Large-scale coherent Ising machine based on optoelectronic parametric oscillator

A monolithically integrated optical Ising machine

The growing demand for enhanced computational power and energy efficiency has driven the development of optical Ising machines for solving combinatorial optimization problems. However, existing implementations face challenges in integration density and energy efficiency. Here, we propose a monolithically integrated four-spin Ising machine based on optoelectronic coupled oscillators. This system integrates a custom-designed Mach-Zehnder interferometer (MZI) symmetric matrix with a high-efficiency optical-electrical coupled (OEC) nonlinear unit. The OEC unit has an ultra-compact 0.01 mm² footprint and achieves a power efficiency of 4 mW per unit, ensuring scalability. The reconfigurable real-valued coupling matrix achieves a mean fidelity of 0.986. The spin evolution time is measured as 150 ns, with a 1.71 ns round-trip time confirmed through bandwidth measurements. The system successfully finds ground states for various four-spin Ising problems, demonstrating its effectiveness. This work represents a significant step toward monolithic integration of all-optical physical annealing systems, minimizing footprint, power consumption, and convergence time.

Impacts of graph structure on the computational properties of oscillator Ising machines

Many combinatorial optimization problems (COPs) can be mapped to Ising Hamiltonians. Oscillator Ising machines (OIMs) are built to minimize the Ising Hamiltonians, thereby indirectly solving the COPs. Specifically, the dynamics of an OIM evolve in the direction of decreasing value of the Ising Hamiltonian, and its state converges to a steady state, known as an asymptotically stable equilibrium point. Such an equilibrium point represents a candidate solution of the underlying COP. Although OIMs show great potential in solving COPs, they, like many heuristic algorithms, are prone to getting stuck in local minima. Even for trivial problems, e.g., unfrustrated Ising Hamiltonians, an OIM may get trapped in local minima if its parameters and its initial condition are not properly selected. In this work, we examine the impacts of the graph structure underlying an OIM on the computational properties of the OIM. In particular, we identify graph structures of OIMs for which the parameters of the OIMs can be properly designed such that only the equilibrium points representing global minima are asymptotically stable and all other equilibrium points are unstable. Thus, an OIM with such a graph structure, possibly with a small perturbation, can converge to the globally optimal solutions from any initial condition.

Fully Programmable Spatial Photonic Ising Machine by Focal Plane Division

Ising machines are an emerging class of hardware that promises ultrafast and energy-efficient solutions to NP-hard combinatorial optimization problems. Spatial photonic Ising machines (SPIMs) exploit optical computing in free space to accelerate the computation, showcasing parallelism, scalability, and low power consumption. However, current SPIMs can implement only a restricted class of problems. This partial programmability is a critical limitation that hampers their benchmark. Achieving full programmability of the device while preserving its scalability is an open challenge. Here, we report a fully programmable SPIM achieved through a novel operation method based on the division of the focal plane. In our scheme, a general Ising problem is decomposed into a set of Mattis Hamiltonians, whose energies are simultaneously computed optically by measuring the intensity on different regions of the camera sensor. Exploiting this concept, we experimentally demonstrate the computation with high success probability of ground-state solutions of up to 32-spin Ising models on unweighted maximum cut graphs with and without ferromagnetic bias. Simulations of the hardware prove a favorable scaling of the accuracy with the number of spin. Our fully programmable SPIM enables the implementation of many quadratic unconstrained binary optimization problems, further establishing SPIMs as a leading paradigm in non-von Neumann hardware.

An Approach to Reduce Tuning Sensitivity in the PIC-Based Optoelectronic Oscillator by Controlling the Phase Shift in Its Feedback Loop

Radio photonic technologies have emerged as a promising solution for addressing microwave frequency synthesis challenges in current and future communication and sensing systems. One particularly effective approach is the optoelectronic oscillator (OEO), a simple and cost-effective electro-optical system. The OEO can generate microwave signals with low phase noise and high oscillation frequencies, often outperforming traditional electrical methods. However, a notable disadvantage of the OEO compared to conventional signal generation methods is its significant frequency tuning step. This paper presents a novel approach for continuously controlling the output frequency of an optoelectronic oscillator (OEO) based on integrated photonics. This is achieved by tuning an integrated optical delay line within a feedback loop. The analytical model developed in this study calculates the OEO's output frequency while accounting for nonlinear errors, enabling the consideration of various control schemes. Specifically, this study examines delay lines based on the Mach-Zehnder interferometer and microring resonators, which can be controlled by either the thermo-optic or electro-optic effect. To evaluate the model, we conducted numerical simulations using Ansys Lumerical software. The OEO that utilized an MRR-based electro-optical delay line demonstrated a tuning sensitivity of 174.5 MHz/V. The calculated frequency tuning sensitivity was as low as 6.98 kHz when utilizing the precision digital-to-analog converter with a minimum output voltage step of 40 μV. The proposed approach to controlling the frequency of the OEO can be implemented using discrete optical components; however, this approach restricts the minimum frequency tuning sensitivity. It provides an additional degree of freedom for frequency tuning within the OEO's operating range, which is ultimately limited by the amplitude-frequency characteristic of the notch filter. Thus, the proposed approach opens up new opportunities for increasing the accuracy and flexibility in generating microwave signals, which can be significant for various communications and radio engineering applications.

A control theoretic analysis of oscillator Ising machines

This work advances the understanding of oscillator Ising machines (OIMs) as a nonlinear dynamic system for solving computationally hard problems. Specifically, we classify the infinite number of all possible equilibrium points of an OIM, including non-0/π ones, into three types based on their structural stability properties. We then employ the stability analysis techniques from control theory to analyze the stability property of all possible equilibrium points and obtain the necessary and sufficient condition for their stability. As a result of these analytical results, we establish, for the first time, the threshold of the binarization in terms of the coupling strength and strength of the second harmonic signal. Furthermore, we provide an estimate of the domain of attraction of each asymptotically stable equilibrium point by employing the Lyapunov stability theory. Finally, we illustrate our theoretical conclusions by numerical simulation.

General spatial photonic Ising machine based on the interaction matrix eigendecomposition method

The spatial photonic Ising machine has achieved remarkable advancements in solving combinatorial optimization problems. However, it still remains a huge challenge to flexibly map an arbitrary problem to the Ising model. In this paper, we propose a general spatial photonic Ising machine based on the interaction matrix eigendecomposition method. The arbitrary interaction matrix can be configured in the two-dimensional Fourier transformation based spatial photonic Ising model by using values generated by matrix eigendecomposition. The error in the structural representation of the Hamiltonian decreases substantially with the growing number of eigenvalues utilized to form the Ising machine. In combination with the optimization algorithm, as low as ∼65% of the eigenvalues are required by intensity modulation to guarantee the best probability of optimal solution for a 20-vertex graph Max-cut problem, and this percentage decreases to below ∼20% for near-zero probability. The 4-spin experiments and error analysis demonstrate the Hamiltonian linear mapping and ergodic optimization. Our work provides a viable approach for spatial photonic Ising machines to solve arbitrary combinatorial optimization problems with the help of the multi-dimensional optical property.

Parametric Frequency Divider Based Ising Machines

We report on a new class of Ising machines (IMs) that rely on coupled parametric frequency dividers (PFDs) as macroscopic artificial spins. Unlike the IM counterparts based on subharmonic-injection locking (SHIL), PFD IMs do not require strong injected continuous-wave signals or applied dc voltages. Therefore, they show a significantly lower power consumption per spin compared to SHIL-based IMs, making it feasible to accurately solve large-scale combinatorial optimization problems that are hard or even impossible to solve by using the current von Neumann computing architectures. Furthermore, using high quality factor resonators in the PFD design makes PFD IMs able to exhibit a nanowatt-level power per spin. Also, it remarkably allows a speedup of the phase synchronization among the PFDs, resulting in shorter time to solution and lower energy to solution despite the resonators' longer relaxation time. As a proof of concept, a 4-node PFD IM has been demonstrated. This IM correctly solves a set of Max-Cut problems while consuming just 600 nanowatts per spin. This power consumption is 2 orders of magnitude lower than the power per spin of state-of-the-art SHIL-based IMs operating at the same frequency.

Parallel edge extraction operators on chip speed up photonic convolutional neural networks

We experimentally establish a 3 × 3 cross-shaped micro-ring resonator (MRR) array-based photonic multiplexing architecture relying on silicon photonics to achieve parallel edge extraction operations in images for photonic convolution neural networks. The main mathematical operations involved are convolution. Precisely, a faster convolutional calculation speed of up to four times is achieved by extracting four feature maps simultaneously with the same photonic hardware's structure and power consumption, where a maximum computility of 0.742 TOPS at an energy cost of 48.6 mW and a convolution accuracy of 95.1% is achieved in an MRR array chip. In particular, our experimental results reveal that this system using parallel edge extraction operators instead of universal operators can improve the imaging recognition accuracy for CIFAR-10 dataset by 6.2% within the same computing time, reaching a maximum of 78.7%. This work presents high scalability and efficiency of parallel edge extraction chips, furnishing a novel, to the best of our knowledge, approach to boost photonic computing speed.

Point convolutional neural network algorithm for Ising model ground state research based on spring vibration

The ground state search of the Ising model can be used to solve many combinatorial optimization problems. Under the current computer architecture, an Ising ground state search algorithm suitable for hardware computing is necessary for solving practical problems. Inspired by the potential energy conversion of the springs, we propose the Spring-Ising Algorithm, a point convolutional neural network algorithm for ground state search based on the spring vibration model. Spring-Ising Algorithm regards the spin as a moving mass point connected to a spring and establishes the equation of motion for all spins. Spring-Ising Algorithm can be mapped on AI chips through the basic structure of the neural network for fast and efficient parallel computing. The algorithm has shown promising results in solving the Ising model and has been tested in the recognized test benchmark K2000. The optimal results of this algorithm after 10,000 steps of iteration are 2.9% of all results. The algorithm introduces the concept of dynamic equilibrium to achieve a more detailed local search by dynamically adjusting the weight of the Ising model in the spring oscillation model. Spring-Ising Algorithm offers the possibility to calculate the Ising model on a chip which focuses on accelerating neural network calculations.

Wavelength-division multiplexing optical Ising simulator enabling fully programmable spin couplings and external magnetic fields

Recently various physical systems have been proposed for modeling Ising spin Hamiltonians appealing to solve combinatorial optimization problems with remarkable performance. However, how to implement arbitrary spin-spin interactions is a critical and challenging problem in unconventional Ising machines. Here, we propose a general gauge transformation scheme to enable arbitrary spin-spin interactions and external magnetic fields as well, by decomposing an Ising Hamiltonian into multiple Mattis-type interactions. With this scheme, a wavelength-division multiplexing spatial photonic Ising machine (SPIM) is developed to show the programmable capability of general spin coupling interactions. We exploit the wavelength-division multiplexing SPIM to simulate three spin systems: ±J models, Sherrington-Kirkpatrick models, and only locally connected J1-J2 models and observe the phase transitions. We also demonstrate the ground-state search for solving Max-Cut problem with the wavelength-division multiplexing SPIM. These results promise the realization of ultrafast-speed and high-power efficiency Boltzmann sampling to a generalized large-scale Ising model.

Phase-diagram investigation of frustrated 1D and 2D Ising models in OEO-based Ising machine

Ising machines have emerged as promising solvers for combinatorial optimization problems in recent years. In practice, these problems are often mapped into a frustrated Ising model due to randomness or competing interactions, which reduces the success ratio for finding the optimal solution. In this study, we simulate one-dimensional and two-dimensional frustrated Ising models in an Ising machine based on the optoelectronic oscillator. Our experiment aims to show the relationship between the Fourier mode of the coupling matrix and the spin distribution under frustration. The results prove the validity of the theoretical predictions and provide insights into the behavior of Ising machines in the presence of frustration. We believe it would help to develop a better strategy to improve the performance of Ising machines.

Hybrid photonic deep convolutional residual spiking neural networks for text classification

Spiking neural networks (SNNs) offer powerful computation capability due to its event-driven nature and temporal processing. However, it is still limited to shallow structure and simple tasks due to the training difficulty. In this work, we propose a deep convolutional residual spiking neural network (DCRSNN) for text classification tasks. In the DCRSNN, the feature extraction is achieved via a convolution SNN with residual connection, using the surrogate gradient direct training technique. Classification is performed by a fully-connected network. We also suggest a hybrid photonic DCRSNN, in which photonic SNNs are used for classification with a converted training method. The accuracy of hard and soft reset methods, as well as three different surrogate functions, were evaluated and compared across four different datasets. Results indicated a maximum accuracy of 76.36% for MR, 91.03% for AG News, 88.06% for IMDB and 93.99% for Yelp review polarity. Soft reset methods used in the deep convolutional SNN yielded slightly better accuracy than their hard reset counterparts. We also considered the effects of different pooling methods and observation time windows and found that the convergence accuracy achieved by convolutional SNNs was comparable to that of convolutional neural networks under the same conditions. Moreover, the hybrid photonic DCRSNN also shows comparable testing accuracy. This work provides new insights into extending the SNN applications in the field of text classification and natural language processing, which is interesting for the resources-restrained scenarios.