  We develop an instantonic calculus to derive an analytical expression for the
thermally-assisted tunneling decay rate of a metastable state in a fully
connected quantum spin model. The tunneling decay problem can be mapped onto
the Kramers escape problem of a classical random dynamical field. This
dynamical field is simulated efficiently by path integral Quantum Monte Carlo
(QMC). We show analytically that the exponential scaling with the number of
spins of the thermally-assisted quantum tunneling rate and the escape rate of
the QMC process are identical. We relate this effect to the existence of a
dominant instantonic tunneling path. The instanton trajectory is described by
nonlinear dynamical mean-field theory equations for a single site magnetization
vector, which we solve exactly. Finally, we derive scaling relations for the
"spiky" barrier shape when the spin tunnelling and QMC rates scale polynomially
with the number of spins $N$ while a purely classical over-the-barrier
activation rate scales exponentially with $N$.
