Annealing machines such as quantum annealing machines and semiconductor-based annealing machines have been attracting attention as an efficient computing alternative for solving combinatorial optimization problems. They solve original combinatorial optimization problems by transforming them into a data structure called an Ising model. At that time, the bit-widths of the coefficients of the Ising model have to be kept within the range that an annealing machine can deal with. However, by reducing the Ising-model bit-widths, its minimum energy state, or ground state, may become different from that of the original one, and hence the targeted combinatorial optimization problem cannot be well solved. This paper proposes an effective method for reducing Ising model's bit-widths. The proposed method is composed of two processes: First, given an Ising model with large coefficient bit-widths, the shift method is applied to reduce its bit-widths roughly. Second, the spin-adding method is applied to further reduce its bit-widths to those that annealing machines can deal with. Without adding too many extra spins, we efficiently reduce the coefficient bit-widths of the original Ising model. Furthermore, the ground state before and after reducing the coefficient bit-widths is not much changed in most of the practical cases. Experimental evaluations demonstrate the effectiveness of the proposed method, compared to existing methods.
Annealing machines such as quantum annealing machines and semiconductor-based annealing machines have been attracting attention as an efficient computing alternative for solving combinatorial optimization problems. They solve original combinatorial optimization problems by transforming them into a data structure called an Ising model. At that time, the bit-widths of the coefficients of the Ising model have to be kept within the range that an annealing machine can deal with. However, by reducing the Ising-model bit-widths, its minimum energy state, or ground state, may become different from that of the original one, and hence the targeted combinatorial optimization problem cannot be well solved. This paper proposes an effective method for reducing Ising model's bit-widths. The proposed method is composed of two processes: First, given an Ising model with large coefficient bit-widths, the shift method is applied to reduce its bit-widths roughly. Second, the spin-adding method is applied to further reduce its bit-widths to those that annealing machines can deal with. Without adding too many extra spins, we efficiently reduce the coefficient bit-widths of the original Ising model. Furthermore, the ground state before and after reducing the coefficient bit-widths is not much changed in most of the practical cases. Experimental evaluations demonstrate the effectiveness of the proposed method, compared to existing methods.
Ising machines such as quantum annealing machines and semiconductor-based annealing machines can solve various combinatorial optimization problems very efficiently by transforming it into a data structure called an Ising model. At that time, the bit-widths of the coefficients of the Ising model have to be kept within the range that an Ising machine can deal with. However, by reducing the Ising-model bit-widths, its minimum energy state, or ground state, may become different from that of the original one and hence the targeted combinatorial optimization problem cannot be well solved. This paper proposes an effective method for reducing Ising-model bit-widths. The proposed method is composed of the two processes: First, given an Ising model with large coefficient bit-widths, the shift method is applied to reduce its bit-widths roughly. Second, the spin-adding method is applied to further reduce its bit-widths to those that Ising machines can deal with. Without adding too many extra spins, we efficiently reduce the coefficient bit-widths of the original Ising model. Furthermore, the ground state before and after reducing the coefficient bit-widths is not much changed in most of the practical cases. Experimental evaluations demonstrate the effectiveness of the proposed method, compared to existing methods.
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.
Annealing machines have been developed as non-von Neumann computers aimed at solving combinatorial optimization problems efficiently. To use annealing machines for solving combinatorial optimization problems, we have to represent the objective function and constraints by an Ising model, which is a theoretical model in statistical physics. Further, it is necessary to transform the Ising model according to the hardware limitations. In the transformation, the process of effectively reducing the bit-widths of coefficients in the Ising model has hardly been studied so far. Thus, when we consider the Ising model with a large bit-width, a naive method, which means right bit-shift, has to be applied. Since it is expected that obtaining highly accurate solutions is difficult by the naive method, it is necessary to construct a method for efficiently reducing the bit-width. This article proposes methods for reducing the bit-widths of interaction and external magnetic field coefficients in the Ising model and proves that the reduction gives theoretically the same ground state of the original Ising model. The experimental evaluations also demonstrate the effectiveness of our proposed methods.
Many consumer applications can be formulated as combinatorial optimization problems (COPs) and Ising machines can efficiently solve them, once they are transformed into Ising models. However, Ising machines have a limitation on the coefficient bit width of the input Ising model, requiring a reduction of these coefficient bit widths. In this paper, we propose a bit-width reduction method based on a topologically isomorphic n-partition. The proposed method first calculates the number, n, of partitions required to satisfy the bit-width limit, and then partitions the original Ising model into n topologically isomorphic Ising models. Furthermore, we prove that, if these n Ising models share a common ground state, that state is also the ground state of the original Ising model. Experimental results confirm the empirical correctness of the theorem and demonstrate the effectiveness of the proposed method.
Combinatorial optimization problems with a large solution space are difficult to solve just using von Neumann computers. Ising machines or annealing machines have been developed to tackle these problems as a promising Non-von Neumann computer. In order to use these annealing machines, every combinatorial optimization problem is mapped onto the physical Ising model, which consists of spins, interactions between them, and their external magnetic fields. Then the annealing machines operate so as to search the ground state of the physical Ising model, which corresponds to the optimal solution of the original combinatorial optimization problem. A combinatorial optimization problem can be firstly described by an ideal fully-connected Ising model but it is very hard to embed it onto the physical Ising model topology of a particular annealing machine, which causes one of the largest issues in annealing machines. In this paper, we propose a fully-connected Ising model embedding method targeting for CMOS annealing machine. The key idea is that the proposed method replicates every logical spin in a fully-connected Ising model and embeds each logical spin onto the physical spins with the same chain length. Experimental results through an actual combinatorial problem show that the proposed method obtains spin embeddings superior to the conventional de facto standard method, in terms of the embedding time and the probability of obtaining a feasible solution.
Ising machines have attracted attention as fast and high-precision solvers for combinatorial optimization problems, which are often formulated as ground-state (lowest-energy) search problems of the Ising model. Since the bit-width of coefficients on Ising machines is limited, the Ising model must be transformed into a bit-width reduced (BWR) Ising model which reduces the coefficients. In a bit-width reduction method, which adds auxiliary spins, the ground state of the BWR Ising model is theoretically the same as the Ising model before bit-width reduction (original Ising model). However, the dynamical process of the BWR Ising model towards the ground state remains to be elucidated. Therefore, using simulated annealing (SA), we compared the dynamical processes of these models. The result shows that the dynamical process differs significantly by model. Analysis from the viewpoint of statistical mechanics found that the BWR Ising model has two characteristic properties: an effective temperature and a slow relaxation. It is suggested that these properties alter the effective temperature and spin flip probability in the BWR Ising model, contributing to the dynamical differences. Therefore, to obtain the same dynamical process as the original Ising model, we proposed SA parameters for the BWR Ising model. We demonstrated the proposed SA parameter using a square lattice Ising model, in which the interaction and magnetic field were set uniformly to the same positive values or randomly. As experimental evaluations, the dynamical process of the BWR Ising model became closer to that of the original Ising model.