ABSTRACT Combinatorial optimization problems (COPs) are fundamental to numerous applications but remain challenging due to their exponentially large solution spaces and the limitations of conventional computing architectures. Ising machines have emerged as a promising alternative by mapping optimization problems onto the energy minimization of the Ising Hamiltonian. Various hardware implementations, including quantum annealers, digital annealers, and coherent Ising machines, have been explored based on distinct physical mechanisms. More recently, emerging device‐based approaches have gained significant attention for their potential to enable scalable and energy‐efficient optimization. In this review, we present a comprehensive overview of Ising machines from both theoretical and hardware perspectives, with particular emphasis on three emerging‐device‐based platforms: crossbar array‐based, stochastic p‐bit, and oscillator‐based Ising machines. We establish a comparative device‐to‐system perspective that traces how distinct physical and device properties translate into computational primitives, hardware architectures, and ultimately system‐level scalability. Platform‐specific bottlenecks, including array nonidealities and peripheral overhead in crossbar‐based systems, device variability and probability‐generation overhead in p‐bit systems, and synchronization and coupling‐network complexity in oscillator‐based systems, are quantitatively and qualitatively analyzed. Finally, we discuss future directions toward scalable and energy‐efficient Ising hardware through device‐to‐system co‐optimization across devices, circuits, architectures, and algorithms.
ABSTRACT Combinatorial optimization problems (COPs) are fundamental to numerous applications but remain challenging due to their exponentially large solution spaces and the limitations of conventional computing architectures. Ising machines have emerged as a promising alternative by mapping optimization problems onto the energy minimization of the Ising Hamiltonian. Various hardware implementations, including quantum annealers, digital annealers, and coherent Ising machines, have been explored based on distinct physical mechanisms. More recently, emerging device‐based approaches have gained significant attention for their potential to enable scalable and energy‐efficient optimization. In this review, we present a comprehensive overview of Ising machines from both theoretical and hardware perspectives, with particular emphasis on three emerging‐device‐based platforms: crossbar array‐based, stochastic p‐bit, and oscillator‐based Ising machines. We establish a comparative device‐to‐system perspective that traces how distinct physical and device properties translate into computational primitives, hardware architectures, and ultimately system‐level scalability. Platform‐specific bottlenecks, including array nonidealities and peripheral overhead in crossbar‐based systems, device variability and probability‐generation overhead in p‐bit systems, and synchronization and coupling‐network complexity in oscillator‐based systems, are quantitatively and qualitatively analyzed. Finally, we discuss future directions toward scalable and energy‐efficient Ising hardware through device‐to‐system co‐optimization across devices, circuits, architectures, and algorithms.
Ising machines have received growing interest as efficient and hardware-friendly solvers for combinatorial optimization problems (COPs). They search for the absolute or approximate ground states of the Ising model with a proper annealing process. In contrast to Ising machines built with superconductive or optical circuits, complementary metal-oxide-semiconductor (CMOS) Ising machines offer inexpensive fabrication, high scalability, and easy integration with mainstream semiconductor chips. As low-energy and CMOS-compatible emerging technologies, spintronics and phase-transition devices offer functionalities that can enhance the scalability and sampling performance of Ising machines. In this article, we survey various approaches in the process flow for solving COPs using CMOS, hybrid CMOS-spintronic, and phase-transition devices. First, the methods for formulating COPs as Ising problems and embedding Ising formulations to the topology of the Ising machine are reviewed. Then, Ising machines are classified by their underlying operational principles and reviewed from a perspective of hardware implementation. CMOS solutions are advantageous with denser connectivity, whereas hybrid CMOS-spintronic and phase-transition device-based solutions show great potential in energy efficiency and high performance. Finally, the challenges and prospects are discussed for the Ising formulation, embedding process, and implementation of Ising machines.
Combinatorial optimization problems are central to many challenges in logistics, finance, engineering, and the life sciences, yet they remain among the most computationally demanding. Many of these problems can be mapped onto the Ising model, in which binary spins interact through a network of couplings, and solutions correspond to low-energy, ideally ground-state, spin configurations. Photonic Ising machines have the potential to be fast and energy-efficient heuristic solvers of optimization problems by leveraging the low latency, high bandwidth, and inherent parallelism of optics. However, current photonic implementations remain limited in scalability, connectivity, reconfigurability, and time-to-solution, preventing their use in many practical applications. In this perspective, we examine the current landscape of photonic Ising machines, discuss the challenges and limitations of existing platforms, and identify the scientific and technological advances needed to realize large-scale systems. These developments could establish photonic Ising machines as useful hardware platforms for practical optimization.
Ising machines are hardware solvers which aim to find the absolute or approximate ground states of the Ising model. The Ising model is of fundamental computational interest because it is possible to formulate any problem in the complexity class NP as an Ising problem with only polynomial overhead. A scalable Ising machine that outperforms existing standard digital computers could have a huge impact for practical applications for a wide variety of optimization problems. In this review, we survey the current status of various approaches to constructing Ising machines and explain their underlying operational principles. The types of Ising machines considered here include classical thermal annealers based on technologies such as spintronics, optics, memristors, and digital hardware accelerators; dynamical-systems solvers implemented with optics and electronics; and superconducting-circuit quantum annealers. We compare and contrast their performance using standard metrics such as the ground-state success probability and time-to-solution, give their scaling relations with problem size, and discuss their strengths and weaknesses.
The demand for efficient solvers of complicated combinatorial optimization problems, especially those classified as NP-complete or NP-hard, has recently led to increased exploration of novel computing architectures. One prominent collective state computing paradigm embodied in the so-called Ising machines has recently attracted considerable research attention due to its ability to optimize complex problems with large numbers of interacting variables. Ising model-inspired solvers, thus named due to mathematical similarities to the well-known model from solid-state physics, represent a promising alternative to traditional von Neumann computer architectures due to their high degree of inherent parallelism. While there are many possible physical realizations of Ising solvers, just as there are many possible implementations of any binary computer, photonic Ising machines (PIMs) use primarily optical components for computation, taking advantage of features like lower power consumption, fast calculation speeds, the leveraging of physical optics to perform the calculations themselves, possessing decent scalability and noise tolerance. Photonic computing in the form of PIMs may offer certain computational advantages that are not easily achieved with non-photonic approaches and is nonetheless an altogether fascinating application of photonics to computing. In this review, we provide an overview of Ising machines generally, introducing why they are useful, what types of problems they can tackle, and how different Ising solvers can be compared and benchmarked. We delineate their various operational mechanisms, advantages, and limitations vis-à-vis non-photonic Ising machines. We describe their scalability, interconnectivity, performance, and physical dimensions. As research in PIMs continues to progress, there is a potential that photonic computing could well emerge as a way to handle large and challenging optimization problems across diverse domains. This review serves as a comprehensive resource for researchers and practitioners interested in understanding capabilities and potential of PIMs in addressing such complex optimization problems.
ABSTRACT Many real‐world applications can be categorized as combinatorial optimization problems (COPs), yet their inherent NP‐hardness and exponential computational complexity pose significant barriers to efficient solutions. Designed specifically for solving COPs, the Ising machine (IM) demonstrates superior performance over traditional Von Neumann‐based computing architectures. Among different IM implementations, optical IMs (OIMs) are especially promising due to their inherent high‐speed and energy efficiency properties. Nevertheless, current OIMs are constrained by serial processing architectures, posing significant challenges for solving large‐scale COPs. To address this challenge, we propose and experimentally demonstrate a multiplexed parallel OIM architecture spanning the optical and electrical domains. The system solved a square lattice MAX‐CUT instance with over 100,000 spins and achieved a fourfold speedup with four‐channel parallelism. In addition, channel‐wise calibration mitigated device mismatch and aligned the nonlinear responses across channels, enabling the four‐channel system to maintain solution quality comparable to the single‐channel system across small benchmark problems of varying difficulty. We also explored a method combining hierarchical clustering and noise injection, which identified the known optimal route within the clustering‐constrained search space of the 14‐city TSP. These results demonstrate the potential of the proposed architecture for larger and more complex optimization problems.
Abstract Coherent Ising machines (CIM) have shown promising potential as hardware solvers for tackling combinatorial optimization problems, where discrete variables are encoded onto physical states to form an Ising Hamiltonian. Various platforms, starting from optical parametric oscillators, have been proposed and demonstrated to implement an actual physical CIM. In this review article, we briefly introduce the basic operation principle of CIMs and review physical schemes focusing on MEMS-based nanophotonics.