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

Quantum Annealing with Trigger Hamiltonians: Application to 2-SAT and Nonstoquastic Problems