Completing the physical representation of quantum algorithms provides a retrocausal explanation of their speedup

In previous works, we showed that an optimal quantum algorithm can always be seen as a sum over classical histories in each of which the problem solver knows in advance one of the possible halves of the solution she will read in the future and performs the computation steps (oracle queries) still needed to reach it. Given an oracle problem, this retrocausal explanation of the speedup yields the order of magnitude of the number of oracle queries needed to solve it in an optimal quantum way. Presently, we provide a fundamental justification for the explanation in question and show that it comes out by just completing the physical representation of quantum algorithms. Since the use of retrocausality in quantum mechanics is controversial, showing that it answers the well accepted requirement of the completeness of the physical description should be an important pass.