  We study first-order quantum phase transitions in models where the mean-field
traitment is exact, and the exponentially fast closure of the energy gap with
the system size at the transition. We consider exactly solvable ferromagnetic
models, and show that they reduce to the Grover problem in a particular limit.
We compute the coefficient in the exponential closure of the gap using an
instantonic approach, and discuss the (dire) consequences for quantum
annealing.
