  Reverse annealing is a variant of quantum annealing that starts from a given
classical configuration of spins (qubits). In contrast to the conventional
formulation, where one starts from a uniform superposition of all possible
states (classical configurations), quantum fluctuations are first increased and
only then decreased. One then reads out the state as a proposed solution to the
given combinatorial optimization problem. We formulate a mean-field theory of
reverse annealing using the fully-connected ferromagnetic $p$-spin model, with
and without random longitudinal fields, and analyze it in order to understand
how and when reverse annealing is effective at solving this problem. We find
that the difficulty arising from the existence of a first-order quantum phase
transition, which leads to an exponentially long computation time in
conventional quantum annealing, is circumvented in the context of this
particular problem by reverse annealing if the proximity of the initial state
to the (known) solution exceeds a threshold. Even when a first-order transition
is unavoidable, the difficulty is mitigated due to a smaller jump in the order
parameter at a first-order transition, which implies a larger rate of quantum
tunneling. This is the first analytical study of reverse annealing and paves
the way toward a systematic understanding of this relatively unexplored
protocol in a broader context.
