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Chinese Physics B

Non-Markovian dynamical solver for efficient combinatorial optimization

Haijie 海杰 Xu 徐, Zhe Yuan

Published Nov 17, 2025• 0 citations
Publisher

Abstract

Abstract We incorporate a non-Markovian feedback mechanism into the simulated bifurcation method for dynamical solvers addressing combinatorial optimization problems. By reinjecting a portion of dissipated kinetic energy into each spin in a history-dependent and trajectory-informed manner, the method effectively suppresses early freezing induced by inelastic boundaries and enhances the system’s ability to explore complex energy landscapes. Numerical results on the maximum cut (MAX-CUT) instances of fully connected Sherrington–Kirkpatrick (SK) spin glass models, including the 2000-spin K 2000 benchmark, demonstrate that the non-Markovian algorithm significantly improves both solution quality and convergence speed. Tests on randomly generated SK instances with 100 to 1000 spins further indicate favorable scalability and substantial gains in computational efficiency. Moreover, the proposed scheme is well suited for massively parallel hardware implementations, such as field-programmable gate arrays, providing a practical and scalable approach for solving large-scale combinatorial optimization problems.

DOI
10.1088/1674-1056/ae1fea

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Abstract We incorporate a non-Markovian feedback mechanism into the simulated bifurcation method for dynamical solvers addressing combinatorial optimization problems. By reinjecting a portion of dissipated kinetic energy into each spin in a history-dependent and trajectory-informed manner, the method effectively suppresses early freezing induced by inelastic boundaries and enhances the system’s ability to explore complex energy landscapes. Numerical results on the maximum cut (MAX-CUT) instances of fully connected Sherrington–Kirkpatrick (SK) spin glass models, including the 2000-spin K 2000 benchmark, demonstrate that the non-Markovian algorithm significantly improves both solution quality and convergence speed. Tests on randomly generated SK instances with 100 to 1000 spins further indicate favorable scalability and substantial gains in computational efficiency. Moreover, the proposed scheme is well suited for massively parallel hardware implementations, such as field-programmable gate arrays, providing a practical and scalable approach for solving large-scale combinatorial optimization problems.
Abstract Combinatorial optimization problems and ground state problems of spin glasses are crucial in various fields of science and technology. However, they often belong to the computational class of NP-hard, presenting significant computational challenges. Traditional algorithms inspired by statistical physics like simulated annealing have been widely adopted. Recently, advancements in Ising machines, such as quantum annealers and coherent Ising machines, offer new paradigms for solving these problems efficiently by embedding them into the analog evolution of nonlinear dynamical systems. However, existing dynamics-based algorithms often suffer from low convergence rates and local minima traps. In this work, we introduce the dual mean-field dynamics into Ising machines. The approach integrates the gradient force and the transverse force into the dynamics of Ising machines in solving combinatorial optimization problems, making it easier for the system to jump out of the local minimums and allowing the dynamics to explore wider in configuration space. We conduct extensive numerical experiments using the Sherrington–Kirkpatrick spin glass up to 10 000 spins and the maximum cut problems with the standard G-set benchmarks. The numerical results demonstrate that our dual mean-field dynamics approach enhances the performance of base Ising machines, providing a more effective solution for large-scale combinatorial optimization problems.
Simulated Bifurcation (SB) algorithms, inspired by quantum annealing, can efficiently solve large-scale combinatorial optimization problems on classical hardware, often outperforming traditional approaches such as simulated annealing. However, their tendency to be trapped in local optima limits global solution quality. In this work, we introduce Tabu-Enhanced Simulated Bifurcation (TESB), an improved SB variant that incorporates a Tabu Search-inspired mechanism. By leveraging a dynamic penalty guided by early search history, TESB can naturally avoid revisiting suboptimal regions. On Max-Cut benchmarks, TESB achieves up to a three-order-of-magnitude reduction in Time-to-Solution compared to standard SB. When applied to particle track reconstruction in high-energy physics, TESB identifies lower-energy configurations on problems exceeding 100,000 spin variables, demonstrating enhanced scalability and performance across a wide range of combinatorial tasks. Combinatorial optimization is a challenging yet crucial class of problems, and Simulated Bifurcation (SB) is a promising approximate solver developed in recent years. The authors enhance SB with ideas from Tabu Search to improve its performance, demonstrating clear gains on Max-Cut and particle track reconstruction problems.
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.
Combinatorial optimization problems (COPs) constitute a fundamental class of computational challenges with extensive applications across scientific and industrial domains. The emergence of the coherent Ising machine (CIM) as a computational paradigm has demonstrated exceptional capabilities in rapidly generating arbitrary spin configurations and effectively solving large-scale COPs. Despite its promising potential, the CIM framework encounters an inherent limitation related to amplitude heterogeneity. Current approaches to this limitation primarily rely on Gaussian approximations to model quantum noise in the system's amplitude dynamics within the original CIM framework. However, these methods generally fail to account for the temporal and periodic non-Markovian characteristics of pulses in practical environments. In this study, we introduce temporally correlated non-Markovian noise into the dynamics of both conventional and spiking neural network-based CIM systems, investigating their performance through comprehensive hyperparameter analysis. Our method incorporates a detailed examination of hyperparameter selection and its influence on algorithmic efficiency. Through extensive numerical simulations, we demonstrate that the integration of non-Markovian noise significantly accelerates variance evolution in both canonical coordinates and amplitude dynamics over time. The proposed approach not only exhibits enhanced computational performance but also shows superior scalability for large-scale problem instances. These findings suggest that our approach offers distinct advantages in addressing complex COPs, potentially opening new avenues for efficient optimization in various applications.
Developed from the Ising model, Ising machines are promising to be efficient domain-specific accelerators for solving combinatorial optimization problems (COPs). A simulated bifurcation (SB) algorithm enables an Ising machine to achieve massive parallelism in the simulation of Hamiltonian dynamics. Although SB significantly accelerates the search, more resources are required due to the use of continuous variables for the position of oscillators to obtain discrete spin states, compared to conventional simulated annealing. This article proposes ternary and multiple-value quantized SB (qSB) algorithms by discretizing the position variables used for the hardware-consuming multiply-accumulate (MAC) operations in SB. These quantization schemes do not only reduce the computational complexity, but also improve the solution quality for COPs. Specifically, the ternary qSB with dynamic threshold settings converts the MAC into addition and subtraction. To improve the precision in number representation when solving large-scale COPs, a uniform quantization scheme is applied to provide multiple-valued quantization. Alternatively, a logarithmic qSB leverages the evolution characteristics of position variables and implements multiplication by using simple shift operations. We demonstrate that using the proposed qSB improves the solution quality in a long search and accelerates energy convergence in a short search for solving COPs tackled by up to 2000 fully connected spins.
Combinatorial optimization problems are economically valuable but computationally hard to solve. Many practical combinatorial optimizations can be converted to the ground-state search problems of Ising spin models. Simulated bifurcation (SB) is a quantum-inspired algorithm to solve these Ising problems. One of the remarkable features of SB is the high-degree parallelism, providing an opportunity for quickly solving those problems by massively parallel processing. In this article, we review our recent works on the design and implementation of high-performance FPGA-based accelerators for SB and their applications toward innovative real-time systems that make optimal responses to ever-changing situations. An example of such an application is an ultrafast financial transaction machine that detects the most profitable cross-currency arbitrage opportunities at microsecond speeds. Also, we show a scale-out architecture for SB-based Ising machines with all-to-all spin-spin couplings that allows continued scaling of both machine size and computational throughput by connecting multiple chips, rather than scaling up a single chip.