A note about claw function with a small range

TL;DR

This paper investigates the quantum query complexity of the claw detection problem, establishing bounds that depend on the sizes of the domain and range, with implications for quantum algorithms.

Contribution

It provides new bounds on the quantum query complexity for the claw detection problem with a small range, refining previous understanding.

Findings

Quantum query complexity is between (n^{1/2}k^{1/6}) and O(n^{1/2+\u03b5}k^{1/4})

Bounds depend on the sizes of domain and range for the problem

Results improve understanding of quantum complexity in claw detection

Abstract

In the claw detection problem we are given two functions $f:D\rightarrow R$ and $g:D\rightarrow R$ ($|D|=n$, $|R|=k$), and we have to determine if there is exist $x,y\in D$ such that $f(x)=g(y)$. We show that the quantum query complexity of this problem is between $\Omega\left(n^{1/2}k^{1/6}\right)$ and $O\left(n^{1/2+\varepsilon}k^{1/4}\right)$ when $2\leq k<n$.