Improved Quantum Algorithm for Triangle Finding via Combinatorial Arguments

TL;DR

This paper introduces a new quantum algorithm for triangle detection in unweighted graphs with improved query complexity, demonstrating that unweighted triangle finding is easier than the weighted case and highlighting limitations of previous approaches.

Contribution

The paper presents a novel quantum algorithm with better query complexity for unweighted triangle finding, using combinatorial methods that surpass previous bounds and approaches.

Findings

Query complexity improved to  O(n^{5/4})

Unweighted triangle finding is easier than weighted case

Limitations of non-adaptive learning graph approach are demonstrated

Abstract

In this paper we present a quantum algorithm solving the triangle finding problem in unweighted graphs with query complexity $\tilde O(n^{5/4})$, where $n$ denotes the number of vertices in the graph. This improves the previous upper bound $O(n^{9/7})=O(n^{1.285...})$ recently obtained by Lee, Magniez and Santha. Our result shows, for the first time, that in the quantum query complexity setting unweighted triangle finding is easier than its edge-weighted version, since for finding an edge-weighted triangle Belovs and Rosmanis proved that any quantum algorithm requires $\Omega(n^{9/7}/\sqrt{\log n})$ queries. Our result also illustrates some limitations of the non-adaptive learning graph approach used to obtain the previous $O(n^{9/7})$ upper bound since, even over unweighted graphs, any quantum algorithm for triangle finding obtained using this approach requires $\Omega(n^{9/7}/\sqrt{\log n})$ queries as well. To bypass the obstacles characterized by these lower bounds, our quantum algorithm uses combinatorial ideas exploiting the graph-theoretic properties of triangle finding, which cannot be used when considering edge-weighted graphs or the non-adaptive learning graph approach.