The QAOA gets stuck starting from a good classical string

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) is designed to maximize a cost function over bit strings. While the initial state is traditionally a superposition over all strings, it is natural to try expediting the QAOA: first use a classical algorithm to produce some good string, and then run the ordinary QAOA starting in the computational basis state associated with that string. Here we report numerical experiments that this method of initializing the QAOA fails dramatically, exhibiting little to no improvement of the cost function. We investigate criteria for the rare instances in which there is any improvement at all, and we provide a statistical argument for the more typical case of no improvement. The statistical argument holds for any string that locally mimics the thermal ensemble at the appropriate temperature. Our numerical experiments indicate this property holds for typical good strings. We emphasize that our negative results only apply to our simple incarnation of the warm-start QAOA and may not apply to other approaches in the literature. We hope that our theoretical analysis will inform future algorithm design.Comment: 26 pages, 1 figure, 6 table