Extended Learning Graphs for Triangle Finding

TL;DR

This paper introduces extended learning graphs to develop new quantum algorithms for triangle finding, achieving improved query complexities for dense and sparse graphs, and simplifying previous approaches based on quantum walks.

Contribution

It presents a novel model of extended learning graphs that simplifies the design and analysis of quantum algorithms for triangle detection, improving upon prior complexities.

Findings

Query complexity for dense graphs: O(n^{5/4})

Query complexity for sparse graphs: O(n^{11/12}m^{1/6}\sqrt{\log n})

New framework enables easier combination and analysis of algorithms.

Abstract

We present new quantum algorithms for Triangle Finding improving its best previously known quantum query complexities for both dense and spare instances.For dense graphs on $n$ vertices, we get a query complexity of $O(n^{5/4})$ without any of the extra logarithmic factors present in the previous algorithm of Le Gall [FOCS'14]. For sparse graphs with $m\geq n^{5/4}$ edges, we get a query complexity of $O(n^{11/12}m^{1/6}\sqrt{\log n})$, which is better than the one obtained by Le Gall and Nakajima [ISAAC'15] when $m \geq n^{3/2}$. We also obtain an algorithm with query complexity ${O}(n^{5/6}(m\log n)^{1/6}+d_2\sqrt{n})$ where $d_2$ is the variance of the degree distribution. Our algorithms are designed and analyzed in a new model of learning graphs that we call extended learning graphs. In addition, we present a framework in order to easily combine and analyze them. As a consequence we get much simpler algorithms and analyses than previous algorithms of Le Gall {\it et al} based on the MNRS quantum walk framework [SICOMP'11].