problem. As a result, we get the main contribution of this work, an asymptotic lower bound on the quantile achieved by the expectation value that formalizes the notion that the Grover mixer, at most, reflects a quadratic Grover-style speed-up over classical brute force. We apply that bound on the max-cut problem to the particular class of complete bipartite graphs and argue that the number of rounds required to achieve guarantees for any approximation ratio must grow exponentially with the number of vertices or edges, a severe limitation on the performance of the algorithm.