Mechanism of the quantum speed-up

TL;DR

This paper explains the quantum speed-up mechanism by showing how quantum correlations and superpositions enable algorithms to effectively know part of the solution in advance, reducing computation steps.

Contribution

It introduces a novel explanation of quantum speed-up based on quantum correlations and measurement sharing, offering new insights into quantum algorithm efficiency.

Findings

Quantum correlations replace classical randomness in algorithms.

Quantum superpositions encode partial knowledge of the solution.

The explanation clarifies how quantum algorithms outperform classical ones.

Abstract

We explain the mechanism of the quantum speed-up - quantum algorithms requiring fewer computation steps than their classical equivalent - for a family of algorithms. Bob chooses a function and gives to Alice the black box that computes it. Alice, without knowing Bob's choice, should find a character of the function (e. g. its period) by computing its value for different arguments. There is naturally correlation between Bob's choice and the solution found by Alice. We show that, in quantum algorithms, this correlation becomes quantum. This highlights an overlooked measurement problem: sharing between two measurements the determination of correlated (thus redundant) measurement outcomes. Solving this problem explains the speed-up. All is like Alice, by reading the solution at the end of the algorithm, contributed to the initial choice of Bob, for half of it in quantum superposition for all the possible ways of taking this half. This contribution, back evolved to before running the algorithm, where Bob's choice is located, becomes Alice knowing in advance half of this choice. The quantum algorithm is the quantum superposition of all the possible ways of taking half of Bob's choice and, given the advanced knowledge of it, classically computing the missing half. This yields a speed-up with respect to the classical case where, initially, Bob's choice is completely unknown to Alice.