Generalized Ramsey numbers through adiabatic quantum optimization

TL;DR

This paper introduces a quantum algorithm using adiabatic quantum optimization to compute generalized Ramsey numbers, successfully determining several previously unknown values for trees of order 6 to 8.

Contribution

The paper presents the first quantum algorithm for generalized Ramsey numbers and computes new values for specific tree graphs, advancing both quantum computing and combinatorics.

Findings

Quantum algorithm successfully computes generalized Ramsey numbers.

Determined new Ramsey numbers for trees of order 6, 7, and 8.

Most of these Ramsey numbers were previously unknown.

Abstract

Ramsey theory is an active research area in combinatorics whose central theme is the emergence of order in large disordered structures, with Ramsey numbers marking the threshold at which this order first appears. For generalized Ramsey numbers $r(G,H)$, the emergent order is characterized by graphs $G$ and $H$. In this paper we: (i) present a quantum algorithm for computing generalized Ramsey numbers by reformulating the computation as a combinatorial optimization problem which is solved using adiabatic quantum optimization; and (ii) determine the Ramsey numbers $r(\mathcal{T}_{m},\mathcal{T}_{n})$ for trees of order $m,n = 6,7,8$, most of which were previously unknown.