  We have proposed the energy landscape transformation of Ising problems
(ELTIP), which changes the combination of the state and eigenvalue without
changing all the original eigenvalues [arXiv:2202.05927]. We study how the
ELTIP affects the anti-crossing between two levels of the ground and first
excited states during quantum annealing. We use a 5-spin maximum-weighted
independent set for the problem to numerically investigate the anticrossing.
For comparison, we introduce a non-stoquastic Hamiltonian that adds
antiferromagnetic interaction to the normal transverse magnetic field.
Annealing with the non-stoquastic Hamiltonian is effective for difficult
problems. The non-stoquastic Hamiltonian mitigates the anti-crossing when only
the energy gap between the ground state and the first excited state of the
final state is small. When the ELTIP is used, the anti-crossing disappears. For
the problems investigated in this paper, the ELTIP shortens the annealing time
to guarantee adiabatic change more than the non-stoquastic Hamiltonian.
