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arXiv:2012.10430 · Communications Physics

Order-of-magnitude differences in computational performance of analog Ising machines induced by the choice of nonlinearity

Fabian Böhm, Thomas Van Vaerenbergh, Guy Verschaffelt, Guy Van der Sande

Published Jul 1, 2021• 63 citations
arXivHTMLPDF

Abstract

Abstract Ising machines based on nonlinear analog systems are a promising method to accelerate computation of NP-hard optimization problems. Yet, their analog nature is also causing amplitude inhomogeneity which can deteriorate the ability to find optimal solutions. Here, we investigate how the system’s nonlinear transfer function can mitigate amplitude inhomogeneity and improve computational performance. By simulating Ising machines with polynomial, periodic, sigmoid and clipped transfer functions and benchmarking them with MaxCut optimization problems, we find the choice of transfer function to have a significant influence on the calculation time and solution quality. For periodic, sigmoid and clipped transfer functions, we report order-of-magnitude improvements in the time-to-solution compared to conventional polynomial models, which we link to the suppression of amplitude inhomogeneity induced by saturation of the transfer function. This provides insights into the suitability of nonlinear systems for building Ising machines and presents an efficient way for overcoming performance limitations.

DOI
10.1038/s42005-021-00655-8

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Abstract Ising machines based on nonlinear analog systems are a promising method to accelerate computation of NP-hard optimization problems. Yet, their analog nature is also causing amplitude inhomogeneity which can deteriorate the ability to find optimal solutions. Here, we investigate how the system’s nonlinear transfer function can mitigate amplitude inhomogeneity and improve computational performance. By simulating Ising machines with polynomial, periodic, sigmoid and clipped transfer functions and benchmarking them with MaxCut optimization problems, we find the choice of transfer function to have a significant influence on the calculation time and solution quality. For periodic, sigmoid and clipped transfer functions, we report order-of-magnitude improvements in the time-to-solution compared to conventional polynomial models, which we link to the suppression of amplitude inhomogeneity induced by saturation of the transfer function. This provides insights into the suitability of nonlinear systems for building Ising machines and presents an efficient way for overcoming performance limitations.
Ising machines based on nonlinear analog systems are a promising method to accelerate computation of NP-hard optimization problems. Yet, their analog nature is also causing amplitude inhomogeneity which can deteriorate the ability to find optimal solutions. Here, we investigate if the shape of the system's nonlinear transfer function can mitigate amplitude inhomogeneity and improve computational performance. By simulating Ising machines with polynomial, periodic, sigmoid and clipped transfer functions and benchmarking them with MaxCut optimization problems, we find the choice of transfer function to have a significant influence on the calculation time and solution quality. For periodic, sigmoid and clipped transfer functions, we report order-ofmagnitude improvements in the time-to-solution compared to conventional polynomial models, which we link to the suppression of amplitude inhomogeneity induced by saturation of the transfer function. This provides insights into the suitability of systems for building Ising machines and presents an efficient way for overcoming performance limitations.
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.
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.
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.
Ising machines are specialized devices designed to efficiently solve combinatorial optimization problems. Among such problems, Boolean Satisfiability (SAT) is particularly relevant in industrial applications. To solve SAT problems using Ising machines, it is crucial to incorporate higher-order interactions. However, in analog Ising machines, interactions of different orders scale unevenly with the continuous spin amplitudes, introducing imbalances that can significantly degrade performance. Here we present a numerical comparison of methods to mitigate these imbalances, evaluating time-to-solution and success rate on uniform random 3-SAT and 4-SAT instances. Our results show that the most effective approach employs spin interactions that are proportional to the signs of spins, rather than their continuous amplitudes. We further demonstrate that smooth approximations of this method make it compatible with analog hardware. These findings underscore the central role of spin-sign-based interactions in enabling robust and scalable analog Ising machine dynamics. Ising machines (IMs) offer promising solutions for combinatorial optimization problems yet face challenges with imbalances in higher-order interactions when applied to Boolean Satisfiability (SAT) tasks. Here, the authors demonstrate that employing spin-sign-based interactions effectively mitigates these imbalances, enhancing performance and compatibility with analog hardware, thus advancing robust and scalable IM dynamics.