  Quantum annealing is an innovative idea and method for avoiding the increase
of the calculation cost of the combinatorial optimization problem. Since the
combinatorial optimization problems are ubiquitous, quantum annealing machine
with high efficiency and scalability will give an immeasurable impact on many
fields. However, the conventional quantum annealing machine may not have a high
success probability for finding the solution because the energy gap closes
exponentially as a function of the system size. To propose an idea for finding
high success probability is one of the most important issues. Here we show that
a degenerate two-level system provides the higher success probability than the
conventional spin-1/2 model in a weak longitudinal magnetic field region. The
physics behind this is that the quantum annealing in this model can be reduced
into that in the spin-1/2 model, where the effective longitudinal magnetic
field may open the energy gap, which suppresses the Landau--Zener tunneling
providing leakage of the ground state. We also present the success probability
of the $\Lambda$-type system, which may show the higher success probability
than the conventional spin-1/2 model.
