  Quantum annealing is a heuristic algorithm for searching the ground state of
an Ising model. Heuristic algorithms aim to obtain near-optimal solutions with
a reasonable computation time. Accordingly, many algorithms have so far been
proposed. In general, the performance of heuristic algorithms strongly depends
on the instance of the combinatorial optimization problem to be solved because
they escape the local minima in different ways. Therefore, combining several
algorithms to exploit their complementary strength is effective for obtaining
highly accurate solutions for a wide range of combinatorial optimization
problems. However, quantum annealing cannot be used to improve a candidate
solution obtained by other algorithms because it starts from an initial state
where all spin configurations are found with a uniform probability. In this
study, we propose an Ising formulation of integer optimization problems to
utilize quantum annealing in the iterative improvement strategy. Our
formulation exploits the biased sampling of degenerated ground states in
transverse magnetic field quantum annealing. We also analytically show that a
first-order phase transition is successfully avoided for a fully connected
ferromagnetic Potts model if the overlap between a ground state and a candidate
solution exceeds a threshold. The proposed formulation is applicable to a wide
range of integer optimization problems and enables us to hybridize quantum
annealing with other optimization algorithms.
