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arXiv:2402.15090 · Physical Review Letters

Frustration Elimination and Excited State Search in Coherent Ising Machines

Zheng-Yang Zhou, Clemens Gneiting, J. Q. You, Franco Nori

Published Mar 5, 2025• 7 citations
arXivHTMLPDF

Abstract

Frustration, that is, the impossibility of satisfying the energetic preferences between all spin pairs simultaneously, underlies the complexity of many fundamental properties in spin systems, including the computational difficulty in determining their ground states. Coherent Ising machines (CIMs) have been proposed as a promising analog computational approach to efficiently find different degenerate ground states of large and complex Ising models. However, CIMs also face challenges in solving frustrated Ising models: frustration not only reduces the probability of finding good solutions, but it also prohibits the leveraging of quantum effects in doing so. To circumvent these detrimental effects of frustration, we show how frustrated Ising models can be mapped to frustration-free CIM configurations by including ancillary modes and modifying the coupling protocol used in current CIM designs. Such frustration elimination may empower current CIMs to benefit from quantum effects in dealing with frustrated Ising models. In addition, these ancillary modes can also enable error detection and searching for excited states.

DOI
10.1103/physrevlett.134.090401

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Frustration, that is, the impossibility of satisfying the energetic preferences between all spin pairs simultaneously, underlies the complexity of many fundamental properties in spin systems, including the computational difficulty in determining their ground states. Coherent Ising machines (CIMs) have been proposed as a promising analog computational approach to efficiently find different degenerate ground states of large and complex Ising models. However, CIMs also face challenges in solving frustrated Ising models: frustration not only reduces the probability of finding good solutions, but it also prohibits the leveraging of quantum effects in doing so. To circumvent these detrimental effects of frustration, we show how frustrated Ising models can be mapped to frustration-free CIM configurations by including ancillary modes and modifying the coupling protocol used in current CIM designs. Such frustration elimination may empower current CIMs to benefit from quantum effects in dealing with frustrated Ising models. In addition, these ancillary modes can also enable error detection and searching for excited states.
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.
Abstract Frustrated Ising models play a fundamental role in understanding complex magnetic phenomena and are directly relevant to quantum annealing applications. However, determining the ground state energies of these systems remains computationally challenging due to the NP-hard nature of the problem. In this study, we propose and demonstrate a novel approach using a network of three coupled parametric oscillators to simulate frustrated Ising spins. By establishing a mapping between oscillator amplitudes, spin states, coupling constants with interaction strengths, and noise power, we show that the steady-state amplitude distribution of the oscillators corresponds to the ground state spin configuration. Numerical simulations for various coupling strengths successfully reproduce the ground state energies for triangular frustrated lattice systems. Our simulations reveal the existence of degenerate ground states that satisfy local energy minima for the frustrated Ising spin models. Our results suggest that coupled parametric oscillator networks may provide a viable physical implementation for studying frustrated Ising spins, with potential applications in quantum annealing and combinatorial optimization. Our work opens new avenues for exploring analog computing approaches to NP-hard optimization problems.
Today, world-wide efforts are being undertaken to explore non-von Neumann computation methods to overcome "the end of Moore's law". A promising approach is the coherent Ising machine (CIM), which utilizes a network of degenerate optical parametric oscillators for solving ground-state-search problems of the Ising model. Since its proposal in 2013 and the first proof-of-principle experiment in 2014, research on experimental CIMs has made rapid progress and now reached the level where we can find solutions to large-scale combinatorial optimization problems using a CIM system. In this paper, we review recent progress on CIMs, mainly focusing on the experimental side.
A coherent Ising machine (CIM) is a network of optical parametric oscillators (OPOs), in which the “strongest” collective mode of oscillation at well above threshold corresponds to an optimum solution of a given Ising problem. When a pump rate or network coupling rate is increased from below to above threshold, however, the eigenvectors with the smallest eigenvalue of the Ising coupling matrix [Jij] appear near threshold and impede the machine to relax to true ground states. Two complementary approaches to attack this problem are described here. One approach is to utilize the squeezed/anti-squeezed vacuum noise of OPOs below threshold to produce coherent spreading over numerous local minima via quantum noise correlation, which could enable the machine to access either true ground states or excited states with eigen-energies close enough to that of ground states above threshold. The other approach is to implement a real-time error correction feedback loop so that the machine migrates from one local minimum to another during an explorative search for ground states. Finally, a set of qualitative analogies connecting the CIM and traditional computer science techniques are pointed out. In particular, belief propagation and survey propagation used in combinatorial optimization are touched upon.
Coherent Ising machine (CIM) is a hardware solver that simulates the Ising model and finds optimal solutions to combinatorial optimization problems. However, for practical tasks, the computational process may be trapped in local minima, which is a key challenge for CIM. In this work, we design a CIM structure with a spiking neural network by adding dissipative pulses, which are anti-symmetrically coupled to the degenerate optical parametric oscillator pulses in CIM with a measurement feedback system. We find that the unstable oscillatory region of the spiking neural network could assist the CIM to escape from the trapped local minima. Moreover, we show that the machine has a different search mechanism than CIM, which can achieve a higher solution success probability and speed-up effect.
The analysis and optimization of complex systems can be reduced to mathematical problems collectively known as combinatorial optimization. Many such problems can be mapped onto ground-state search problems of the Ising model, and various artificial spin systems are now emerging as promising approaches. However, physical Ising machines have suffered from limited numbers of spin-spin couplings because of implementations based on localized spins, resulting in severe scalability problems. We report a 2000-spin network with all-to-all spin-spin couplings. Using a measurement and feedback scheme, we coupled time-multiplexed degenerate optical parametric oscillators to implement maximum cut problems on arbitrary graph topologies with up to 2000 nodes. Our coherent Ising machine outperformed simulated annealing in terms of accuracy and computation time for a 2000-node complete graph.