There is no an exact solution to three-dimensional (3D) finite-size Ising model (referred to as the Ising model hereafter for simplicity) and even two-dimensional (2D) Ising model with non-zero external field to our knowledge. Here by using an elementary but rigorous method, we obtain an exact solution to the partition function of the Ising model with
N lattice sites. It is a sum of
2N exponential functions and holds for
D-dimensional (
D=1,2,3,...) Ising model with or without the external field. This solution provides a new insight into the problem of the Ising model and the related difficulties, and new understanding of the classic exact solutions for one-dimensional (1D) (Kramers and Wannier, 1941) or 2D Ising model (Onsager, 1944). With this solution, the specific heat and magnetisation of a simple 3D Ising model are calculated, which are consistent with the results from experiments and/or numerical simulations. Furthermore, the solution here and the related approaches, can also be available to other models like the percolation and/or the Potts model.
Ising machines are being developed as an efficient computing alternative for solving combinatorial optimization problems. Ising machines solve a combinatorial optimization problem by transforming it into a data structure called an Ising model. However, Ising machines have hardware limitations and hence the input Ising model coefficients must be limited to a certain range. Existing coefficient bit-width reducing methods do not guarantee that the output solution is optimum without adding too many extra spins. In this paper, we propose a new bit-width reducing method for Ising models. In this method, we first partition an original Ising model into several Ising models, each of which has the same graph topology, and the coefficient bit-width is reduced to be dealt with by the target Ising machine. Next, we obtain a ground-state solution for every partitioned Ising model. At that time, we can theoretically prove that, if all the partitioned Ising models have a common ground-state solution, it also gives a ground-state solution for the original non-bit-width-reduced Ising model. Experimental results demonstrate that the theorem is empirically true.
Various methods for formulating practical combinatorial optimization problems as an Ising model have been developed, expanding real-world applications of Ising machines. Nevertheless, the usability of Ising machines remains limited since the formulation process involves complicated tasks, such as fine-tuning parameters or selecting options, which can be challenging for non-expert users. Moreover, the existing formulation technique is often insufficient to design an Ising model that facilitates efficient solution search by Ising machines. This paper proposes a novel approach in which the Boltzmann machine, an energy-based machine learning model, autonomously optimizes the Ising model based on solutions obtained from an Ising machine. The proposed combinatorial optimization flow iteratively alternates between solution search by the Ising machine and model refinement by the Boltzmann machine, gradually improving solution quality without manual adjustments of parameters or options. Experimental results demonstrate that the proposed method consistently outperforms the standalone Ising machine in several benchmark problems, achieving superior solution quality under a comparable computational budget.