Ramsey theory is a highly active research area in mathematics that studies the emergence of order in large disordered structures. It has found applications in mathematics, theoretical computer science, information theory, and classical error correcting codes. Ramsey numbers mark the threshold at which order first appears and are notoriously difficult to calculate due to their explosive rate of growth. Recently, a quantum algorithm has been proposed that calculates the two-color Ramsey numbers R(m,n). Here we present results of an experimental implementation of this algorithm based on quantum annealing 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 quantum 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. It has found applications in mathematics, theoretical computer science, information theory, and classical error correcting codes. Ramsey numbers mark the threshold at which order first appears and are notoriously difficult to calculate due to their explosive rate of growth. Recently, a quantum algorithm has been proposed that calculates the two-color Ramsey numbers R(m,n). Here we present results of an experimental implementation of this algorithm based on quantum annealing 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 quantum 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 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.
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.
Critical decision-making issues in science, engineering, and industry are based on combinatorial optimization; however, its application is inherently limited by the NP-hard nature of the problem. A specialized paradigm of analogue quantum computing, quantum annealing (QA), has been proposed to solve these problems by encoding optimization problems into physical energy landscapes and solving them by quantum tunnelling systematically through exploration of solution space. This is a critical review that summarizes the current applications of quantum annealing to combinatorial optimization and includes a theoretical background, hardware designs, algorithm implementation strategies, encoding and embedding schemes, protocols to benchmark quantum annealing, areas of implementation, and links with the quantum algorithms implementation with gate-based hardware and classical solvers. We develop a unified framework, relating adiabatic quantum dynamics, Ising and QUBO models, stoquastic and non-stoquastic Hamiltonians, and diabatic transitions to modern flux-qubit annealers (Chimera, Pegasus, Zephyr topologies), and emergent architectures (Lechner-Hauke-Zoller systems, Rydberg atom platforms), and hybrids of quantum and classical computation. Through our analysis, we find that overhead in embedding and encoding is the largest determinant of the scalability and performance (this is not just the number of qubits). Minor embeddings also usually have a physical qubit count per logical variable of between 5 and 12 qubits, which limits effective problem capacity by 80-92% and, due to chain-breaking errors, compromises the quality of solutions.
Quantum annealing has recently been used to determine the Ramsey numbers R(m,2) for 3 < m < 9 and R(3,3) [Bian et al. (2013) PRL 111, 130505]. This was greatly celebrated as the largest experimental implementation of an adiabatic evolution algorithm to that date. However, in that computation, more than 66% of the qubits used were auxiliary qubits, so the sizes of the Ramsey number Hamiltonians used were tremendously smaller than the full 128-qubit capacity of the device used. The reason these auxiliary qubits were needed was because the best quantum annealing devices at the time (and still now) cannot implement multi-qubit interactions beyond 2-qubit interactions, and they are also limited in their capacity for 2-qubit interactions. We present a method which allows the full qubit capacity of a quantum annealing device to be used, by reducing multi-qubit and 2-qubit interactions. With our method, the device used in the 2013 Ramsey number quantum computation could have determined R(16,2) and R(4,3) with under 10 minutes of runtime.
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.