1 research outputs found
Mathematical Mechanism on Dynamical System Algorithms of the Ising Model
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 -spin and -spin Ising models is presented as application.Comment: 39 pages, 2 figures(including 8 sub-figures