Heuristic recurrent algorithms for photonic Ising machines

Exploring Types of Photonic Neural Networks for Imaging and Computing-A Review

Photonic neural networks (PNNs), utilizing light-based technologies, show immense potential in artificial intelligence (AI) and computing. Compared to traditional electronic neural networks, they offer faster processing speeds, lower energy usage, and improved parallelism. Leveraging light's properties for information processing could revolutionize diverse applications, including complex calculations and advanced machine learning (ML). Furthermore, these networks could address scalability and efficiency challenges in large-scale AI systems, potentially reshaping the future of computing and AI research. In this comprehensive review, we provide current, cutting-edge insights into diverse types of PNNs crafted for both imaging and computing purposes. Additionally, we delve into the intricate challenges they encounter during implementation, while also illuminating the promising perspectives they introduce to the field.

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.

Photonic Hopfield neural network for the Ising problem

The Ising problem, a vital combinatorial optimization problem in various fields, is hard to solve by traditional Von Neumann computing architecture on a large scale. Thus, lots of application-specific physical architectures are reported, including quantum-based, electronics-based, and optical-based platforms. A Hopfield neural network combined with a simulated annealing algorithm is considered one of the effective approaches but is still limited by large resource consumption. Here, we propose to accelerate the Hopfield network on a photonic integrated circuit composed of the arrays of Mach-Zehnder interferometer. Our proposed Photonic Hopfield Neural Network (PHNN), utilizing the massively parallel operations and integrated circuit with ultrafast iteration rate, converges to a stable ground state solution with high probability. The average success probabilities for the MaxCut problem with a problem size of 100 and the Spin-glass problem with a problem size of 60 can both reach more than 80%. Moreover, our proposed architecture is inherently robust to the noise induced by the imperfect characteristics of components on chip.

Ultrafast dynamic machine vision with spatiotemporal photonic computing

Ultrafast dynamic machine vision in the optical domain can provide unprecedented perspectives for high-performance computing. However, owing to the limited degrees of freedom, existing photonic computing approaches rely on the memory's slow read/write operations to implement dynamic processing. Here, we propose a spatiotemporal photonic computing architecture to match the highly parallel spatial computing with high-speed temporal computing and achieve a three-dimensional spatiotemporal plane. A unified training framework is devised to optimize the physical system and the network model. The photonic processing speed of the benchmark video dataset is increased by 40-fold on a space-multiplexed system with 35-fold fewer parameters. A wavelength-multiplexed system realizes all-optical nonlinear computing of dynamic light field with a frame time of 3.57 nanoseconds. The proposed architecture paves the way for ultrafast advanced machine vision free from the limits of memory wall and will find applications in unmanned systems, autonomous driving, ultrafast science, etc.

Fixed-point iterative linear inverse solver with extended precision

Solving linear systems, often accomplished by iterative algorithms, is a ubiquitous task in science and engineering. To accommodate the dynamic range and precision requirements, these iterative solvers are carried out on floating-point processing units, which are not efficient in handling large-scale matrix multiplications and inversions. Low-precision, fixed-point digital or analog processors consume only a fraction of the energy per operation than their floating-point counterparts, yet their current usages exclude iterative solvers due to the cumulative computational errors arising from fixed-point arithmetic. In this work, we show that for a simple iterative algorithm, such as Richardson iteration, using a fixed-point processor can provide the same convergence rate and achieve solutions beyond its native precision when combined with residual iteration. These results indicate that power-efficient computing platforms consisting of analog computing devices can be used to solve a broad range of problems without compromising the speed or precision.

Low Power Coherent Ising Machine Based on Mechanical Kerr Nonlinearity

Finding a reliable Ising machine for solving nondeterministic polynomial-class problems has attracted great attention in recent years, where an authentic system can be expanded with polynomial-scaled resources to find the ground state Ising Hamiltonian. In this Letter, we propose an extremely low power optomechanical coherent Ising machine based on a new enhanced symmetry breaking mechanism and highly nonlinear mechanical Kerr effect. The mechanical movement of an optomechanical actuator induced by the optical gradient force greatly increases the nonlinearity by a few orders and significantly reduces the power threshold using conventional structures capable of fabrication via photonic integrated circuit platforms. With the simple but strong bifurcation mechanism and remarkably low power requirement, our optomechanical spin model opens a path for chip-scale integration of large-size Ising machine implementations with great stability.

CMOS-compatible ising machines built using bistable latches coupled through ferroelectric transistor arrays

Realizing compact and scalable Ising machines that are compatible with CMOS-process technology is crucial to the effectiveness and practicality of using such hardware platforms for accelerating computationally intractable problems. Besides the need for realizing compact Ising spins, the implementation of the coupling network, which describes the spin interaction, is also a potential bottleneck in the scalability of such platforms. Therefore, in this work, we propose an Ising machine platform that exploits the novel behavior of compact bi-stable CMOS-latches (cross-coupled inverters) as classical Ising spins interacting through highly scalable and CMOS-process compatible ferroelectric-HfO₂-based Ferroelectric FETs (FeFETs) which act as coupling elements. We experimentally demonstrate the prototype building blocks of this system, and evaluate the scaling behavior of the system using simulations. Our work not only provides a pathway to realizing CMOS-compatible designs but also to overcoming their scaling challenges.

Noise-injected analog Ising machines enable ultrafast statistical sampling and machine learning

Ising machines are a promising non-von-Neumann computational concept for neural network training and combinatorial optimization. However, while various neural networks can be implemented with Ising machines, their inability to perform fast statistical sampling makes them inefficient for training neural networks compared to digital computers. Here, we introduce a universal concept to achieve ultrafast statistical sampling with analog Ising machines by injecting noise. With an opto-electronic Ising machine, we experimentally demonstrate that this can be used for accurate sampling of Boltzmann distributions and for unsupervised training of neural networks, with equal accuracy as software-based training. Through simulations, we find that Ising machines can perform statistical sampling orders-of-magnitudes faster than software-based methods. This enables the use of Ising machines beyond combinatorial optimization and makes them into efficient tools for machine learning and other applications.

Quadrature photonic spatial Ising machine

As a computing accelerator, a large-scale photonic spatial Ising machine has great advantages and potential due to its excellent scalability and compactness. However, the current fundamental limitation of a photonic spatial Ising machine is the configuration flexibility for problem implementation in the accelerator model. Arbitrary spin interactions are highly desired for solving various non-deterministic polynomial (NP)-hard problems. In this paper, we propose a novel quadrature photonic spatial Ising machine to break through the limitation of the photonic Ising accelerator by synchronous phase manipulation in two sections. The max-cut problem solution with a graph order of 100 and density from 0.5 to 1 is experimentally demonstrated after almost 100 iterations. Our work suggests flexible problem solving by the large-scale photonic spatial Ising machine.

Photonic matrix multiplication lights up photonic accelerator and beyond

Matrix computation, as a fundamental building block of information processing in science and technology, contributes most of the computational overheads in modern signal processing and artificial intelligence algorithms. Photonic accelerators are designed to accelerate specific categories of computing in the optical domain, especially matrix multiplication, to address the growing demand for computing resources and capacity. Photonic matrix multiplication has much potential to expand the domain of telecommunication, and artificial intelligence benefiting from its superior performance. Recent research in photonic matrix multiplication has flourished and may provide opportunities to develop applications that are unachievable at present by conventional electronic processors. In this review, we first introduce the methods of photonic matrix multiplication, mainly including the plane light conversion method, Mach-Zehnder interferometer method and wavelength division multiplexing method. We also summarize the developmental milestones of photonic matrix multiplication and the related applications. Then, we review their detailed advances in applications to optical signal processing and artificial neural networks in recent years. Finally, we comment on the challenges and perspectives of photonic matrix multiplication and photonic acceleration.

Experimental Observation of Phase Transitions in Spatial Photonic Ising Machine

Statistical spin dynamics plays a key role in understanding the working principle for novel optical Ising machines. Here, we propose the gauge transformation for a spatial photonic Ising machine, where a single spatial phase modulator simultaneously encodes spin configurations and programs interaction strengths. Using gauge transformation, we experimentally evaluate the phase diagram of a high-dimensional spin-glass equilibrium system with 100 fully connected spins. We observe the presence of paramagnetic, ferromagnetic, and spin-glass phases and determine the critical temperature T_{c} and the critical probability p_{c} of the phase transitions, which agree well with the mean-field theory predictions. Thus, the approximation of the mean-field model is experimentally verified in the spatial photonic Ising machine. Furthermore, we discuss the phase transition in parallel with solving combinatorial optimization problems during the cooling process and identify that the spatial photonic Ising machine is robust with sufficient many-spin interactions even when the system is associated with optical aberrations and measurement uncertainty.

Accuracy-enhanced coherent Ising machine using the quantum adiabatic theorem

The coherent Ising machine (CIM) implemented by degenerate optical parametric oscillator (DOPO) networks is a novel optical platform to accelerate computation of hard combinatorial optimization problems. Nevertheless, with the increase of the problem size, the probability of the machine being trapped by local minima increases exponentially. According to the quantum adiabatic theorem, a physical system will remain in its instantaneous ground state if the time-dependent Hamiltonian varies slowly enough. Here, we propose a method to help the machine partially avoid getting stuck in local minima by introducing quantum adiabatic evolution to the ground-state-search process of the CIM, which we call A-CIM. Numerical simulation results demonstrate that A-CIM can obtain improved solution accuracy in solving MAXCUT problems of vertices ranging from 10 to 2000 than CIM. The proposed machine that is based on quantum adiabatic theorem is expected to solve optimization problems more correctly.

Collective and synchronous dynamics of photonic spiking neurons

Nonlinear dynamics of spiking neural networks have recently attracted much interest as an approach to understand possible information processing in the brain and apply it to artificial intelligence. Since information can be processed by collective spiking dynamics of neurons, the fine control of spiking dynamics is desirable for neuromorphic devices. Here we show that photonic spiking neurons implemented with paired nonlinear optical oscillators can be controlled to generate two modes of bio-realistic spiking dynamics by changing optical-pump amplitude. When the photonic neurons are coupled in a network, the interaction between them induces an effective change in the pump amplitude depending on the order parameter that characterizes synchronization. The experimental results show that the effective change causes spontaneous modification of the spiking modes and firing rates of clustered neurons, and such collective dynamics can be utilized to realize efficient heuristics for solving NP-hard combinatorial optimization problems.

High-speed all-optical processing for spectrum

Data-processing techniques in spectroscopy are fundamental and powerful analytical tools for lots of practical applications. In the age of big data, high-speed data-processing in spectroscopy is in urgent need, especially for the real-time analysis/feedback of data stream of spectroscopy or the capture of non-repetitive/rare phenomena in fast dynamic process. So far, intensive researches focus on high-speed processing of light signal in time/spatial domain but few people find a way to do it in spectral domain. Here, we report an optical computing technology for high-speed optical spectrum processing with features of real time, multiple functions, all-fiber configuration and immunity to electromagnetic interference. The software-controlled system could perform as, but not limited to, the first-order (or arbitrary fractional-order) differentiator/integrator/Hilbert transformer and tunable band-pass filter, respectively, to handle spectral data rapidly. High-speed processing of optical spectrum at a rate of 10,000,000 times per second is demonstrated.

Taperless Si hybrid optical phase shifter based on a metal-oxide-semiconductor capacitor using an ultrathin InP membrane

We propose a III-V/Si hybrid metal-oxide-semiconductor (MOS) optical phase shifter using an ultrathin InP membrane, which allows us to eliminate the III-V taper required for mode conversion between Si and hybrid waveguides. We numerically revealed that thinning a III-V membrane can reduce the insertion loss of the phase shifter while maintaining high modulation efficiency because the optical phase shift is induced by carrier accumulation at the MOS interface. We experimentally demonstrated the proposed optical phase shifter with an ultrathin InP membrane and achieved the modulation efficiency of 0.54 Vcm and the insertion loss of 0.055 dB. Since the taperless structure makes the hybrid integration easier and more flexible, the hybrid MOS optical phase shifter with an ultrathin III-V membrane is promising for large-scale Si programmable photonic integrated circuits.

Physics successfully implements Lagrange multiplier optimization

Optimization is a major part of human effort. While being mathematical, optimization is also built into physics. For example, physics has the Principle of Least Action; the Principle of Minimum Power Dissipation, also called Minimum Entropy Generation; and the Variational Principle. Physics also has Physical Annealing, which, of course, preceded computational Simulated Annealing. Physics has the Adiabatic Principle, which, in its quantum form, is called Quantum Annealing. Thus, physical machines can solve the mathematical problem of optimization, including constraints. Binary constraints can be built into the physical optimization. In that case, the machines are digital in the same sense that a flip-flop is digital. A wide variety of machines have had recent success at optimizing the Ising magnetic energy. We demonstrate in this paper that almost all those machines perform optimization according to the Principle of Minimum Power Dissipation as put forth by Onsager. Further, we show that this optimization is in fact equivalent to Lagrange multiplier optimization for constrained problems. We find that the physical gain coefficients that drive those systems actually play the role of the corresponding Lagrange multipliers.

Theory of Neuromorphic Computing by Waves: Machine Learning by Rogue Waves, Dispersive Shocks, and Solitons

We study artificial neural networks with nonlinear waves as a computing reservoir. We discuss universality and the conditions to learn a dataset in terms of output channels and nonlinearity. A feed-forward three-layered model, with an encoding input layer, a wave layer, and a decoding readout, behaves as a conventional neural network in approximating mathematical functions, real-world datasets, and universal Boolean gates. The rank of the transmission matrix has a fundamental role in assessing the learning abilities of the wave. For a given set of training points, a threshold nonlinearity for universal interpolation exists. When considering the nonlinear Schrödinger equation, the use of highly nonlinear regimes implies that solitons, rogue, and shock waves do have a leading role in training and computing. Our results may enable the realization of novel machine learning devices by using diverse physical systems, as nonlinear optics, hydrodynamics, polaritonics, and Bose-Einstein condensates. The application of these concepts to photonics opens the way to a large class of accelerators and new computational paradigms. In complex wave systems, as multimodal fibers, integrated optical circuits, random, topological devices, and metasurfaces, nonlinear waves can be employed to perform computation and solve complex combinatorial optimization.

Demonstration of chip-based coupled degenerate optical parametric oscillators for realizing a nanophotonic spin-glass

The need for solving optimization problems is prevalent in various physical applications, including neuroscience, network design, biological systems, socio-economics, and chemical reactions. Many of these are classified as non-deterministic polynomial-time hard and thus become intractable to solve as the system scales to a large number of elements. Recent research advances in photonics have sparked interest in using a network of coupled degenerate optical parametric oscillators (DOPOs) to effectively find the ground state of the Ising Hamiltonian, which can be used to solve other combinatorial optimization problems through polynomial-time mapping. Here, using the nanophotonic silicon-nitride platform, we demonstrate a spatial-multiplexed DOPO system using continuous-wave pumping. We experimentally demonstrate the generation and coupling of two microresonator-based DOPOs on a single chip. Through a reconfigurable phase link, we achieve both in-phase and out-of-phase operation, which can be deterministically achieved at a fast regeneration speed of 400 kHz with a large phase tolerance.