  We study the performance of quantum annealing for the simple $p$-body
infinite-range ferromagnetic Ising model. In particular, we generalize the
transverse antiferromagnetic interactions proposed by Seki and Nishimori as a
quantum driver to many-body transverse interactions to understand if the
two-body interactions are essential to allow the system to avoid troublesome
first-order quantum phase transitions. We conclude that the general many-body
interactions are effective to let the system evolve only through second-order
transitions as long as a few minor conditions are satisfied. It is also
discussed whether the overlap of the ground-state wave function of the new
driver term with the target ground state is an essential factor for the
success.
