  We investigate the quantum annealing of the ferromagnetic $ p $-spin model in
a dissipative environment ($ p = 5 $ and $ p = 7 $). This model, in the large $
p $ limit, codifies the Grover's algorithm for searching in an unsorted
database. The dissipative environment is described by a phonon bath in thermal
equilibrium at finite temperature. The dynamics is studied in the framework of
a Lindblad master equation for the reduced density matrix describing only the
spins. Exploiting the symmetries of our model Hamiltonian, we can describe many
spins and extrapolate expected trends for large $ N $, and $ p $. While at weak
system bath coupling the dissipative environment has detrimental effects on the
annealing results, we show that in the intermediate coupling regime, the phonon
bath seems to speed up the annealing at low temperatures. This improvement in
the performance is likely not due to thermal fluctuation but rather arises from
a correlated spin-bath state and persists even at zero temperature. This result
may pave the way to a new scenario in which, by appropriately engineering the
system-bath coupling, one may optimize quantum annealing performances below
either the purely quantum or classical limit.
