On connection among quantum-inspired algorithms of the ising model

Various combinatorial optimization problems can be reduced to find the minimizer of an Ising model without external fields. This Ising problem is NP-hard and discrete. It is an intellectual challenge to develop algorithms for solving the problem. Over the past decades, many quantum and classical computations have been proposed from physical, mathematical or computational views for finding the minimizer of the Ising model, such as quantum annealing, coherent Ising machine, simulated annealing, adiabatic Hamiltonian systems, etc.. Especially, some of them can be described by differential equations called quantum-inspired algorithms, which use the signum vector of a solution of the differential equations to approach the minimizer of the Ising model. However, the mathematical principle why the quantum-inspired algorithms can work, to the best of our knowledge, is far from being well understood.In this paper, using Morse's theory we reveal the mathematical principle of these quantum-inspired algorithms for the Ising model. It shows that the Ising problem can be designed by minimizing a smooth function, and those quantum-inspired algorithms are to find a global minimum point of the smooth function. In view of this mathematical principle, it can be proved that several known quantum-inspired algorithms: coherent Ising machine, Kerr-nonlinear parametric oscillators and simulated bifurcation algorithm, can reach the minimizers of the Ising model under suitable conditions. Moreover, we discuss the uniqueness of minimizers for the Ising problem in some senses, and give a sufficient condition to guarantee that the Ising model has a unique minimizer, that is, there is only a pair of minimizers with opposite signs.