While rigorous quantum dynamical simulations of many-body systems are extremely difficult (or impossible) due to exponential scaling with dimensionality, the corresponding classical simulations 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 fidelity in up to 100 dimensions with 21700-dimensional Hilbert space.