Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Simulation of Frustrated Ising Spins Using Three Coupled Parametric Oscillators: A Ground State Energy Approach

韩骋晓, Yingming Yan

Published Apr 17, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.21203/rs.3.rs-9211211/v1

Related papers

A coherent Ising machine for 2000-node optimization problems

Takahiro Inagaki, Yoshitaka Haribara, Koji Igarashi +14 more

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.

Science•Oct 21, 2016•799 citations

Three Strongly Coupled Kerr Parametric Oscillators Forming a Boltzmann Machine

Gabriel Margiani, Orjan Ameye, Oded Zilberberg +1 more

Coupled Kerr parametric oscillators (KPOs) are a promising resource for classical and quantum analog computation, for example to find the ground state of Ising Hamiltonians. Yet, the state space of strongly coupled KPO networks is very involved. As such, their phase diagram sometimes features either too few or too many states, including some that cannot be mapped to Ising spin configurations. This complexity makes it challenging to find and meet the conditions under which an analog optimization algorithm can be successful. Here, we demonstrate how to use three strongly coupled KPOs as a simulator for an Ising Hamiltonian, and estimate its ground state using a Boltzmann sampling measurement. While fully classical, our Letter is directly relevant for quantum systems operating on coherent states.

arXiv:2504.04254•Physical Review Letters•Jul 29, 2025•3 citations

Ising selector machine by Kerr parametric oscillators

Jacopo Tosca, Cristiano Ciuti, Claudio Conti +1 more

Ising machines are physical platforms designed to minimize the energy of classical Ising Hamiltonians, yet accessing specific excited states remains an open challenge of both fundamental and practical relevance. In this letter we show that a network of Kerr parametric oscillators (KPOs) naturally implements an Ising selector machine. By tuning the frequency detuning between the parametric pump and the oscillator resonances, the system can be steered to converge close to the ground state, the highest-energy configuration, or targeted intermediate excited states. Beyond mean field, numerical simulations based on the truncated Wigner approximation demonstrate that noise insertion preserves the energetic structure of the landscape. The targeted state emerges with an exponentially enhanced probability over the rest of the Ising spectrum. Our results establish the pump-cavity detuning as a control knob for navigating the full Ising energy landscape, opening a route to applications in Boltzmann sampling, hardness characterization, and spectral analysis of combinatorial problems.

quant-pharXiv:2604.12718•Apr 15, 2026•0 citations

Simulations of frustrated Ising Hamiltonians using quantum approximate optimization

Phillip C. Lotshaw, Hanjing Xu, Bilal Khalid +3 more

Novel magnetic materials are important for future technological advances. Theoretical and numerical calculations of ground-state properties are essential in understanding these materials, however, computational complexity limits conventional methods for studying these states. Here we investigate an alternative approach to preparing materials ground states using the quantum approximate optimization algorithm (QAOA) on near-term quantum computers. We study classical Ising spin models on unit cells of square, Shastry-Sutherland and triangular lattices, with varying field amplitudes and couplings in the material Hamiltonian. We find relationships between the theoretical QAOA success probability and the structure of the ground state, indicating that only a modest number of measurements ([Formula: see text]) are needed to find the ground state of our nine-spin Hamiltonians, even for parameters leading to frustrated magnetism. We further demonstrate the approach in calculations on a trapped-ion quantum computer and succeed in recovering each ground state of the Shastry-Sutherland unit cell with probabilities close to ideal theoretical values. The results demonstrate the viability of QAOA for materials ground state preparation in the frustrated Ising limit, giving important first steps towards larger sizes and more complex Hamiltonians where quantum computational advantage may prove essential in developing a systematic understanding of novel materials. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.

arXiv:2206.05343•Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences•Dec 4, 2022•15 citations

Frustration Elimination and Excited State Search in Coherent Ising Machines

Zheng-Yang Zhou, Clemens Gneiting, J. Q. You +1 more

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.

arXiv:2402.15090•Physical Review Letters•Mar 5, 2025•7 citations

Josephson parametric oscillator based Ising machine

Sasan Razmkhah, Mehdi Kamal, Nobuyuki Yoshikawa +1 more

Ising machines (IMs) have emerged as a promising solution for rapidly solving NP-complete combinatorial optimization problems, surpassing the capabilities of traditional computing methods. By efficiently determining the ground state of the Hamiltonian during the annealing process, IMs can effectively complement CPUs in tackling optimization challenges. To realize these IMs, a bistable oscillator is essential to emulate the atomic spins and interactions of the Ising model. This study introduces a Josephson parametric oscillator (JPO) based tile structure, serving as a fundamental unit for scalable superconductor-based IMs. Leveraging the bistable nature of JPOs, which are superconductor-based oscillators, the proposed machine can operate at frequencies of $\ensuremath{\sim}7.5$ GHz while consuming significantly less power (by three orders of magnitude) than complementary metal-oxide semiconductor based systems. Furthermore, the compatibility of the proposed tile structure with the Lechner-Hauke-Zoller architecture ensures its viability for large-scale integration. We conducted simulations of the tile in a noisy environment to validate its functionality. We verified its operational characteristics by comparing the results with the analytical solution of its Hamiltonian model. This verification demonstrates the feasibility and effectiveness of the JPO-based tile in implementing IMs, opening new avenues for efficient and scalable combinatorial optimization in quantum computing.

arXiv:2309.03407•Physical review. B./Physical review. B•Jan 16, 2024•11 citations
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.
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.
Coupled Kerr parametric oscillators (KPOs) are a promising resource for classical and quantum analog computation, for example to find the ground state of Ising Hamiltonians. Yet, the state space of strongly coupled KPO networks is very involved. As such, their phase diagram sometimes features either too few or too many states, including some that cannot be mapped to Ising spin configurations. This complexity makes it challenging to find and meet the conditions under which an analog optimization algorithm can be successful. Here, we demonstrate how to use three strongly coupled KPOs as a simulator for an Ising Hamiltonian, and estimate its ground state using a Boltzmann sampling measurement. While fully classical, our Letter is directly relevant for quantum systems operating on coherent states.
Ising machines are physical platforms designed to minimize the energy of classical Ising Hamiltonians, yet accessing specific excited states remains an open challenge of both fundamental and practical relevance. In this letter we show that a network of Kerr parametric oscillators (KPOs) naturally implements an Ising selector machine. By tuning the frequency detuning between the parametric pump and the oscillator resonances, the system can be steered to converge close to the ground state, the highest-energy configuration, or targeted intermediate excited states. Beyond mean field, numerical simulations based on the truncated Wigner approximation demonstrate that noise insertion preserves the energetic structure of the landscape. The targeted state emerges with an exponentially enhanced probability over the rest of the Ising spectrum. Our results establish the pump-cavity detuning as a control knob for navigating the full Ising energy landscape, opening a route to applications in Boltzmann sampling, hardness characterization, and spectral analysis of combinatorial problems.
Novel magnetic materials are important for future technological advances. Theoretical and numerical calculations of ground-state properties are essential in understanding these materials, however, computational complexity limits conventional methods for studying these states. Here we investigate an alternative approach to preparing materials ground states using the quantum approximate optimization algorithm (QAOA) on near-term quantum computers. We study classical Ising spin models on unit cells of square, Shastry-Sutherland and triangular lattices, with varying field amplitudes and couplings in the material Hamiltonian. We find relationships between the theoretical QAOA success probability and the structure of the ground state, indicating that only a modest number of measurements ([Formula: see text]) are needed to find the ground state of our nine-spin Hamiltonians, even for parameters leading to frustrated magnetism. We further demonstrate the approach in calculations on a trapped-ion quantum computer and succeed in recovering each ground state of the Shastry-Sutherland unit cell with probabilities close to ideal theoretical values. The results demonstrate the viability of QAOA for materials ground state preparation in the frustrated Ising limit, giving important first steps towards larger sizes and more complex Hamiltonians where quantum computational advantage may prove essential in developing a systematic understanding of novel materials. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.
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.
Ising machines (IMs) have emerged as a promising solution for rapidly solving NP-complete combinatorial optimization problems, surpassing the capabilities of traditional computing methods. By efficiently determining the ground state of the Hamiltonian during the annealing process, IMs can effectively complement CPUs in tackling optimization challenges. To realize these IMs, a bistable oscillator is essential to emulate the atomic spins and interactions of the Ising model. This study introduces a Josephson parametric oscillator (JPO) based tile structure, serving as a fundamental unit for scalable superconductor-based IMs. Leveraging the bistable nature of JPOs, which are superconductor-based oscillators, the proposed machine can operate at frequencies of $\ensuremath{\sim}7.5$ GHz while consuming significantly less power (by three orders of magnitude) than complementary metal-oxide semiconductor based systems. Furthermore, the compatibility of the proposed tile structure with the Lechner-Hauke-Zoller architecture ensures its viability for large-scale integration. We conducted simulations of the tile in a noisy environment to validate its functionality. We verified its operational characteristics by comparing the results with the analytical solution of its Hamiltonian model. This verification demonstrates the feasibility and effectiveness of the JPO-based tile in implementing IMs, opening new avenues for efficient and scalable combinatorial optimization in quantum computing.