Quantum annealing for hard 2-satisfiability problems: Distribution and scaling of minimum energy gap and success probability

Abstract

In recent years, quantum annealing has gained the status of being a promising candidate for solving various optimization problems. Using a set of hard 2-satisfiability (2-SAT) problems, consisting of problems of up to 18 variables, we analyze the scaling complexity of the quantum annealing algorithm and study the distributions of the minimum energy gap and the success probability. We extend the analysis of the standard quantum annealing Hamiltonian by introducing an additional term, the trigger Hamiltonian, which can be of two types: ferromagnetic and antiferromagnetic. We use these trigger Hamiltonians to study their influence on the success probability for solving the selected 2-SAT problems. We find that although the scaling of the runtime is exponential for the standard and modified quantum annealing Hamiltonians, the scaling constant in the case of adding the trigger Hamiltonians can be significantly smaller. Furthermore, certain choices for the trigger Hamiltonian and annealing times can result in a better scaling than that for simulated annealing. Finally, we also use the quantum annealers of D-Wave Systems Inc. to study their performance in solving the 2-SAT problems and compare it with the simulation results.