An exact relation between number of black box computations required to solve an oracle problem quantumly and quantum retrocausality

TL;DR

This paper establishes a precise relationship between the number of black box computations needed for quantum solutions to oracle problems and the concept of quantum retrocausality, offering insights into quantum speedup mechanisms.

Contribution

It introduces a retrocausality-based model that links the quantum speedup to the number of black box computations and provides a novel interpretation of quantum algorithms.

Findings

R=1/2 matches existing quantum algorithms' complexity

The model explains quantum speedup through retrocausality

Provides a method to determine the number of black box steps needed

Abstract

We investigate the reason for the quantum speedup -- quantum algorithms requiring fewer computation steps than their classical counterparts. We extend their representation to the process of setting the problem. The initial measurement selects a setting at random, Bob (the problem setter) unitarily changes it into the desired one. This representation is to Bob and any external observer, it cannot be to Alice (the problem solver). It would tell her the function computed by the black box, which to her should be hidden inside it. We resort to relational quantum mechanics. To Alice, the projection of the quantum state due to the initial measurement is retarded at the end of her problem solving action. To her, the algorithm input state remains one of complete ignorance of the setting. By black box computations, she unitarily sends it into the output state that, for each possible setting, encodes the corresponding solution, acquired by the final measurement. We show that we can ascribe to the final measurement the selection of any part -- say the R-th part -- of the random outcome of the initial measurement. This projects the input state to Alice on a state of lower entropy where she knows a corresponding part of the problem setting. The quantum algorithm is a sum over classical histories in each of which Alice, knowing in advance one of the R-th parts of the setting, performs the black box computations still required to identify the solution. Given an oracle problem and a value of R, this retrocausality model provides the number of black box computations required to solve it. Conversely, given a known quantum algorithm, it yields the value of R that explains its speed up. R = 1/2 always yields the number of black box computations required by an existing quantum algorithm and the order of magnitude of the number required by optimal one.