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Bob hides a ball in one of four drawers. Alice is to locate it. Classicallyshe has to open up to three drawers, quantally just one. The fundamental reasonfor this quantum speedup is not known. The usual representation of the quantumalgorithm is limited to the process of solving the problem. We extend it to theprocess of setting the problem. The number of the drawer with the ball becomesa unitary transformation of the random outcome of the preparation measurement.This extended, time-symmetric, representation brings in relational quantummechanics. It is with respect to Bob and any external observer and cannot bewith respect to Alice. It would tell her the number of the drawer with the ballbefore she opens any drawer. To Alice, the projection of the quantum state dueto the preparation measurement should be retarded at the end of her search; inthe input state of the search, the drawer number is determined to Bob andundetermined to Alice. We show that, mathematically, one can ascribe any partof the selection of the random outcome of the preparation measurement to thefinal Alice's measurement. Ascribing half of it explains the speedup of thepresent algorithm. This projects the input state to Alice on a state of lowerentropy where she knows half of the number of the drawer with the ball inadvance. The quantum algorithm turns out to be a sum over histories in each ofwhich Alice knows in advance that the ball is in a pair of drawers and locatesit by opening one of the two. In the sample of quantum algorithms examined, thepart of the random outcome of the initial measurement selected by the finalmeasurement is one half or slightly above it. Conversely, given an oracleproblem, the assumption it is one half always corresponds to an existingquantum algorithm and gives the order of magnitude of the number of oraclequeries required by the optimal one.

Highlighting the mechanism of the quantum speedup by time-symmetric and relational quantum mechanics