  We construct the first complete exact numerical solution of a mean field
quantum spin glass model, the transverse field Sherrington-Kirkpatrick model,
by implementing a continuous-time quantum Monte Carlo method in the presence of
full replica symmetry breaking. We extract the full numerically exact phase
diagram, displaying a glassy phase with continuous replica symmetry breaking at
small transverse fields and low temperatures. A paramagnetic phase emerges once
thermal and quantum fluctuations melt the spin glass. We characterize both
phases by extracting the order parameter, as well as the static and dynamical
local spin susceptibilities. The static susceptibility shows a plateau in the
glassy phase, but remains smooth across the phase boundary, while the shape of
dynamical susceptibility varies upon crossing the glass transition by reducing
quantum fluctuations. We qualitatively compare these results to a.c.
susceptibility measurements on dipole-coupled Ising magnets in a transverse
magnetic field. Our work provides a general framework for the exact numerical
solution of mean field quantum glass models, constituting an important step
towards understanding glassiness in realistic systems.
