  Classical simulations of quantum computations are vital for the future
development of this emerging technology. To this end, decision diagrams have
been proposed as a complementary technique which frequently allows to tackle
the inherent exponential complexity of these simulations. In the worst case,
however, they still cannot escape this complexity. Additionally, while other
techniques make use of all the available processing power, decision
diagram-based simulation to date cannot exploit the many processing units of
today's systems. In this work, we show that both problems can be tackled
together by employing a hybrid Schr\"odinger-Feynman scheme for the simulation.
More precisely, we show that realizing such a scheme with decision diagrams is
indeed possible, we discuss the resulting problems in its realization, and
propose solutions how they can be handled. Experimental evaluations confirm
that this significantly advances the state of the art in decision diagram-based
simulation -- allowing to simulate certain hard circuits within minutes that
could not be simulated in a whole day thus far.
