Ramsey Numbers and Adiabatic Quantum Computing

Abstract

The graph-theoretic Ramsey numbers are notoriously difficult to calculate. In fact, for the twocolor Ramsey numbers R(m, n) with m, n ≥ 3, only nine are currently known. We present a quantum algorithm for the computation of the Ramsey numbers R(m, n). We show how the computation of R(m, n) can be mapped to a combinatorial optimization problem whose solution can be found using adiabatic quantum evolution. We numerically simulate this adiabatic quantum algorithm and show that it correctly determines the Ramsey numbers R(3, 3) and R(2, s) for 5 ≤ s ≤ 7. We then discuss the algorithm's experimental implementation, and close by showing that Ramsey number computation belongs to the quantum complexity class QMA.PACS numbers: 03.67. Ac,02.10.Ox,89.75.Hc In an arbitrary party of N people one might ask whether there is a group of m people who are all mutually acquainted, or a group of n people who are all mutual strangers. Using Ramsey theory [1,2], it can be shown that a threshold value R(m, n) exists for the party size N so that when N ≥ R(m, n), all parties of N people will either contain m mutual acquaintances, or n mutual strangers. The threshold value R(m, n) is an example of a two-color Ramsey number. Other types of Ramsey numbers exist, though we will focus on two color Ramsey numbers in this paper.One can represent the N -person party problem by an N -vertex graph. Here each person is associated with a vertex, and an edge is drawn between a pair of vertices only when the corresponding people know each other. In the case where m people are mutual acquaintances, there will be an edge connecting any pair of the m corresponding vertices. Similarly, if n people are mutual strangers, there will be no edge between any of the n corresponding vertices. In the language of graph theory [3], the m vertices form an m-clique, and the n vertices form an nindependent set. The party problem is now a statement in graph theory: if N ≥ R(m, n), every graph with N vertices will contain either an m-clique, or an n-independent set. Ramsey numbers can also be introduced using colorings of complete graphs, and R(m, n) corresponds to the case where only two colors are used.Ramsey theory has found applications in mathematics, information theory, and theoretical computer science [6]. An application of fundamental significance appears in the Paris-Harrington (PH) theorem of mathematical logic [4] which established that a particular statement in Ramsey theory related to graph colorings and natural numbers is true, though unprovable within the axioms of Peano arithmetic. Such statements are known to exist as a consequence of Godel's incompleteness theorem, though the PH theorem provided the first natural example. Deep connections have also been shown to exist between Ramsey theory, topological dynamics, and ergodic theory [5].Ramsey numbers grow extremely quickly and so are notoriously difficult to calculate. In fact, for two color Ramsey numbers R(m, n) with m, n ≥ 3, only nine are presently known [3]. To check whether N ? = R(m, n) requir...

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