Edge-of-chaos enhanced quantum-inspired algorithm for combinatorial optimization

Nonlinear dynamical systems with continuous variables can be used for solving combinatorial optimization problems with discrete variables. Numerical simulations of them are also useful as heuristic algorithms with a desirable property, namely, parallelizability, which allows us to execute them in a massively parallel manner, leading to ultrafast performance. However, the dynamical-system approaches with continuous variables are usually less accurate than conventional approaches with discrete variables such as simulated annealing. To improve the solution accuracy of a quantum-inspired algorithm called simulated bifurcation (SB), which was found from classical simulation of a quantum nonlinear oscillator network exhibiting quantum bifurcation, here we generalize it by introducing nonlinear control of individual bifurcation parameters and show that the generalized SB (GSB) can achieve surprisingly high performance, namely, almost 100% success probabilities for some large-scale problems. As a result, the time to solution for a 2,000-variable problem is shortened to 10 ms by a GSB-based machine, which is two orders of magnitude shorter than the best known value, 1.3 s, previously obtained by an SB-based machine. To examine the reason for the ultrahigh performance, we investigated chaos in the GSB changing the nonlinear-control strength and found that the dramatic increase of success probabilities happens near the edge of chaos. That is, the GSB can find a solution with high probability by harnessing the edge of chaos. This finding suggests that dynamical-system approaches to combinatorial optimization will be enhanced by harnessing the edge of chaos, opening a broad possibility for physics-inspired approaches to combinatorial optimization.

💡 Research Summary

The paper addresses a longstanding limitation of physics‑inspired combinatorial optimization methods that rely on continuous‑variable nonlinear dynamical systems. Simulated bifurcation (SB), a classical simulation of a quantum nonlinear oscillator network, offers excellent parallelizability and ultrafast execution on GPUs and FPGAs, but its solution quality lags behind discrete‑variable heuristics such as simulated annealing (SA). To bridge this gap, the authors propose a Generalized Ballistic Simulated Bifurcation (GBSB), which extends the conventional SB by assigning an individual bifurcation parameter p_i(t) to each oscillator and controlling its evolution with a nonlinear term. Specifically, the update rule for p_i(t) includes a factor A·(1−x_i²) that slows the decay of p_i(t) when the corresponding position x_i approaches the hard walls at ±1. This mechanism prevents oscillators from becoming trapped at the walls, thereby reducing premature convergence to local minima.

The authors evaluate GBSB on several large‑scale benchmark problems: (i) K2000, a 2,000‑spin all‑to‑all MAX‑CUT instance; (ii) a set of 100 random 700‑spin all‑to‑all Ising instances; and (iii) ten instances from the G‑set, which are sparse MAX‑CUT problems equivalent to 800‑spin Ising models. By sweeping the total number of integration steps M and the nonlinear‑control strength A, they find regions where the success probability P_S of obtaining the best‑known cut value approaches 100 % for K2000 and exceeds 90 % for many 700‑spin and G‑set instances. Notably, the dramatic rise in P_S occurs precisely when a normalized distance metric δ(t_M) between two trajectories that start from infinitesimally different initial conditions reaches ≈1/√2, a hallmark of chaotic dynamics. The authors therefore identify an “edge‑of‑chaos” regime where the system is neither fully regular nor fully chaotic; in this regime the dynamics appear to avoid trapping in local minima while still preserving enough structure to guide the search toward global optima.

To confirm that the observed effect is intrinsic to the continuous dynamics rather than an artifact of numerical discretization, the authors vary the time step Δt over several orders of magnitude. Both the success probability and the chaos indicator δ(t_M) remain qualitatively unchanged, demonstrating robustness of the edge‑of‑chaos phenomenon. Moreover, unlike SA, the success probability does not monotonically increase with the number of steps M; instead, it exhibits a “valley of chaos” where optimal performance aligns with specific values of M that modulate the effective decay rate of the individual p_i(t).

The paper also presents a hardware implementation of GBSB on a field‑programmable gate array (FPGA). Because each p_i(t) depends only on its own state x_i(t), the algorithm is embarrassingly parallel and maps naturally onto the fine‑grained parallelism of modern FPGAs. A 2,048‑spin GBSB machine (GBSBM) was built, achieving a time‑to‑solution of roughly 10 ms for the K2000 instance—two orders of magnitude faster than the best previously reported SB‑based FPGA machine (1.3 s). This demonstrates that the algorithm’s theoretical advantages translate into practical ultrafast performance without sacrificing solution quality.

In summary, the key contributions are: (1) introduction of individual, nonlinearly controlled bifurcation parameters that dramatically improve SB’s solution accuracy; (2) empirical identification of an edge‑of‑chaos regime that correlates with near‑perfect success rates, providing a new physical insight into why dynamical‑system‑based heuristics can outperform traditional stochastic methods; (3) validation of the phenomenon’s independence from discretization artifacts; and (4) a scalable FPGA implementation that showcases the algorithm’s parallelizability and real‑time capability. The work opens several avenues for future research, including analytical modeling of the optimal A‑M schedule, exploration of alternative nonlinear control functions, comparison with genuine quantum‑hardware implementations, and extension to other NP‑hard problem classes such as constraint satisfaction and routing. By harnessing the edge of chaos, the authors suggest a promising new paradigm for physics‑inspired combinatorial optimization.