How significant are the known collision and element distinctness quantum algorithms?

TL;DR

This paper evaluates the significance of quantum algorithms for collision and element distinctness, arguing that their speedups are not substantial relative to their hardware requirements, thus questioning their practical impact.

Contribution

It introduces a criterion for assessing quantum algorithms' significance based on hardware and speedup, and applies this to show existing algorithms are not non-trivial.

Findings

Known algorithms do not provide a significant speedup relative to hardware used.

Speedup of at least O(sqrt(P)) is necessary to be considered meaningful.

Current algorithms for collision and element distinctness are considered trivial under this criterion.

Abstract

Quantum search is a technique for searching N possibilities in only O(sqrt(N)) steps. It has been applied in the design of quantum algorithms for several structured problems. Many of these algorithms require significant amount of quantum hardware. In this paper we observe that if an algorithm requires O(P) hardware, it should be considered significant if and only if it produces a speedup of at least O(sqrt(P)) over a simple quantum search algorithm. This is because a speedup of $O(sqrt(P)) $ can be trivially obtained by dividing the search space into $O(P)$ separate parts and handing the problem to independent processors that do a quantum search. We argue that the known algorithms for collision and element distinctness fail to be non-trivial in this sense.