Photonic Ising machines are a recent and promising non-von Neuman architecture to efficiently solve large NP-hard optimization problems. This is done by letting spins evolve to the ground state of the Ising system. However, the Ising machine does not always relax to this ground state and can get trapped in local minima, resulting in suboptimal solutions. Here, we show that Ising-encoded optimization problems implemented on analog Ising machines can be divided in three difficulty classes and we propose an annealing scheme to efficiently solve two of them. Furthermore, we show that changing the physical implementation of the Ising machine can cause a problem to change difficulty class, rendering it potentially a lot more easy to solve.
Photonic Ising machines are a recent and promising non-von Neuman architecture to efficiently solve large NP-hard optimization problems. This is done by letting spins evolve to the ground state of the Ising system. However, the Ising machine does not always relax to this ground state and can get trapped in local minima, resulting in suboptimal solutions. Here, we show that Ising-encoded optimization problems implemented on analog Ising machines can be divided in three difficulty classes and we propose an annealing scheme to efficiently solve two of them. Furthermore, we show that changing the physical implementation of the Ising machine can cause a problem to change difficulty class, rendering it potentially a lot more easy to solve.
Ising machines are dedicated hardware solvers of NP-hard optimization problems. However, they do not always find the most optimal solution. The probability of finding this optimal solution depends on the problem at hand. Using continuation methods, we show that this is closely linked to how the ground state emerges from other states when a system parameter is changed, i.e. its bifurcation sequence. From this analysis, we can determine the effectiveness of solution schemes. Moreover, we find that the proper choice of implementation of the Ising machine can drastically change this bifurcation sequence and therefore vastly increase the probability of finding the optimal solution. Lastly, we also show that continuation methods themselves can be used directly to solve optimization problems. An Ising machine is a piece of hardware that tries to solve quadratic unconstrained binary optimization problems. The authors explain why some problems are significantly easier to tackle than others using Ising machines and demonstrate that different physical implementations can render some challenging problems a lot easier to solve.
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.
Analog Ising machines are dedicated hardware solvers designed to solve NP hard optimization problems. However, the global optimum is often not found as the system gets stuck in local minima. While several strategies exist to increase the chance of escaping local minima, these methods often require extensive parameter tuning. In this work, we investigate the injection of large noise as a standalone scheme and in combination with annealing to improve the success rate and the time-to-target (TTT) of analog Ising machines for MaxCut problems. We demonstrate that optimizing the noise improves the TTT by several orders of magnitude and renders both approaches competitive with state-of-the-art schemes, such as chaotic amplitude control. Moreover, we are able to predict an optimal noise value based on the problem connectivity, statistics of the coupling matrix and the coupling strength, thereby eliminating the need for costly parameter optimization.
Abstract Solving large-scale Quadratic Unconstrained Binary Optimization (QUBO) problems is critical in various fields, including finance, physics, biology and engineering. However, these problems remain intractable on conventional computing architectures. As a result, alternative solvers—such as Ising Machines (IMs) based on networks of coupled electronic, mechanical or photonic parametric oscillators (POs)—have recently been developed. PO-based Ising Machines (IMs) aim to find the ground state of an Ising Hamiltonian, which encodes the solution to a QUBO problem. However, they rely on an energy minimization process based on gradient descent, making them inherently susceptible to getting trapped in local minima and, as a result, to identifying inaccurate solutions. In this work, we introduce and validate a QUBO solver—the Analog Floquet Solver (AFS)—which enhances the dynamics of PO-based Ising machines (IMs) by leveraging Floquet states that emerge spontaneously in POs coupled to high quality-factor resonances. These states enable the AFS to embed periodic time modulation into its energy minimization process, allowing it to escape local minima during the search for QUBO problems’ solutions. As a result, the AFS significantly increases the likelihood of identifying accurate solutions compared to conventional PO-based IMs. More generally, this work establishes a new paradigm in analog computing that can be physically implemented using existing technologies across a variety of physical domains.
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.
With increasing computational demands, the importance of efficient computer architectures rises. Photonic Ising machines offer a promising approach to address complex binary optimization problems by leveraging the speed and bandwidth inherent in photonic systems. When the cost function of an optimization problem can be mapped to the energy of an Ising problem, the ground state of the latter will represent the optimal solution. We conduct a stability analysis on this ground state of various benchmark problems. This analysis reveals that some benchmark problems are manageable for all Ising machine implementations. However, there are other problems for which the probability of achieving the ground state depends on the physical implementation. Our analysis shows the reasons for this advantage, leading us to formulate strategies to further enhance Ising machines and pave the way for future research. In future work, we will investigate if photonic Ising machines utilizing a different nonlinearity possess a distinct advantage in this context.