Quantum Algorithm for Triangle Finding in Sparse Graphs

TL;DR

This paper introduces a quantum algorithm that efficiently finds triangles in sparse graphs, surpassing previous quantum methods by leveraging recent dense graph algorithms and achieving sublinear query complexity.

Contribution

It extends quantum triangle finding algorithms to sparse graphs, improving query complexity over prior dense graph algorithms and demonstrating efficiency in graphs with fewer edges.

Findings

Achieves $O(n^{5/4- ext{constant}})$ query complexity for sparse graphs.

Shows quantum advantage in triangle detection for graphs with $O(n^{2-c})$ edges.

Builds on Le Gall's dense graph algorithm to improve sparse graph performance.

Abstract

This paper presents a quantum algorithm for triangle finding over sparse graphs that improves over the previous best quantum algorithm for this task by Buhrman et al. [SIAM Journal on Computing, 2005]. Our algorithm is based on the recent $\tilde O(n^{5/4})$-query algorithm given by Le Gall [FOCS 2014] for triangle finding over dense graphs (here $n$ denotes the number of vertices in the graph). We show in particular that triangle finding can be solved with $O(n^{5/4-\epsilon})$ queries for some constant $\epsilon>0$ whenever the graph has at most $O(n^{2-c})$ edges for some constant $c>0$.