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We study first-order quantum phase transitions in models where the mean-fieldtraitment is exact, and the exponentially fast closure of the energy gap withthe system size at the transition. We consider exactly solvable ferromagneticmodels, 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 aninstantonic approach, and discuss the (dire) consequences for quantumannealing.

Energy gaps in quantum first-order mean-field-like transitions: The problems that quantum annealing cannot solve