  Reverse annealing is a relatively new variant of quantum annealing, in which
one starts from a classical state and increases and then decreases the
amplitude of the transverse field, in the hope of finding a better classical
state than the initial state for a given optimization problem. We numerically
study the unitary quantum dynamics of reverse annealing for the mean-field-type
$p$-spin model and show that the results are consistent with the predictions of
equilibrium statistical mechanics. In particular, we corroborate the
equilibrium analysis prediction that reverse annealing provides an exponential
speedup over conventional quantum annealing in terms of solving the $p$-spin
model. This lends support to the expectation that equilibrium analyses are
effective at revealing essential aspects of the dynamics of quantum annealing.
We also compare the results of quantum dynamics with the corresponding
classical dynamics, to reveal their similarities and differences. We
distinguish between two reverse annealing protocols we call adiabatic and
iterated reverse annealing. We further show that iterated reverse annealing, as
has been realized in the D-Wave device, is ineffective in the case of the
$p$-spin model, but note that a recently-introduced protocol ("$h$-gain"),
which implements adiabatic reverse annealing, may lead to improved performance.
