  Quantum annealing (QA) is a promising method for solving combinatorial
optimization problems whose solutions are embedded into a ground state of the
Ising Hamiltonian. This method employs two types of Hamiltonians: a driver
Hamiltonian and a problem Hamiltonian. After a sufficiently slow change from
the driver Hamiltonian to the problem Hamiltonian, we can obtain the target
ground state that corresponds to the solution. The inclusion of non-stoquastic
terms in the driver Hamiltonian is believed to enhance the efficiency of the
QA. Meanwhile, decoherence is regarded as of the main obstacles for QA. Here,
we present examples showing that non-stoaquastic Hamiltonians can lead to
catastrophic failure of QA, whereas a certain decoherence process can be used
to avoid such failure. More specifically, when we include anti-ferromagnetic
interactions (i.e., typical non-stoquastic terms) in the Hamiltonian, we are
unable to prepare the target ground state even with an infinitely long
annealing time for some specific cases. In our example, owing to a symmetry,
the Hamiltonian is block-diagonalized, and a crossing occurs during the QA,
which leads to a complete failure of the ground-state search. Moreover, we show
that, when we add a certain type of decoherence, we can obtain the ground state
after QA for these cases. This is because, even when symmetry exists in
isolated quantum systems, the environment breaks the symmetry. Our counter
intuitive results provide a deep insight into the fundamental mechanism of QA.
