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

Nature Communications

Efficient optimization accelerator framework for multi-state spin Ising problems

Chirag Garg, Sayeef Salahuddin

Published Oct 30, 2025• 3 citations
Publisher

Abstract

Abstract Ising Machines are emerging hardware architectures that efficiently solve NP-hard combinatorial optimization problems. Generally, combinatorial problems are transformed into quadratic unconstrained binary optimization (QUBO) form, but this transformation often complicates the solution landscape, degrading performance, especially for multi-state problems. To address this challenge, we model spin interactions as generalized boolean logic function to significantly reduce the exploration space. We demonstrate the effectiveness of our approach on graph coloring problem using probabilistic Ising solvers, achieving similar accuracy compared to state-of-the-art heuristics and machine learning algorithms. It also shows significant improvement over state-of-the-art QUBO-based Ising solvers, including probabilistic Ising and simulated bifurcation machines. We also design 1024-neuron all-to-all connected probabilistic Ising accelerator on FPGA with the proposed approach that shows $$\sim$$ ~ 10000 $$\times$$ × performance acceleration compared to GPU-based Tabucol heuristics and reducing physical neurons by 1.5 $$-$$ − 4 $$\times$$ × over baseline Ising frameworks. Thus, this work establishes superior efficiency, scalability and solution quality for multi-state optimization problems.

DOI
10.1038/s41467-025-64625-2

Related papers

Efficient Optimization Accelerator Framework for Multistate Ising Problems

Chirag Garg, Sayeef Salahuddin

Ising Machines are emerging hardware architectures that efficiently solve NP-Hard combinatorial optimization problems. Generally, combinatorial problems are transformed into quadratic unconstrained binary optimization (QUBO) form, but this transformation often complicates the solution landscape, degrading performance, especially for multi-state problems. To address this challenge, we model spin interactions as generalized boolean logic function to significantly reduce the exploration space. We demonstrate the effectiveness of our approach on graph coloring problem using probabilistic Ising solvers, achieving similar accuracy compared to state-of-the-art heuristics and machine learning algorithms. It also shows significant improvement over state-of-the-art QUBO-based Ising solvers, including probabilistic Ising and simulated bifurcation machines. We also design 1024-neuron all-to-all connected probabilistic Ising accelerator on FPGA with the proposed approach that shows ~10000x performance acceleration compared to GPU-based Tabucol heuristics and reducing physical neurons by 1.5-4x over baseline Ising frameworks. Thus, this work establishes superior efficiency, scalability and solution quality for multi-state optimization problems.

arXiv:2505.20250•May 26, 2025•0 citations

Scalable Optimization with GIS-PIM: A Generalized Integer-State Probabilistic Ising Machine

Chirag Garg, Sayeef Salahuddin

Physics-inspired hardware solvers based on Ising machines have gained significant attention for addressing combinatorial optimization problems across diverse domains, including logistics, communication networks, and financial optimization. While the problems in these domains typically involve integer states, they are often reformulated to fit the quadratic unconstrained binary optimization (QUBO) framework supported by Ising machines. This is usually achieved by introducing penalty terms, which can degrade solution quality. In this work, we present a generalized integer-state probabilistic Ising machine (GIS-PIM) that natively supports integer-state encoding using compact binary representations. By avoiding one-hot encoding and associated constraints, GIS-PIM efficiently explores the solution space with fewer spin variables. The framework is implemented on a GPU and evaluated on publicly available graph coloring benchmarks, including both synthetic and realworld datasets. It demonstrates strong scalability, solving problem instances with up to 13 colors and graph sizes reaching approximately 20,000 nodes, a level not previously reported for other Ising-based methods. Our results indicate that the proposed approach: i) achieves solution accuracies exceeding 98%, competitive with leading classical methods such as Tabucol and graph neural network-based solver; and ii) significantly performs better over state-of-the-art simulated bifurcation (SBM) and QUBObased probabilistic Ising machines (Qu-PIM), achieving higher success probability and faster time-to-solution.

Asia and South Pacific Design Automation Conference (ASP-DAC)•Jan 19, 2026•0 citations

Linearizing Binary Optimization Problems Using Variable Posets for Ising Machines

Kentaro Ohno, Nozomu Togawa

Ising machines are next-generation computers expected to efficiently sample near-optimal solutions of combinatorial optimization problems. Combinatorial optimization problems are modeled as quadratic unconstrained binary optimization (QUBO) problems to apply an Ising machine. However, current state-of-the-art Ising machines still often fail to output nearoptimal solutions due to the complicated energy landscape of QUBO problems. Furthermore, the physical implementation of Ising machines severely restricts the size of QUBO problems to be input as a result of limited hardware graph structures. In this study, we take a new approach to these challenges by injecting auxiliary penalties preserving the optimum, which reduces quadratic terms in QUBO objective functions. The process simultaneously simplifies the energy landscape of QUBO problems, allowing the search for near-optimal solutions, and makes QUBO problems sparser, facilitating encoding into Ising machines with restriction on the hardware graph structure. We propose linearization of QUBO problems using variable posets as an outcome of the approach. By applying the proposed method to synthetic QUBO instances and to multi-dimensional knapsack problems, we empirically validate the effects on enhancing minorembedding of QUBO problems and the performance of Ising machines.

arXiv:2307.05125•IEEE Transactions on Emerging Topics in Computing•May 30, 2024•4 citations

Accelerating the Max-Cut Problems via Distributed Ising Machine Solvers

Xing-Yu Wu, Yi-Xin Zhang, Chen-rui Fan +1 more

The Ising machine, as a quantum-inspired computing system, can be used to efficiently solve combinatorial optimization problems. Ongoing studies have positioned it to potentially surpass the performance limitations of traditional computers. However, such Ising machines also suffer from scalability as the solution quality becomes sub-optimal when the problem size increases. In this work, we propose an Ising-based framework that uses a distributed Ising algorithm to find the near-optimal solution for large-scale Max-Cut problems through parallel processing via graph segmentation and reassembly. We selected the coherent Ising machine as the hardware solver and successfully found the solution for a 2000-node instance utilizing 20 computational bits on the Gset standard dataset. Additionally, we investigated the key factors influencing the system's performance and discovered that setting the number of reorganizations to one results in higher accuracy. Building on this configuration, we utilized a 100-bit CIM hardware platform (CPQC-1) to solve instances with varying numbers of nodes. The experimental results agree well with the theoretical simulations, confirming the practical feasibility of our method. Moreover, compared to existing Ising-based solvers, our algorithm significantly reduces computational resource requirements while achieving superior performance on large-scale combinatorial optimization problems, highlighting its effectiveness on quantum-inspired hardware.

IEEE Transactions on Quantum Engineering•Jan 1, 2026•0 citations

An Evolutionary Ising Optimization Framework for Unconstrained Binary Quadratic Programming

Wujie Fu, Anupam Trivedi, Guanyu Chen +3 more

An Ising machine (IM), as a type of analog computer tailored for tackling intractable combinatorial optimization problems, has attracted remarkable attention in recent years. In contrast to the blossoming field of bespoke IM hardware, developing metaheuristics from IMs remains largely uninvestigated. Here, we propose a physics-inspired evolutionary computation paradigm, termed the Ising optimization framework (IOF); it comprises a unique Ising algorithm and a hybrid annealing scheme, which together are well-suited for solving quadratic unconstrained binary optimization (QUBO) problems embedded in Ising system energy. The Ising algorithm leverages a set of iterated self-mapping functions to evolve an Ising-spin swarm, enabling efficient energy minimization in artificial Ising systems while mitigating detrimental chaos. Complementing the algorithm, a hybrid annealing scheme integrating singular value dropout, bifurcation control, and a nudging strategy, is devised to augment the overall optimization capacity. The effectiveness of the IOF is validated on various Ising and Max-cut problems with decision variables ranging from 625 to 5000 in number. In comparison to four major types of methods for solving QUBOs, including IM simulations, nature-inspired algorithms, a state-of-the-art heuristic, and the commercial solver Gurobi, the IOF consistently demonstrates notable optimization quality and computational efficiency. This paper provides a theoretical foundation and practical guidelines for bridging Ising-inspired approaches with evolutionary computation, offering an evolutionary perspective on Ising optimizations and suggesting a fertile avenue for future research and application.

IEEE Transactions on Evolutionary Computation•May 7, 2025•2 citations

Physics-Inspired Optimization in the QUBO Framework: Key Concepts and Approaches

Lien-Po Yu, Chin-Fu Nien

Quantum computing promises to have a tremendous advantage over its classical counterpart for solving computationally hard problems, yet remains in a relatively early stage for practical applications owing to the limited capabilities of today’s quantum computers. The approach to the special purposes of quantum computers by exploiting the special-purpose physics-inspired or quantum-inspired computers is emerging as a novel alternative to its quantum counterpart in tackling hard problems in high-performance computing. Inspired by physics, the Ising machine — a type of special-purpose computer that implements or emulates physics or quantum effects of the Ising model to speed up finding solutions to optimization problems — has recently become an active research area in the field of combinatorial optimization. This paper is to address the key enabling software and hardware technology underlying physics-inspired optimization using Ising machines in the unified quadratic unconstrained binary optimization (QUBO) framework for modeling and solving computationally hard combinatorial optimization problems, and with an aim to shed some light on the challenges and opportunities associated with the ever-growing landscape of this novel high-performance computing.

SPIN•Aug 30, 2023•1 citation
Abstract Ising Machines are emerging hardware architectures that efficiently solve NP-hard combinatorial optimization problems. Generally, combinatorial problems are transformed into quadratic unconstrained binary optimization (QUBO) form, but this transformation often complicates the solution landscape, degrading performance, especially for multi-state problems. To address this challenge, we model spin interactions as generalized boolean logic function to significantly reduce the exploration space. We demonstrate the effectiveness of our approach on graph coloring problem using probabilistic Ising solvers, achieving similar accuracy compared to state-of-the-art heuristics and machine learning algorithms. It also shows significant improvement over state-of-the-art QUBO-based Ising solvers, including probabilistic Ising and simulated bifurcation machines. We also design 1024-neuron all-to-all connected probabilistic Ising accelerator on FPGA with the proposed approach that shows ∼\sim∼ ~ 10000 ×\times× × performance acceleration compared to GPU-based Tabucol heuristics and reducing physical neurons by 1.5 −-− − 4 ×\times× × over baseline Ising frameworks. Thus, this work establishes superior efficiency, scalability and solution quality for multi-state optimization problems.
Ising Machines are emerging hardware architectures that efficiently solve NP-Hard combinatorial optimization problems. Generally, combinatorial problems are transformed into quadratic unconstrained binary optimization (QUBO) form, but this transformation often complicates the solution landscape, degrading performance, especially for multi-state problems. To address this challenge, we model spin interactions as generalized boolean logic function to significantly reduce the exploration space. We demonstrate the effectiveness of our approach on graph coloring problem using probabilistic Ising solvers, achieving similar accuracy compared to state-of-the-art heuristics and machine learning algorithms. It also shows significant improvement over state-of-the-art QUBO-based Ising solvers, including probabilistic Ising and simulated bifurcation machines. We also design 1024-neuron all-to-all connected probabilistic Ising accelerator on FPGA with the proposed approach that shows ~10000x performance acceleration compared to GPU-based Tabucol heuristics and reducing physical neurons by 1.5-4x over baseline Ising frameworks. Thus, this work establishes superior efficiency, scalability and solution quality for multi-state optimization problems.
Physics-inspired hardware solvers based on Ising machines have gained significant attention for addressing combinatorial optimization problems across diverse domains, including logistics, communication networks, and financial optimization. While the problems in these domains typically involve integer states, they are often reformulated to fit the quadratic unconstrained binary optimization (QUBO) framework supported by Ising machines. This is usually achieved by introducing penalty terms, which can degrade solution quality. In this work, we present a generalized integer-state probabilistic Ising machine (GIS-PIM) that natively supports integer-state encoding using compact binary representations. By avoiding one-hot encoding and associated constraints, GIS-PIM efficiently explores the solution space with fewer spin variables. The framework is implemented on a GPU and evaluated on publicly available graph coloring benchmarks, including both synthetic and realworld datasets. It demonstrates strong scalability, solving problem instances with up to 13 colors and graph sizes reaching approximately 20,000 nodes, a level not previously reported for other Ising-based methods. Our results indicate that the proposed approach: i) achieves solution accuracies exceeding 98%, competitive with leading classical methods such as Tabucol and graph neural network-based solver; and ii) significantly performs better over state-of-the-art simulated bifurcation (SBM) and QUBObased probabilistic Ising machines (Qu-PIM), achieving higher success probability and faster time-to-solution.
Ising machines are next-generation computers expected to efficiently sample near-optimal solutions of combinatorial optimization problems. Combinatorial optimization problems are modeled as quadratic unconstrained binary optimization (QUBO) problems to apply an Ising machine. However, current state-of-the-art Ising machines still often fail to output nearoptimal solutions due to the complicated energy landscape of QUBO problems. Furthermore, the physical implementation of Ising machines severely restricts the size of QUBO problems to be input as a result of limited hardware graph structures. In this study, we take a new approach to these challenges by injecting auxiliary penalties preserving the optimum, which reduces quadratic terms in QUBO objective functions. The process simultaneously simplifies the energy landscape of QUBO problems, allowing the search for near-optimal solutions, and makes QUBO problems sparser, facilitating encoding into Ising machines with restriction on the hardware graph structure. We propose linearization of QUBO problems using variable posets as an outcome of the approach. By applying the proposed method to synthetic QUBO instances and to multi-dimensional knapsack problems, we empirically validate the effects on enhancing minorembedding of QUBO problems and the performance of Ising machines.
The Ising machine, as a quantum-inspired computing system, can be used to efficiently solve combinatorial optimization problems. Ongoing studies have positioned it to potentially surpass the performance limitations of traditional computers. However, such Ising machines also suffer from scalability as the solution quality becomes sub-optimal when the problem size increases. In this work, we propose an Ising-based framework that uses a distributed Ising algorithm to find the near-optimal solution for large-scale Max-Cut problems through parallel processing via graph segmentation and reassembly. We selected the coherent Ising machine as the hardware solver and successfully found the solution for a 2000-node instance utilizing 20 computational bits on the Gset standard dataset. Additionally, we investigated the key factors influencing the system's performance and discovered that setting the number of reorganizations to one results in higher accuracy. Building on this configuration, we utilized a 100-bit CIM hardware platform (CPQC-1) to solve instances with varying numbers of nodes. The experimental results agree well with the theoretical simulations, confirming the practical feasibility of our method. Moreover, compared to existing Ising-based solvers, our algorithm significantly reduces computational resource requirements while achieving superior performance on large-scale combinatorial optimization problems, highlighting its effectiveness on quantum-inspired hardware.
An Ising machine (IM), as a type of analog computer tailored for tackling intractable combinatorial optimization problems, has attracted remarkable attention in recent years. In contrast to the blossoming field of bespoke IM hardware, developing metaheuristics from IMs remains largely uninvestigated. Here, we propose a physics-inspired evolutionary computation paradigm, termed the Ising optimization framework (IOF); it comprises a unique Ising algorithm and a hybrid annealing scheme, which together are well-suited for solving quadratic unconstrained binary optimization (QUBO) problems embedded in Ising system energy. The Ising algorithm leverages a set of iterated self-mapping functions to evolve an Ising-spin swarm, enabling efficient energy minimization in artificial Ising systems while mitigating detrimental chaos. Complementing the algorithm, a hybrid annealing scheme integrating singular value dropout, bifurcation control, and a nudging strategy, is devised to augment the overall optimization capacity. The effectiveness of the IOF is validated on various Ising and Max-cut problems with decision variables ranging from 625 to 5000 in number. In comparison to four major types of methods for solving QUBOs, including IM simulations, nature-inspired algorithms, a state-of-the-art heuristic, and the commercial solver Gurobi, the IOF consistently demonstrates notable optimization quality and computational efficiency. This paper provides a theoretical foundation and practical guidelines for bridging Ising-inspired approaches with evolutionary computation, offering an evolutionary perspective on Ising optimizations and suggesting a fertile avenue for future research and application.
Quantum computing promises to have a tremendous advantage over its classical counterpart for solving computationally hard problems, yet remains in a relatively early stage for practical applications owing to the limited capabilities of today’s quantum computers. The approach to the special purposes of quantum computers by exploiting the special-purpose physics-inspired or quantum-inspired computers is emerging as a novel alternative to its quantum counterpart in tackling hard problems in high-performance computing. Inspired by physics, the Ising machine — a type of special-purpose computer that implements or emulates physics or quantum effects of the Ising model to speed up finding solutions to optimization problems — has recently become an active research area in the field of combinatorial optimization. This paper is to address the key enabling software and hardware technology underlying physics-inspired optimization using Ising machines in the unified quadratic unconstrained binary optimization (QUBO) framework for modeling and solving computationally hard combinatorial optimization problems, and with an aim to shed some light on the challenges and opportunities associated with the ever-growing landscape of this novel high-performance computing.