Ramsey theory is a highly active research area in mathematics that studies the emergence of order in large disordered structures. Ramsey numbers mark the threshold at which order first appears and are extremely difficult to calculate due to their explosive rate of growth. Recently, an algorithm that can be implemented using adiabatic quantum evolution has been proposed that calculates the two-color Ramsey numbers R(m,n). Here we present results of an experimental implementation of this algorithm and show that it correctly determines the Ramsey numbers R(3,3) and R(m,2) for 4≤m≤8. The R(8,2) computation used 84 qubits of which 28 were computational qubits. This computation is the largest experimental implementation of a scientifically meaningful adiabatic evolution algorithm that has been done to date.

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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In the Graph Isomorphism (GI) problem two N -vertex graphs G and G are given and the task is to determine whether there exists a permutation of the vertices of G that preserves adjacency and transforms G → G . If yes, then G and G are said to be isomorphic; otherwise they are non-isomorphic. The GI problem is an important problem in computer science and is thought to be of comparable difficulty to integer factorization. In this paper we present a quantum algorithm that solves arbitrary instances of GI and which also provides a novel approach to determining all automorphisms of a given graph. We show how the GI problem can be converted to a combinatorial optimization problem that can be solved using adiabatic quantum evolution. We numerically simulate the algorithm's quantum dynamics and show that it correctly: (i) distinguishes non-isomorphic graphs; (ii) recognizes isomorphic graphs and determines the permutation(s) that connect them; and (iii) finds the automorphism group of a given graph G. We then discuss the GI quantum algorithm's experimental implementation, and close by showing how it can be leveraged to give a quantum algorithm that solves arbitrary instances of the NP-Complete Sub-Graph Isomorphism problem. The computational complexity of an adiabatic quantum algorithm is largely determined by the minimum energy gap ∆(N ) separating the ground-and first-excited states in the limit of large problem size N 1. Calculating ∆(N ) in this limit is a fundamental open problem in adiabatic quantum computing, and so it is not possible to determine the computational complexity of adiabatic quantum algorithms in general, nor consequently, of the specific adiabatic quantum algorithms presented here. Adiabatic quantum computing has been shown to be equivalent to the circuit-model of quantum computing, and so development of adiabatic quantum algorithms continues to be of great interest.