Improved Quantum Query Algorithms for Triangle Finding and Associativity Testing

TL;DR

This paper presents improved quantum query algorithms for detecting triangles in graphs and testing associativity of binary operations, utilizing a flexible learning graph framework to achieve better complexity bounds.

Contribution

It introduces novel quantum algorithms with enhanced query complexities for triangle detection and associativity testing, using a generalized learning graph approach.

Findings

Quantum triangle detection complexity improved to O(n^{9/7})

Associativity testing complexity improved to O(|S|^{10/7})

Framework allows for detection of complex subgraphs with flexible parameters

Abstract

We show that the quantum query complexity of detecting if an $n$-vertex graph contains a triangle is $O(n^{9/7})$. This improves the previous best algorithm of Belovs making $O(n^{35/27})$ queries. For the problem of determining if an operation $\circ : S \times S \rightarrow S$ is associative, we give an algorithm making $O(|S|^{10/7})$ queries, the first improvement to the trivial $O(|S|^{3/2})$ application of Grover search. Our algorithms are designed using the learning graph framework of Belovs. We give a family of algorithms for detecting constant-sized subgraphs, which can possibly be directed and colored. These algorithms are designed in a simple high-level language; our main theorem shows how this high-level language can be compiled as a learning graph and gives the resulting complexity. The key idea to our improvements is to allow more freedom in the parameters of the database kept by the algorithm. As in our previous work, the edge slots maintained in the database are specified by a graph whose edges are the union of regular bipartite graphs, the overall structure of which mimics that of the graph of the certificate. By allowing these bipartite graphs to be unbalanced and of variable degree we obtain better algorithms.