  While rigorous quantum dynamical simulations of many-body systems are
extremely difficult (or impossible) due to the exponential scaling with
dimensionality, corresponding classical simulations completely ignore quantum
effects. Semiclassical methods are generally more efficient but less accurate
than quantum methods, and more accurate but less efficient than classical
methods. We find a remarkable exception to this rule by showing that a
semiclassical method can be both more accurate and faster than a classical
simulation. Specifically, we prove that for the semiclassical dephasing
representation the number of trajectories needed to simulate quantum fidelity
is independent of dimensionality and also that this semiclassical method is
even faster than the most efficient corresponding classical algorithm.
Analytical results are confirmed with simulations of quantum fidelity in up to
100 dimensions with 2^1700-dimensional Hilbert space.
