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arXiv:1204.2483 · International Journal of Quantum Information

HYPERGRAPH RAMSEY NUMBERS AND ADIABATIC QUANTUM ALGORITHM

Ri Qu, Su-li Zhao, Yan-ru Bao, Xiaochun Cao

Published Sep 1, 2012• 1 citation
arXivHTMLPDF

Abstract

Gaitan and Clark [Phys. Rev. Lett.108 (2012) 010501] have recently presented a quantum algorithm for the computation of the Ramsey numbers R(m, n) using adiabatic quantum evolution (AQE). We consider that the two-color Ramsey numbers R(m, n; r) for r-uniform hypergraphs can be computed by using the similar ways in Phys. Rev. Lett.108 (2012) 010501. In this paper, we show how the computation of R(m, n; r) can be mapped to a combinatorial optimization problem whose solution be found using AQE.

DOI
10.1142/s0219749912500670

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Gaitan and Clark [Phys. Rev. Lett.108 (2012) 010501] have recently presented a quantum algorithm for the computation of the Ramsey numbers R(m, n) using adiabatic quantum evolution (AQE). We consider that the two-color Ramsey numbers R(m, n; r) for r-uniform hypergraphs can be computed by using the similar ways in Phys. Rev. Lett.108 (2012) 010501. In this paper, we show how the computation of R(m, n; r) can be mapped to a combinatorial optimization problem whose solution be found using AQE.
The graph-theoretic Ramsey numbers are notoriously difficult to calculate. In fact, for the two-color 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 quantum Merlin Arthur.
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.
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 Rðm; nÞ.Here we present results of an experimental implementation of this algorithm and show that it correctly determines the Ramsey numbers Rð3; 3Þ and Rðm; 2Þ for 4 m 8.The Rð8; 2Þ 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.
Abstract Quantum annealing is a powerful tool for solving and approximating combinatorial optimization problems, such as graph partitioning, community detection, centrality, routing problems, and more. In this paper we explore the use of quantum annealing as a tool for use in exploring combinatorial mathematics research problems. We consider the monochromatic triangle problem and the Ramsey number problem, both examples of graph coloring. Conversion to quadratic unconstrained binary optimization (QUBO) form is required to run on quantum hardware. While the monochromatic triangle problem is quadratic by nature, the Ramsey number problem requires the use of order reduction methods for a quadratic formulation. The goal is to provide a method for producing special colorings of graphs which if successful would provide lower bounds for certain Ramsey numbers. We discuss implementations, limitations, and results when running on the D-Wave Advantage quantum annealer.
We have developed a general technique to study the dynamics of the quantum adiabatic evolution algorithm applied to random combinatorial optimization problems in the asymptotic limit of large problem size nnn. We use as an example the NP-complete Number Partitioning problem and map the algorithm dynamics to that of an auxilary quantum spin glass system with the slowly varying Hamiltonian. We use a Green function method to obtain the adiabatic eigenstates and the minimum excitation gap, gmin=O(n2−n/2)g_{\rm min}={\cal O}(n 2^{-n/2})gmin​=O(n2−n/2), corresponding to the exponential complexity of the algorithm for Number Partitioning. The key element of the analysis is the conditional energy distribution computed for the set of all spin configurations generated from a given (ancestor) configuration by simulteneous fipping of a fixed number of spins. For the problem in question this distribution is shown to depend on the ancestor spin configuration only via a certain parameter related to the energy of the configuration. As the result, the algorithm dynamics can be described in terms of one-dimenssional quantum diffusion in the energy space. This effect provides a general limitation on the power of a quantum adiabatic computation in random optimization problems. Analytical results are in agreement with the numerical simulation of the algorithm.
The design of a good algorithm to solve NP-hard combinatorial approximation problems requires specific domain knowledge about the problems and often needs a trial-and-error problem solving approach. Graph coloring is one of the essential fields to provide an efficient solution for combinatorial applications such as flight scheduling, frequency allocation in networking, and register allocation. In particular, some optimization algorithms have been proposed to solve the multi-coloring graph problems but most of the cases a simple searching method would be the best approach to find an optimal solution for graph coloring problems. However, this naive approach can increase the computation cost exponentially as the graph size and the number of colors increase. To mitigate such intolerable overhead, we investigate the methods to take the advantages of quantum computing properties to find a solution for multi-coloring graph problems in polynomial time. We utilize the variational quantum eigensolver (VQE) technique and quantum approximate optimization algorithm (QAOA) to find solutions for three combinatorial applications by both transferring each problem model to the corresponding Ising model and by using the calculated Hamiltonian matrices. Our results demonstrate that VQE and QAOA algorithms can find one of the best solutions for each application. Therefore, our modeling approach with hybrid quantum algorithms can be applicable for combinatorial problems in various fields to find an optimal solution in polynomial time.