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IEEE International Conference on Nanotechnology (IEEE-NANO)

A comparison of Oscillatory Ising Machines and Simulated Bifurcation Machines for Solving Maximum Cut Problems

Md Shabaz Ansari, Hemkant Nehete, Andrea Grimaldi, Luciano Mazza, Eleonora Raimondo, Vito Puliafito, Giovanni Finocchio, Brajesh Kumar Kaushik

Published Jul 8, 2024• 2 citations
Publisher

Abstract

Unconventional computing approaches have gained significant attention in the search for hardware-friendly solvers capable of finding the solution of combinatorial optimization problems COPs. Among others, oscillator-based, or oscillatory, Ising machines (OIMs), and simulated bifurcation-based Ising machines (SBMs) are two promising paradigms that are based on dynamical equations. This work presents a preliminary direct software comparison between an OIM and an SBM on instances from the G-set, a well-known benchmark library of maximum cut problems. The performance of the two paradigms is compared in terms of their average optimal approximation within a given number of timesteps using the same integration scheme. Our findings show that OIM performs better when optimizing its parameter for a specific problem or topology, while SBM has better performance in scenarios in which preprocessing is not possible, i.e., when considering large batches of instances with varied topologies and scales.

DOI
10.1109/nano61778.2024.10628622

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Unconventional computing approaches have gained significant attention in the search for hardware-friendly solvers capable of finding the solution of combinatorial optimization problems COPs. Among others, oscillator-based, or oscillatory, Ising machines (OIMs), and simulated bifurcation-based Ising machines (SBMs) are two promising paradigms that are based on dynamical equations. This work presents a preliminary direct software comparison between an OIM and an SBM on instances from the G-set, a well-known benchmark library of maximum cut problems. The performance of the two paradigms is compared in terms of their average optimal approximation within a given number of timesteps using the same integration scheme. Our findings show that OIM performs better when optimizing its parameter for a specific problem or topology, while SBM has better performance in scenarios in which preprocessing is not possible, i.e., when considering large batches of instances with varied topologies and scales.
Solving large-scale combinatorial optimization problems, such as Max-Cut, is crucial for a variety of applications such as network analysis and finance. Ising machines are emerging as unconventional computing paradigms for facing those problems while promising to reduce energy consumption and time-to-solution as compared to traditional methods such as the Goemans-Williamson algorithm. Here, we implement and benchmark a Graphics Processing Unit (GPU)-native software-based implementation of oscillator-based Ising machine (OIM) which can solve Max-Cut problems up to 20 million spins within a regular graph with degree 25. The benchmarked implementation is running on a single GPU, but it is scalable to multi-GPU platforms. Additionally, we identify an annealing strategy that searches the ground state with the help of dynamical checkpoints as bifurcation points in the dynamical phase space of the Max-Cut problems mapped into the Ising Hamiltonian, enabling the algorithm for a fast local search. The OIM consistently achieves accuracy averaging above 99.5% (up to 99.9%) for G-set problems.
Combinatorial optimization problems are ubiquitous in industrial applications. However, finding optimal or close-to-optimal solutions can often be extremely hard. Because some of these problems can be mapped to the ground-state search of the Ising model, tremendous effort has been devoted to developing solvers for Ising-type problems over the past decades. Recent advances in controlling and manipulating both quantum and classical systems have enabled novel computing paradigms such as quantum simulators and coherent Ising machines to tackle hard optimization problems. Here, we examine and benchmark several physics-inspired optimization algorithms, including coherent Ising machines, gain-dissipative algorithms, simulated bifurcation machines, and Hopfield neural networks, which we collectively refer to as stochastic driven nonlinear dynamical systems. Most importantly, we benchmark these algorithms against random Ising problems with planted solutions and compare them to simulated annealing as a baseline leveraging the same software stack for all solvers. We further study how different numerical integration techniques and graph connectivity affect performance. This work provides an overview of a diverse set of new optimization paradigms.
Oscillator-based Ising machines (OIMs) and oscillator-based Potts machines (OPMs) have emerged as promising hardware accelerators for solving NP-hard combinatorial optimization problems by leveraging the phase dynamics of coupled oscillators. In this work, a GPU-accelerated simulated OIM/OPM digital computation framework capable of solving combinatorial optimization problems is presented. The proposed implementation harnesses the parallel processing capabilities of GPUs to simulate large-scale OIM/OPMs, leveraging the advantages of digital computing to offer high precision, programmability, and scalability. The performance of the proposed GPU framework is evaluated on the max-cut problems from the GSET benchmark dataset and graph coloring problems from the SATLIB benchmarks dataset, demonstrating competitive speed and accuracy in tackling large-scale problems. The results from simulations, reaching up to 11295x speed-up over CPUs with up to 99% accuracy, establish this framework as a scalable, massively parallelized, and high-fidelity digital realization of OIM/OPMs.
Abstract This paper benchmarks quantum, classical, and hybrid solvers on NP-hard Max-Cut and QUBO problems, emphasizing solution quality relative to known global optima. We evaluate D-Wave’s fast annealing QPU and Hybrid solver against classical simulated annealing (SA) and Toshiba’s simulated bifurcation machine (SBM) using 139 Max-Cut instances (100 to 10,000 nodes). For small instances ( ≤\le ≤ ≤ 250 nodes) with known global optima, Hybrid and SA consistently achieve optimal solutions, outperforming the QPU. For larger instances, SBM and slower SA yield superior solutions, while Hybrid and faster SA perform less effectively. Computation time varies across solvers.
Since many combinatorial optimization problems can be mapped onto ground-state search problems of Ising models, special-purpose machines for Ising problems have attracted intense attention. Simulated bifurcation (SB) is a recently proposed algorithm to solve these Ising problems. One of the remarkable features of SB is the high-degree parallelism underlying in the algorithm, providing an opportunity to solve the Ising problems very fast by massively parallel processing. In this work, we implement the SB algorithm on FPGAs by designing massively parallel custom circuits. We then compare the FPGA-based SB machines with a state-of-the-art machine called a coherent Ising machine (CIM), a highly optimized implementation of simulated annealing (SA), and GPU-based SB machines. SB machines with spin size of 2,048/4,096 (2K/4K) on an Arria10 GX1150 FPGA have 8,192 processing elements for the matrix-vector multiplication (MM) modules (the most computationally intensive part) and achieve computation throughput of 1,873/2,027 GMAC/s for the MM modules, outperforming 2K/4K SB machines on an Nvidia Tesla V100 GPU (113/183 GMAC/s). The 2K FPGA-SB solves all-to-all connected 2000-node MAX-CUT problem 14X (/124X) faster than the CIM (/the highly-optimized SA), with much better energy efficiency (288X better than the CIM).
Combinatorial optimization problems are ubiquitous but difficult to solve. Hardware devices for these problems have recently been developed by various approaches, including quantum computers. Inspired by recently proposed quantum adiabatic optimization using a nonlinear oscillator network, we propose a new optimization algorithm simulating adiabatic evolutions of classical nonlinear Hamiltonian systems exhibiting bifurcation phenomena, which we call simulated bifurcation (SB). SB is based on adiabatic and chaotic (ergodic) evolutions of nonlinear Hamiltonian systems. SB is also suitable for parallel computing because of its simultaneous updating. Implementing SB with a field-programmable gate array, we demonstrate that the SB machine can obtain good approximate solutions of an all-to-all connected 2000-node MAX-CUT problem in 0.5 ms, which is about 10 times faster than a state-of-the-art laser-based machine called a coherent Ising machine. SB will accelerate large-scale combinatorial optimization harnessing digital computer technologies and also offer a new application of computational and mathematical physics.