Quantifying computational advantage of Grover’s algorithm with the trace speed

• Main Authors: Valentin Gebhart, Luca Pezzè, Augusto Smerzi
• Format: Article
• Language: English
• Published: Nature Publishing Group 2021-01-01
• Series: Scientific Reports
• Online Access: https://doi.org/10.1038/s41598-020-80153-z

Abstract 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.