Recently, purpose-built analog hardware that can efficiently minimize the Ising energy and thereby solve a variety of combinatorial optimization problems has been receiving widespread attention. In this work, we show how multidimensional, vectorial degrees of freedom, that are either naturally present or can be artificially created in such hardware, could strengthen the capability to find optimal solutions to optimization problems. In order to achieve this, we introduce a simple model of soft vector spins that should be implementable on a variety of analog hardware platforms as well as three different dimensional annealing methods which harness the enlarged phase space of the vectorial degrees of freedom to minimize the Ising energy. We perform simulations on different benchmark problems and show that for all dimensional annealing methods we tested, vectorial degrees of freedom improve upon one-dimensional degrees of freedom when it comes to finding the ground state of the Ising model. In particular, we find that this advantage becomes most pronounced for $d \gtrsim 3$ dimensional degrees of freedom, with diminishing returns as the dimension is increased further. Our results could inspire new analog optimization hardware and algorithms that explicitly incorporate the advantage of vectorial degrees of freedom.
Recently, purpose-built analog hardware that can efficiently minimize the Ising energy and thereby solve a variety of combinatorial optimization problems has been receiving widespread attention. In this work, we show how multidimensional, vectorial degrees of freedom, that are either naturally present or can be artificially created in such hardware, could strengthen the capability to find optimal solutions to optimization problems. In order to achieve this, we introduce a simple model of soft vector spins that should be implementable on a variety of analog hardware platforms as well as three different dimensional annealing methods which harness the enlarged phase space of the vectorial degrees of freedom to minimize the Ising energy. We perform simulations on different benchmark problems and show that for all dimensional annealing methods we tested, vectorial degrees of freedom improve upon one-dimensional degrees of freedom when it comes to finding the ground state of the Ising model. In particular, we find that this advantage becomes most pronounced for d≳3 dimensional degrees of freedom, with diminishing returns as the dimension is increased further. Our results could inspire new analog optimization hardware and algorithms that explicitly incorporate the advantage of vectorial degrees of freedom.
This research is concerned with an annealing processor for efficiently solving a combinatorial optimization problem.\nIn order to realize a smart society, optimization of people and thingssuch as transportation networks, power networks, human resource allocationand routes is indispensable. Many of these problems are called combinatorial optimization problems and are known as very important problems in modern society. However, it is known that the combinatorial optimization problem is NP-hard and it is difficult to efficiently solve it with a conventional Neumann computer.\nIn recent years, the miniaturization of the process of manufacturing computer chips is approaching the physical limit. Therefore, especially in the solution by the full search, the improvement in speed due to the calculation speed of the computer cannot be expected. From such a background, dedicated hardware capable of solving a combination optimization problem at high speed and with low power consumption has been attracting attention.\nThe annealing processor is a general-purpose combination optimization solver that utilizes the property that the ground state of the optimization problem expressed by using the Ising model and the optimal solution of the combination optimization problem match. The Ising model is a ferromagnetic model in statistical physics expressed by spins having an upward and downward state, spin-spin interactions occurring between the spins, and an external magnetic field acting on each spin. The Ising model can express a combinatorial optimization problem by expressing a control target and a cost using these parameters. The annealing processor updates the spin state while slowly lowering the temperature parameters based on simulated annealing,which is an optimization method learned from natural phenomena generally called annealing. Eventually, the spin will reach the ground state with a high probability and obtain the optimal solution.\nAnnealing processors are classified into two types, the nearest neighbor type and the fully connected type, depending on the shape of the Ising model simulated on the hardware. In the nearest neighbor type, the coupling between spins is limited to only between adjacent spins, and in the full coupling type, coupling exists between arbitrary spins. The nearest neighbor type realizes high parallel processing by reducing the data spin required by simulated annealing by reducing the proximity spin. In addition, since data transfer is limited to adjacent spins, the scalability is high. However, since the spin-to-spin interaction is sparse, there is a problem that the class of combinatorial optimization problems that can be dealt with is limited, or that the problem requires deformation processing to match the Ising model of hardware. The fully connected type can solve any class of problems as long as the number of spins permits, however it requires connections between arbitrary spins.Therefore, scalability is low. Because there there is a dependency on all spins,the number of spins that can be updated at one time is always limited to only one regardless of the total number of spins. Therefore, it takes time to converge to the ground state.\nIn this research, we propose a method of increasing the number of spins for this annealing processor by using (1) a spatially-expanded time-division processing mechanism architecture based on a loosely-coupled type, (2) An algorithm for applying a fully coupled Ising network to a more complex Ising network using the nearest neighbor Ising network and an annealing method that can be updated in parallel, (3)We conducted a study on the architecture of a fully coupled annealing processor for parallel updating annealing.\nAs described above, the research described in this paper contributes to the improvement of the efficiency of the annealing processor for solving the large-scale combinatorial optimization problem from both the fundamental algorithm and the hardware architecture. I hope that this research will solve the combinatorial optimization problem that is expected to increase in demand in the future and contribute to the development of human society.
From condensed matter to quantum chromodynamics, multidimensional spins are a fundamental paradigm, with a pivotal role in combinatorial optimization and machine learning. Machines formed by coupled parametric oscillators can simulate spin models, but only for Ising or low-dimensional spins. Currently, machines implementing arbitrary dimensions remain a challenge. Here, we introduce and validate a hyperspin machine to simulate multidimensional continuous spin models. We realize high-dimensional spins by pumping groups of parametric oscillators, and show that the hyperspin machine finds to a very good approximation the ground state of complex graphs. The hyperspin machine can interpolate between different dimensions by tuning the coupling topology, a strategy that we call "dimensional annealing". When interpolating between the XY and the Ising model, the dimensional annealing substantially increases the success probability compared to conventional Ising simulators. Hyperspin machines are a new computational model for combinatorial optimization. They can be realized by off-the-shelf hardware for ultrafast, large-scale applications in classical and quantum computing, condensed-matter physics, and fundamental studies.
Combinatorial optimization problems are crucial for widespread applications but remain difficult to solve on a large scale with conventional hardware. Novel optical platforms, known as coherent or photonic Ising machines, are attracting considerable attention as accelerators on optimization tasks formulable as Ising models. Annealing is a well-known technique based on adiabatic evolution for finding optimal solutions in classical and quantum systems made by atoms, electrons, or photons. Although various Ising machines employ annealing in some form, adiabatic computing on optical settings has been only partially investigated. Here, we realize the adiabatic evolution of frustrated Ising models with 100 spins programmed by spatial light modulation. We use holographic and optical control to change the spin couplings adiabatically, and exploit experimental noise to explore the energy landscape. Annealing enhances the convergence to the Ising ground state and allows to find the problem solution with probability close to unity. Our results demonstrate a photonic scheme for combinatorial optimization in analogy with adiabatic quantum algorithms and classical annealing methods but enforced by optical vector-matrix multiplications and scalable photonic technology.
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.
Combinatorial optimization problems ubiquitously appear in AI applications, e.g. machine learning, operational planning, drug discovery, etc. Yet, their NP-hardness makes them notoriously difficult to solve on present computers. To address this problem, new hardware architectures [1, 4], called annealers, have emerged in recent years. Annealers exploit the fact that combinatorial optimization problems can be mapped to the ground state search of an Ising model, where the combination of N spins (σ- ∈ f-1}. x=1, ...,N) with lowest Ising energy represents the optimal solution to the problem (Fig. 7.3.1).
Abstract Combinatorial optimization problems and ground state problems of spin glasses are crucial in various fields of science and technology. However, they often belong to the computational class of NP-hard, presenting significant computational challenges. Traditional algorithms inspired by statistical physics like simulated annealing have been widely adopted. Recently, advancements in Ising machines, such as quantum annealers and coherent Ising machines, offer new paradigms for solving these problems efficiently by embedding them into the analog evolution of nonlinear dynamical systems. However, existing dynamics-based algorithms often suffer from low convergence rates and local minima traps. In this work, we introduce the dual mean-field dynamics into Ising machines. The approach integrates the gradient force and the transverse force into the dynamics of Ising machines in solving combinatorial optimization problems, making it easier for the system to jump out of the local minimums and allowing the dynamics to explore wider in configuration space. We conduct extensive numerical experiments using the Sherrington–Kirkpatrick spin glass up to 10 000 spins and the maximum cut problems with the standard G-set benchmarks. The numerical results demonstrate that our dual mean-field dynamics approach enhances the performance of base Ising machines, providing a more effective solution for large-scale combinatorial optimization problems.