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We study the energy gap between the ground state and the first excited stateof a mean-field-type non-stoquastic Hamiltonian by a semi-classical analysis.The fully connected mean-field model with $p$-body ferromagnetic interactionsunder a transverse field has a first-order quantum phase transition for $p\ge3$. This first-order transition is known to be reduced to second order for$p\ge 5$ by an introduction of antiferromagnetic transverse interactions, whichmakes the Hamiltonian non-stoquastic. This reduction of the order of transitionmeans an exponential speedup of quantum annealing by adiabatic processesbecause the first-order transition is shown to have an exponentially smallenergy gap whereas the second order case does not. We apply a semi-classicalmethod to analytically derive the explicit expression of the rate of theexponential decay of the energy gap at first-order transitions. The resultreveals how the property of first-order transition changes as a function of thesystem parameters. We also derive the exact closed-form expression for thecritical point where the first-order transition line disappears within theferromagnetic phase. These results help us understand how the antiferromagnetictransverse interactions affect the performance of quantum annealing bycontrolling the effects of non-stoquasticity in the Hamiltonian.

Exact expression of the energy gap at first-order quantum phase transitions of a non-stoquastic Hamiltonian