  Despite intensive research, the physical origin of the speed-up offered by
quantum algorithms remains mysterious. No general physical quantity, like, for
instance, entanglement, can be singled out as the essential useful resource.
Here we report a close connection between the trace speed and the quantum
speed-up in Grover's search algorithm implemented with pure and pseudo-pure
states. For a noiseless algorithm, we find a one-to-one correspondence between
the quantum speed-up and the polarization of the pseudo-pure state, which can
be connected to a wide class of quantum statistical speeds. For time-dependent
partial depolarization and for interrupted Grover searches, the speed-up is
specifically bounded by the maximal trace speed that occurs during the
algorithm operations. Our results quantify the quantum speed-up with a physical
resource that is experimentally measurable and related to multipartite
entanglement and quantum coherence.
