Ramsey numbers and adiabatic quantum computing

TL;DR

This paper introduces a quantum algorithm leveraging adiabatic quantum computing to compute Ramsey numbers, a notoriously difficult graph-theoretic problem, and demonstrates its effectiveness through simulations and complexity analysis.

Contribution

The paper presents the first quantum algorithm for Ramsey number computation, mapping it to a combinatorial optimization problem solvable via adiabatic quantum evolution.

Findings

Successfully simulated R(3,3) and R(2,s) for 5≤s≤7

Shows Ramsey number computation is in the QMA complexity class

Proposes a feasible approach for quantum computation of complex graph invariants

Abstract

The graph-theoretic Ramsey numbers are notoriously difficult to calculate. In fact, for the two-color Ramsey numbers $R(m,n)$ with $m,n\geq 3$, 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 $5\leq s\leq 7$. We then discuss the algorithm's experimental implementation, and close by showing that Ramsey number computation belongs to the quantum complexity class QMA.