Quantum Distributed Algorithms for Detection of Cliques

TL;DR

This paper introduces a quantum distributed algorithm framework for efficiently detecting larger cliques in networks, extending previous results on triangle detection and employing nested quantum searches.

Contribution

It presents a novel framework for quantum distributed clique detection, generalizing triangle detection to larger cliques and developing nested quantum search techniques.

Findings

Improved quantum algorithms for clique detection beyond triangles.

Framework for nested quantum searches with quantum checking procedures.

Circuit complexity barrier established for classical lower bounds on clique detection.

Abstract

The possibilities offered by quantum computing have drawn attention in the distributed computing community recently, with several breakthrough results showing quantum distributed algorithms that run faster than the fastest known classical counterparts, and even separations between the two models. A prime example is the result by Izumi, Le Gall, and Magniez [STACS 2020], who showed that triangle detection by quantum distributed algorithms is easier than triangle listing, while an analogous result is not known in the classical case. In this paper we present a framework for fast quantum distributed clique detection. This improves upon the state-of-the-art for the triangle case, and is also more general, applying to larger clique sizes. Our main technical contribution is a new approach for detecting cliques by encapsulating this as a search task for nodes that can be added to smaller cliques. To extract the best complexities out of our approach, we develop a framework for nested distributed quantum searches, which employ checking procedures that are quantum themselves. Moreover, we show a circuit-complexity barrier on proving a lower bound of the form $\Omega(n^{3/5+\epsilon})$ for $K_p$-detection for any $p \geq 4$, even in the classical (non-quantum) distributed CONGEST setting.