Ising machines show promise as ultrafast hardware for optimizations encoded in Ising Hamiltonians but fall short in terms of success rate and performance scaling. Here, we propose a novel Ising machine that exploits the three-dimensional nature of nonlinear polarization oscillators to counteract these limitations. Based on the evolution of the optical polarization in third-order nonlinear media, the high-dimensional machine reaches the Ising ground state by the mechanism of "dimensional collapse": the dynamics on the Poincaré sphere undergoes a self-induced collapse into polarization fixed points mapping an Ising spin. Collapse from a spherical to a binary spin occurs when the polarization oscillator experiences iterative loss and anisotropic feedback. The photonic setup consists of polarization modulated pulses in a χ^{(3)} crystal subject to measurement and feedback. We numerically demonstrate the polarization machine achieves enhanced success probability on benchmark graphs and an exponential improvement in performance scaling with respect to coherent Ising machines due to its high-dimensional operation. The proposed Ising machine paves the way for a new class of Ising solvers with enhanced computing capabilities.
Ising machines show promise as ultrafast hardware for optimizations encoded in Ising Hamiltonians but fall short in terms of success rate and performance scaling. Here, we propose a novel Ising machine that exploits the three-dimensional nature of nonlinear polarization oscillators to counteract these limitations. Based on the evolution of the optical polarization in third-order nonlinear media, the high-dimensional machine reaches the Ising ground state by the mechanism of "dimensional collapse": the dynamics on the Poincaré sphere undergoes a self-induced collapse into polarization fixed points mapping an Ising spin. Collapse from a spherical to a binary spin occurs when the polarization oscillator experiences iterative loss and anisotropic feedback. The photonic setup consists of polarization modulated pulses in a χ^{(3)} crystal subject to measurement and feedback. We numerically demonstrate the polarization machine achieves enhanced success probability on benchmark graphs and an exponential improvement in performance scaling with respect to coherent Ising machines due to its high-dimensional operation. The proposed Ising machine paves the way for a new class of Ising solvers with enhanced computing capabilities.
Time-multiplexed networks of degenerate optical parametric oscillators have demonstrated remarkable success in simulating coupled Ising spins, thus providing a promising route to solving complex combinatorial optimization problems. In these systems $\unicode{x2014}$ referred to as coherent Ising machines $\unicode{x2014}$ the spins are encoded in the phases of the oscillators, and spin states are measured at the system output using phase-sensitive techniques. Here, we present an experimental demonstration of a conceptually new optical Ising machine based upon spontaneous polarization symmetry breaking within a coherently-driven optical fibre Kerr nonlinear resonator. In our scheme, the spin states are encoded using polarization, which allows the state of the network to be robustly read out using straightforward intensity measurements. Furthermore, by operating within a recently-discovered regime where the interplay between nonlinearity and topology fundamentally safeguards the system's symmetry, we ensure that our spins evolve without unwanted biases. This enables continuous Ising machine trials at optical data rates for up to an hour without resetting or manual adjustments. With an all-fibre implementation that relies solely on standard telecommunications components, we believe our work paves the way for substantial advances in the performance and stability of coherent optical Ising machines for applications ranging from financial modeling to drug discovery and machine learning.
Time-multiplexed networks of degenerate optical parametric oscillators have demonstrated remarkable success in simulating coupled Ising spins, thus providing a promising route to solving complex combinatorial optimization problems. In these systems, referred to as coherent Ising machines, spins are encoded in the oscillator phases, and measured at the system output using phase-sensitive techniques, making intricate phase stabilization necessary. Here, we introduce an optical Ising machine based on spontaneous polarization symmetry breaking in a coherently driven fiber Kerr nonlinear resonator. In our architecture, the spins are encoded in the polarization state, allowing robust, all-intensity readout with off-the-shelf telecom components. By operating in a newly-discovered regime where nonlinearity and topology lock the system's symmetry, we eliminate drift and bias, enabling uninterrupted Ising trials at optical speeds for over an hour, without manual intervention. This all-fiber platform not only simplifies the hardware but also opens a path to more stable, high-throughput coherent optical optimization devices for applications from finance to drug design and beyond.
Many tasks in our modern life, such as planning an efficient travel, image processing and optimizing integrated circuit design, are modeled as complex combinatorial optimization problems with binary variables. Such problems can be mapped to finding a ground state of the Ising Hamiltonian, thus various physical systems have been studied to emulate and solve this Ising problem. Recently, networks of mutually injected optical oscillators, called coherent Ising machines, have been developed as promising solvers for the problem, benefiting from programmability, scalability and room temperature operation. Here, we report a 16-bit coherent Ising machine based on a network of time-division-multiplexed femtosecond degenerate optical parametric oscillators. The system experimentally gives more than 99.6% of success rates for one-dimensional Ising ring and nondeterministic polynomial-time (NP) hard instances. The experimental and numerical results indicate that gradual pumping of the network combined with multiple spectral and temporal modes of the femtosecond pulses can improve the computational performance of the Ising machine, offering a new path for tackling larger and more complex instances.
Classical and quantum systems are used to simulate the Ising Hamiltonian, an essential component in large-scale optimization and machine learning. However, as the system size increases, devices like quantum annealers and coherent Ising machines face an exponential drop in their success rate. Here, we introduce a novel approach involving high-dimensional embeddings of the Ising Hamiltonian and a technique called "dimensional annealing" to counteract the decrease in performance. This approach leads to an exponential improvement in the success rate and other performance metrics, slowing down the decline in performance as the system size grows. A thorough examination of convergence dynamics in high-performance computing validates the new methodology. Additionally, we suggest practical implementations using technologies like coherent Ising machines, all-optical systems, and hybrid digital systems. The proposed hyperscaling heuristics can also be applied to other quantum or classical Ising devices by adjusting parameters such as nonlinear gain, loss, and nonlocal couplings.
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.
Abstract Optical Ising machines with two-body interactions have shown potential in solving combinatorial optimization problems which are extremely hard to solve with digital computers. Yet, some physical systems cannot be properly described by only two-body interactions. Here, we propose and demonstrate a nonlinear optics approach to emulate Ising machines containing many spins (up to a million in the absence of optical imperfections) and with tailored all-to-all two and four-body interactions. Our approach employs a spatial light modulator to encode and control the spins in the form of the binary-phase values, and emulates the high-order interaction with frequency conversion in a nonlinear crystal. By implementing adaptive feedback, the system can be evolved into effective spin configurations that well-approximate the ground-states of Ising Hamiltonians with all-to-all connected many-body interactions. Our technique could serve as a tool to probe complex, many-body physics and give rise to exciting applications in big-data optimization, computing, and analytics.