Generalized Ramsey numbers through adiabatic quantum optimization

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.