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Optics Express

Enhancing the performance of coherent Ising machines in the large-noise regime with a fifth-order nonlinearity

Ruqi Shi, Fabian Böhm, Thomas Van Vaerenbergh, Peter Bienstman

Published May 1, 2024• 11 citations
Publisher

Abstract

Coherent Ising machines (CIMs), leveraging the bistable physical properties of coherent light to emulate Ising spins, exhibit great potential as hardware accelerators for tackling complex combinatorial optimization problems. Recent advances have demonstrated that the performance of CIMs can be enhanced either by incorporating large random noise or higher-order nonlinearities, yet their combined effects on CIM performance remain mainly unexplored. In this work, we develop a numerical CIM model that utilizes a tunable fifth-order polynomial nonlinear dynamic function under large noise levels, which has the potential to be implemented in all-optical platforms. We propose a normal form of a CIM model that allows for both supercritical and subcritical pitchfork bifurcation operational regimes, with fifth-order nonlinearity and tunable hyperparameters to control the Ising spin dynamics. In the benchmark studies, we simulate various sets of MaxCut problems using our fifth-order polynomial CIM model. The results show a significant performance improvement, achieving an average of 59.5% improvement in median time-to-solution (TTS) and an average of 6 times improvement in median success rate (SR) for dense Maxcut problems in the BiqMac library, compared to the commonly used third-order polynomial CIM model with low noise. The fifth-order polynomial CIM model in the large-noise regime also shows better performance trends as the problem size scales up. These findings reveal the enhancements on the computational performance of Ising machines in the large-nose regime from fifth-order nonlinearity, showing important implications for both simulation and hardware perspectives.

DOI
10.1364/oe.522474

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Coherent Ising machines (CIMs), leveraging the bistable physical properties of coherent light to emulate Ising spins, exhibit great potential as hardware accelerators for tackling complex combinatorial optimization problems. Recent advances have demonstrated that the performance of CIMs can be enhanced either by incorporating large random noise or higher-order nonlinearities, yet their combined effects on CIM performance remain mainly unexplored. In this work, we develop a numerical CIM model that utilizes a tunable fifth-order polynomial nonlinear dynamic function under large noise levels, which has the potential to be implemented in all-optical platforms. We propose a normal form of a CIM model that allows for both supercritical and subcritical pitchfork bifurcation operational regimes, with fifth-order nonlinearity and tunable hyperparameters to control the Ising spin dynamics. In the benchmark studies, we simulate various sets of MaxCut problems using our fifth-order polynomial CIM model. The results show a significant performance improvement, achieving an average of 59.5% improvement in median time-to-solution (TTS) and an average of 6 times improvement in median success rate (SR) for dense Maxcut problems in the BiqMac library, compared to the commonly used third-order polynomial CIM model with low noise. The fifth-order polynomial CIM model in the large-noise regime also shows better performance trends as the problem size scales up. These findings reveal the enhancements on the computational performance of Ising machines in the large-nose regime from fifth-order nonlinearity, showing important implications for both simulation and hardware perspectives.
Coherent Ising Machines (CIMs) have emerged as promising hardware solvers for NP-hard combinatorial optimization problems, potentially offering energy-efficiency, accuracy, speed, and scalability. These problems span applications including scheduling, routing, resource allocation, and machine learning. Spin amplitude inhomogeneity (an artifact of its analog nature) can hinder CIM convergence to the ground state. Several CIM modifications have been proposed to address this issue. However, no prior work has systematically compared the performance of these concepts. This paper presents the first side-by-side evaluation of six promising CIM architectures using a unified simulation model. We develop stochastic differential equations, find optimal hyperparameters via Bayesian optimization per problem-architecture pair, and assess the performance on multiple metrics using MaxCut instances from the Gset and BiqMac datasets. CIM-SNN outperforms the rest on most metrics, with a 54.21% average success probability (SP) improvement and 1.36× less time-to-solution (TTS) at 96% GS energy compared to the Standard CIM on BiqMac graphs. Other CIMs show some advantages on certain Gset graphs. CIM-QA is usually the worst-performing, with an 18.55% average SP loss and a 1.30× higher TTS at 96% GS compared to the Standard CIM on BiqMac graphs. CIM-SNN demonstrates the greatest noise robustness and promise for further tuning via a linear pump schedule. CIM-QA benefits from high noise levels. This study makes two key contributions: (1) identifying promising CIM architectures to guide targeted research and accelerate practical adoption, and (2) providing a unified framework for evaluating future architectural modifications against existing ones.
Coherent Ising Machines (CIMs) are emerging hardware solvers for approximating solutions to combinatorial optimization problems, and may offer advantages in energy-efficiency, speed, and scalability across applications. However, CIM computing is a nascent and methodologically fragmented field, and substantial research is required to determine its utility for real-world combinatorial optimization problems. The core argument of this thesis is that several complementary efforts are required: (1) developing a unified simulations framework that systematically evaluates CIM variations by isolating implementation and methodological differences from architectural ones; (2) introducing layers of abstraction, analogous to those in mature CMOS-based computing, to enable effort (1); and (3), by extension of efforts (1) and (2), potentially enabling CIM-inspired dynamics -- which, when simulated, are simply classical algorithms that run on conventional computers -- for efficient in-silico approximation of combinatorial optimization problems.As a concrete example of the utility of the unified simulations framework, this thesis compares several CIM modifications proposed to address spin amplitude inhomogeneity (an artifact of the CIM's analog nature) that can hinder convergence to the ground state. These variations have been analyzed under varying assumptions and methodologies, and no prior work has systematically compared their performance by isolating architectural differences. This thesis presents the first side-by-side evaluation of six promising CIM architectures using a unified simulation model. We develop implementation-agnostic stochastic differential equations, find optimal hyperparameters via Bayesian optimization per problem-architecture pair, and assess the performance on multiple metrics using MaxCut instances from the Gset and BiqMac datasets. CIM-SNN outperforms the rest on most metrics, with a 54.21% average success probability (SP) improvement and 1.36× faster time-to-solution (TTS) at 96% GS energy compared to the Standard CIM on BiqMac graphs. Other CIMs show some advantages on certain Gset graphs. CIM-QA is usually the worst-performing, with an 18.55% average SP loss and a 1.30× slower TTS at 96% GS compared to the Standard CIM on BiqMac graphs. CIM-SNN demonstrates the greatest noise robustness and promise for further tuning via a linear pump schedule. CIM-QA benefits from high noise levels. These analyses were made possible only by the unified simulations framework developed in this thesis, which can also be used to evaluate future architectural modifications against existing ones. Lastly, we present a potential abstraction stack that further extends the unified simulations framework, which can benefit future research and introduce a new direction in the field.
Combinatorial optimization problems (COPs) constitute a fundamental class of computational challenges with extensive applications across scientific and industrial domains. The emergence of the coherent Ising machine (CIM) as a computational paradigm has demonstrated exceptional capabilities in rapidly generating arbitrary spin configurations and effectively solving large-scale COPs. Despite its promising potential, the CIM framework encounters an inherent limitation related to amplitude heterogeneity. Current approaches to this limitation primarily rely on Gaussian approximations to model quantum noise in the system's amplitude dynamics within the original CIM framework. However, these methods generally fail to account for the temporal and periodic non-Markovian characteristics of pulses in practical environments. In this study, we introduce temporally correlated non-Markovian noise into the dynamics of both conventional and spiking neural network-based CIM systems, investigating their performance through comprehensive hyperparameter analysis. Our method incorporates a detailed examination of hyperparameter selection and its influence on algorithmic efficiency. Through extensive numerical simulations, we demonstrate that the integration of non-Markovian noise significantly accelerates variance evolution in both canonical coordinates and amplitude dynamics over time. The proposed approach not only exhibits enhanced computational performance but also shows superior scalability for large-scale problem instances. These findings suggest that our approach offers distinct advantages in addressing complex COPs, potentially opening new avenues for efficient optimization in various applications.
Coherent Ising Machines (CIMs) are an emerging class of computational architectures that embed hard combinatorial optimization problems in the continuous dynamics of a physical system with analog degrees of freedom. While crisp theoretical results on the ultimate performance and scaling of such architectures are lacking, large-scale experimental prototypes have begun to exhibit promising results in practice. Our team at Stanford has begun to study the fundamental properties of CIM dynamics using a combination of techniques from statistical physics, random matrices, and dynamical systems theory. Many connections to recent work in neuroscience and deep learning are noted. Our work focuses specifically on CIMs that utilize the nonlinear threshold behavior of optical parametric oscillators to effect a soft (potentially glassy) transition between linear and binary dynamical regimes.
Coherent Ising machines (CIMs) are an experimentally promising class of physics-based computational architectures that embed hard combinatorial optimization problems into systems of coupled nonlinear optical oscillators. The solution-finding mechanisms employed by CIMs feature complicated dynamical bifurcations occurring on a network scale, posing significant challenges to the development of theory and models for their underlying principles of operation. These difficulties are especially pronounced in the ultra-low-power or quantum regimes where the benefits in computational efficiency over conventional optimization algorithms are expected to be largest. We discuss some of our recent approaches and results at this intersection of dynamical systems theory and quantum model reduction, which have highlighted some potentially useful architectures and applications on the horizon for CIMs.
The Coherent Ising Machine (CIM) is a quantum network of optical parametric oscillators (OPOs) intended to find ground states of the Ising model. This is an NP-hard problem, related to several important minimization problems, including the max-cut graph problem. In order to enhance its potential performance, we analyze the coherent coupling strategy for the CIM in a highly quantum regime. To explore this limit, without assuming gaussianity, we employ accurate numerical simulations. Due to the inherent complexity of the system, the maximum network size is limited. While master equation methods can be used, their scalability diminishes rapidly for larger systems. Instead, we use Monte Carlo wave-function methods, which scale as the wave-function dimension, and use large numbers of samples. These simulations involve Hilbert spaces exceeding 10710^{7}107 dimensions. To evaluate success probabilities, we use quadrature probabilities. We demonstrate the potential for quantum computational advantage by reducing the time required to reach maximum success probability in a low-dissipation regime enabled by initial quantum superpositions and entanglement. Furthermore, we demonstrate that tailored time-dependent couplings can amplify these quantum effects. Comparisons with classical CIM models give evidence that quantum tunneling effects in this strong coupling limit can overcome trapping in false minima. This can greatly increase success rates, indicating a potential for quantum advantage. Finally, we perform a coherence analysis based on the state purity to examine the role of quantum coherence in CIM performance and to determine how state purity correlates with improved optimization outcomes.