  We investigate the minimization of the Ising Hamiltonians, comparing the
dynamics of semi-classical soft-spin models with quantum annealing. We
systematically analyze how the energy landscape for the circulant couplings of
a Mobius graph evolves with increased annealing parameters. Our findings
indicate that these semi-classical models face challenges due to a widening
dimensionality landscape. To counteract this issue, we introduce the `manifold
reduction' method, which restricts the soft-spin amplitudes to a defined phase
space region. Concurrently, quantum annealing demonstrates a natural capability
to navigate the Ising Hamiltonian's energy landscape due to its operation
within the comprehensive Hilbert space. Our study indicates that
physics-inspired or physics-enhanced optimizers will likely benefit from a
blend of classical and quantum annealing techniques.
