On the Quantum Query Complexity of Detecting Triangles in Graphs

TL;DR

This paper improves the quantum query complexity bounds for detecting triangles in graphs, reducing the number of queries needed compared to previous methods, and applies to both detection and output of triangles.

Contribution

It establishes a tighter quantum query complexity bound of O(n^{1+3/7} log^2 n) for triangle detection in graphs, advancing quantum graph algorithms.

Findings

Quantum query complexity for triangle detection is reduced to O(n^{1+3/7} log^2 n).

The same bound applies for outputting a triangle if present.

Improves upon previous bound of O(n^{1+1/2}).

Abstract

We show that in the quantum query model the complexity of detecting a triangle in an undirected graph on $n$ nodes can be done using $O(n^{1+{3\over 7}}\log^{2}n)$ quantum queries. The same complexity bound applies for outputting the triangle if there is any. This improves upon the earlier bound of $O(n^{1+{1\over 2}})$.