  We study the performance of quantum annealing for two sets of problems,
namely, 2-satisfiability (2-SAT) problems represented by Ising-type
Hamiltonians, and nonstoquastic problems which are obtained by adding extra
couplings to the 2-SAT problem Hamiltonians. In addition, we add to the
transverse Ising-type Hamiltonian used for quantum annealing a third term, the
trigger Hamiltonian with ferromagnetic or antiferromagnetic couplings, which
vanishes at the beginning and end of the annealing process. We also analyze
some problem instances using the energy spectrum, average energy or overlap of
the state during the evolution with the instantaneous low lying eigenstates of
the Hamiltonian, and identify some non-adiabatic mechanisms which can enhance
the performance of quantum annealing.
