Comparing the hardness of MAX 2-SAT problem instances for quantum and classical algorithms

We investigate whether small sized problem instances, which are commonly used in numerical simulations of quantum algorithms for benchmarking purposes, are a good representation of larger instances in terms of hardness. We approach this through a numerical analysis of the hardness of MAX 2-SAT problem instances for various continuous-time quantum algorithms and a comparable classical algorithm. Our results can be used to predict the viability of hybrid approaches that combine multiple algorithms in parallel to take advantage of the variation in the hardness of instances between the algorithms. We find that, while there are correlations in instance hardness between all of the algorithms considered, they appear weak enough that a hybrid strategy would likely be desirable in practice. Our results also show a widening range of hardness of randomly generated instances as the problem size is increased, which demonstrates both the difference in the distribution of hardness at small sizes and the value of a hybrid approach that can reduce the number of extremely hard instances. We identify specific weaknesses that can be overcome with hybrid techniques, such as the inability of these quantum algorithms to efficiently solve satisfiable instances (which is easy classically).