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Scientific Reports

Accelerating a coherent Ising machine by XY-Ising spin transition

Kyungduk Kim, Y. Yamamoto

Published Feb 24, 2026• 0 citations
Publisher

Abstract

We numerically demonstrate that introducing XY-spin dynamics enabled by phase-insensitive optical parametric gain enhances the performance of coherent Ising machines (CIMs). Whereas conventional CIMs based on phase-sensitive optical parametric gain confine spins to two discrete Ising states, nondegenerate optical parametric oscillators with phase-insensitive gain support continuous-phase XY spins, facilitating spin flips that enable the system to escape local minima. Benchmarking on Wishart-planted problem instances shows that gradually transitioning from XY to Ising spins reduces the time-to-solution by roughly an order of magnitude relative to purely Ising-based dynamics. Furthermore, tailoring the transitions between XY-like and Ising-like regimes yields additional improvements beyond a simple unidirectional XY-to-Ising transition. Our results establish a new framework for engineering CIM dynamics in full phase-quadrature space and point toward fully optical architectures for efficient combinatorial optimization.

DOI
10.1038/s41598-026-41315-7

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We numerically demonstrate that introducing XY-spin dynamics enabled by phase-insensitive optical parametric gain enhances the performance of coherent Ising machines (CIMs). Whereas conventional CIMs based on phase-sensitive optical parametric gain confine spins to two discrete Ising states, nondegenerate optical parametric oscillators with phase-insensitive gain support continuous-phase XY spins, facilitating spin flips that enable the system to escape local minima. Benchmarking on Wishart-planted problem instances shows that gradually transitioning from XY to Ising spins reduces the time-to-solution by roughly an order of magnitude relative to purely Ising-based dynamics. Furthermore, tailoring the transitions between XY-like and Ising-like regimes yields additional improvements beyond a simple unidirectional XY-to-Ising transition. Our results establish a new framework for engineering CIM dynamics in full phase-quadrature space and point toward fully optical architectures for efficient combinatorial optimization.
Coherent Ising machines (CIMs) are increasingly anticipated as a new computer that can solve combinatorial optimization problems. CIMs use degenerate optical parametric oscillator pulses as artificial spins. We constructed a 100,512-spin coherent Ising machine, the world's largest. Our CIM can find good solutions to maximum cut problems of 100,000-node graphs ~1000 times faster than simulated annealing. Furthermore, it provides both very good scores and a broad distribution when we operate it near the phase transition point.
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.
In this work, we study the detection of quantum phase transitions (QPTs) in coherent Ising machines (CIMs) through a spectral mapping between the one-dimensional XY spin model and a network of degenerate optical parametric oscillators (DOPOs). This exact correspondence reveals that the DOPO spectrum faithfully encodes the critical behavior of anisotropic and isotropic XY chains, while a strongly anisotropic regime provides a controlled near-transverse-field-Ising realization, with the energy gap closing precisely at the critical point. The ground-state energy density and its derivatives are analyzed, and the resulting singularities in magnetic susceptibility accurately pinpoint the quantum critical points of the corresponding spin model. These results show that CIMs not only serve as powerful platforms for solving combinatorial optimization problems but also provide a versatile tool for probing universal quantum critical phenomena, bridging quantum-spin models and photonic quantum systems.
Computers based on physical systems are increasingly anticipated to overcome the impending limitations on digital computer performance. One such computer is a coherent Ising machine (CIM) for solving combinatorial optimization problems. Here, we report a CIM with 100,512 degenerate optical parametric oscillator pulses working as the Ising spins. We show that the CIM delivers fine solutions to maximum cut problems of 100,000-node graphs drastically faster than standard simulated annealing. Moreover, the CIM, when operated near the phase transition point, provides some extremely good solutions and a very broad distribution. This characteristic will be useful for applications that require fast random sampling such as machine learning.
We propose and numerically study the performance of an all-optical machine for tackling combinatorial clustering, one of the unsupervised machine learning problems. A problem instance is encoded into the phase of time-multiplexed optical pulses, which are coupled repulsively through optical delay lines. To maintain uniform pulse amplitudes, we utilize a nonlinear amplifier with gain saturation, enabling our optical architecture to emulate the classical XY-spin system. This solver, called the coherent XY machine, leads to the formation of clusters in an optical phase space and allows us to efficiently identify the solution with post-processing. Additionally, we implement momentum in our solver to provide a powerful mechanism for escaping local minima and searching for the global optimum. Benchmarking our approach with a most advanced Ising-spin-based solver reveals a two-orders-of-magnitude improvement in the time-to-solution of the algorithm. Furthermore, our approach exhibits scaling advantages for larger problems, which will facilitate time- and energy-efficient data clustering.
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.