  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-satisfiabilty (2-SAT) problems, consisting of upto 18-variables problems, 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 found that although the scaling of the run-time is exponential for
the standard and modified quantum annealing Hamiltonians, the scaling constant
in 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. Lastly, 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.
