Combinatorial optimization problems can be mapped onto Ising models, and their ground state is generally difficult to find. A lot of heuristics for these problems have been proposed, and one promising approach is to use continuous variables. In recent years, one such algorithm has been implemented by using parametric oscillators known as coherent Ising machines. Although these algorithms have been confirmed to have high performance through many experiments, unlike other familiar algorithms such as simulated annealing, their computational ability has not been fully investigated. In this paper, we propose a simple heuristic based on continuous variables whose static and dynamical properties are easy to investigate. Through the analyses of the proposed algorithm, we find that many local minima in the early stage of the optimization and bifurcation delay reduce its performance in a certain class of Ising models.
Combinatorial optimization problems can be mapped onto Ising models, and their ground state is generally difficult to find. A lot of heuristics for these problems have been proposed, and one promising approach is to use continuous variables. In recent years, one such algorithm has been implemented by using parametric oscillators known as coherent Ising machines. Although these algorithms have been confirmed to have high performance through many experiments, unlike other familiar algorithms such as simulated annealing, their computational ability has not been fully investigated. In this paper, we propose a simple heuristic based on continuous variables whose static and dynamical properties are easy to investigate. Through the analyses of the proposed algorithm, we find that many local minima in the early stage of the optimization and bifurcation delay reduce its performance in a certain class of Ising models.
Quickly obtaining optimal solutions of combinatorial optimization problems has tremendous value but is extremely difficult. Thus, various kinds of machines specially designed for combinatorial optimization have recently been proposed and developed. Toward the realization of higher-performance machines, here, we propose an algorithm based on classical mechanics, which is obtained by modifying a previously proposed algorithm called simulated bifurcation. Our proposed algorithm allows us to achieve not only high speed by parallel computing but also high solution accuracy for problems with up to one million binary variables. Benchmarking shows that our machine based on the algorithm achieves high performance compared to recently developed machines, including a quantum annealer using a superconducting circuit, a coherent Ising machine using a laser, and digital processors based on various algorithms. Thus, high-performance combinatorial optimization is realized by massively parallel implementations of the proposed algorithm based on classical mechanics.
Simulated annealing (SA) attracts more attention among classical heuristic algorithms because the solution of the combinatorial optimization problem can be naturally mapped to the ground state of the Ising Hamiltonian. However, in practical implementation, the annealing process cannot be arbitrarily slow and hence, it may deviate from the expected stationary Boltzmann distribution and become trapped in a local energy minimum. To overcome this problem, this paper proposes a heuristic search algorithm by expanding search space from a Markov chain to a recursive depth limited tree based on SA, where the parent and child nodes represent the current and future spin states. At each iteration, the algorithm will select the best near-optimal solution within the feasible search space by exploring along the tree in the sense of `look ahead'. Furthermore, motivated by coherent Ising machine (CIM), we relax the discrete representation of spin states to continuous representation with a regularization term and utilize the reduced dynamics of the oscillators to explore the surrounding neighborhood of the selected tree nodes. We tested our algorithm on a representative NP-hard problem (MAX-CUT) to illustrate the effectiveness of this algorithm compared to semi-definite programming (SDP), SA, and simulated CIM. Our results show that above the primal heuristics SA and CIM, our high-level tree search strategy is able to provide solutions within fewer epochs for Ising formulated NP-optimization problems.
Various combinatorial optimization NP-hard problems can be reduced to finding the minimizer of an Ising model, which is a discrete mathematical model. It is an intellectual challenge to develop some mathematical tools or algorithms for solving the Ising model. Over the past decades, some continuous approaches or algorithms have been proposed from physical, mathematical or computational views for optimizing the Ising model such as quantum annealing, the coherent Ising machine, simulated annealing, adiabatic Hamiltonian systems, etc.. However, the mathematical principle of these algorithms is far from being understood. In this paper, we reveal the mathematical mechanism of dynamical system algorithms for the Ising model by Morse theory and variational methods. We prove that the dynamical system algorithms can be designed to minimize a continuous function whose local minimum points give all the candidates of the Ising model and the global minimum gives the minimizer of Ising problem. Using this mathematical mechanism, we can easily understand several dynamical system algorithms of the Ising model such as the coherent Ising machine, the Kerr-nonlinear parametric oscillators and the simulated bifurcation algorithm. Furthermore, motivated by the works of C. Conley, we study transit and capture properties of the simulated bifurcation algorithm to explain its convergence by the low energy transit and capture in celestial mechanics. A detailed discussion on 2-spin and 3-spin Ising models is presented as application.
Combinatorial optimization problems are ubiquitous in industrial applications. However, finding optimal or close-to-optimal solutions can often be extremely hard. Because some of these problems can be mapped to the ground-state search of the Ising model, tremendous effort has been devoted to developing solvers for Ising-type problems over the past decades. Recent advances in controlling and manipulating both quantum and classical systems have enabled novel computing paradigms such as quantum simulators and coherent Ising machines to tackle hard optimization problems. Here, we examine and benchmark several physics-inspired optimization algorithms, including coherent Ising machines, gain-dissipative algorithms, simulated bifurcation machines, and Hopfield neural networks, which we collectively refer to as stochastic driven nonlinear dynamical systems. Most importantly, we benchmark these algorithms against random Ising problems with planted solutions and compare them to simulated annealing as a baseline leveraging the same software stack for all solvers. We further study how different numerical integration techniques and graph connectivity affect performance. This work provides an overview of a diverse set of new optimization paradigms.
Nonlinear dynamical systems with continuous variables can be used for solving combinatorial optimization problems with discrete variables. Numerical simulations of them are also useful as heuristic algorithms with a desirable property, namely, parallelizability, which allows us to execute them in a massively parallel manner, leading to ultrafast performance. However, the dynamical-system approaches with continuous variables are usually less accurate than conventional approaches with discrete variables such as simulated annealing. To improve the solution accuracy of a quantum-inspired algorithm called simulated bifurcation (SB), which was found from classical simulation of a quantum nonlinear oscillator network exhibiting quantum bifurcation, here we generalize it by introducing nonlinear control of individual bifurcation parameters and show that the generalized SB (GSB) can achieve surprisingly high performance, namely, almost 100% success probabilities for some large-scale problems. As a result, the time to solution for a 2000-variable problem is shortened to 10 ms by a GSB-based machine, which is 2 orders of magnitude shorter than the best known value, 1.3 s, previously obtained by an SB-based machine. To examine the reason for the ultrahigh performance, we investigated chaos in the GSB changing the nonlinear-control strength and found that the dramatic increase of success probabilities happens near the edge of chaos. That is, the GSB can find a solution with high probability by harnessing the edge of chaos. This finding suggests that dynamical-system approaches to combinatorial optimization will be enhanced by harnessing the edge of chaos, opening a broad possibility for physics-inspired approaches to combinatorial optimization.
Simulating physical dynamics to solve hard combinatorial optimization has proven effective for medium- to large-scale problems. The dynamics of such systems is continuous, with no guarantee of finding optimal solutions of the original discrete problem. We investigate the open question of when simulated physical solvers solve discrete optimizations correctly, with a focus on coherent Ising machines (CIMs). Having established the existence of an exact mapping between CIM dynamics and discrete Ising optimization, we report two fundamentally distinct bifurcation behaviors of the Ising dynamics at the first bifurcation point: either all nodal states simultaneously deviate from zero (synchronized bifurcation) or undergo a cascade of such deviations (retarded bifurcation). For synchronized bifurcation, we prove that when the nodal states are uniformly bounded away from the origin, they contain sufficient information for exactly solving the Ising problem. When the exact mapping conditions are violated, subsequent bifurcations become necessary and often cause slow convergence. Inspired by those findings, we devise a trapping-and-correction (TAC) technique to accelerate dynamics-based Ising solvers, including CIMs and simulated bifurcation. TAC takes advantage of early bifurcated "trapped nodes" which maintain their sign throughout the Ising dynamics to reduce computation time effectively. Using problem instances from open benchmark and random Ising models, we validate the superior convergence and accuracy of TAC.