Different Paths, Same Destination: Designing New Physics-Inspired Dynamical Systems with Engineered Stability to Minimize the Ising Hamiltonian

TL;DR

This work investigates solving COPs via physics-inspired dynamical systems and shows that the same objective, the Ising Hamiltonian $H = - \sum_{i,j}^{N} J_{ij} \sigma_i \sigma_j$, can be minimized by distinct dynamics. It introduces the Dynamical Ising Machine (DIM) with additive-phase interactions and second harmonic injection, establishes that its energy function decreases over time and, at $K=1/2$, maps to the Ising Hamiltonian up to a constant. Through Jacobian-based stability analysis, the paper identifies a pitchfork-like bifurcation where the ground-state configuration shifts from $\phi^* \in \{\tfrac{\pi}{2}\}$ to $\phi^* \in \{0,\pi\}$ as $K_s$ grows, and defines critical thresholds $K^{\{\pi/2\}}_{s,ls}$ and $K_{s,E}$ linked to stability and ground-state energetics. Empirical results illustrate that model performance is graph-dependent and that a diversification strategy—running parallel DIM and OIM trials—yields more robust, high-quality solutions across graphs, with $K_{s,E}$ providing a practical energy estimate that closely tracks the true ground state.

Abstract

Oscillator Ising machines (OIMs) represent an exemplar case of using physics-inspired non-linear dynamical systems to solve computationally challenging combinatorial optimization problems (COPs). The computational performance of such systems is highly sensitive to the underlying dynamical properties, the topology of the input graph, and their relative compatibility. In this work, we explore the concept of designing different dynamical systems that minimize the same objective function but exhibit drastically different dynamical properties. Our goal is to leverage this diversification in dynamics to reduce the sensitivity of the computational performance to the underlying graph, and subsequently, enhance the overall effectiveness of such physics-based computational methods. To this end, we introduce a novel dynamical system, the Dynamical Ising Machine (DIM), which, like the OIM, minimizes the Ising Hamiltonian but offers significantly different dynamical properties. We analyze the characteristic properties of the DIM and compare them with those of the OIM. We also show that the relative performance of each model is dependent on the input graph. Our work illustrates that using multiple dynamical systems with varying properties to solve the same COP enables an effective method that is less sensitive to the input graph, while producing robust solutions.

Figures (3)

• Figure 1: (a) An Illustrative graph with 15 nodes and 56 edges. The edges were randomly generated. (b) Time evoution of the system parameters, $K (=1)$, and $K_s$ . Resulting phase response of the (c) OIM; and (d) DIM, respectively. In (d), the critical $K_s$ at the point of bifurcation is observed to be $K_{s,E}=2.4$.
• Figure 2: (a)(b) Illustrative graphs (graph #1 and graphs#2) with 15 nodes considered in the example, respectively. (c-f) Minimum (blue circles) and maximum (orange diamonds) value of $\lambda_L$ for the various phase configurations at each Ising energy, calculated using the OIM and the DIM dynamics, for each of the two graphs, respectively. Insets magnifying the relative distribution of $\lambda_L$ as a function of energy close to the ground state are also shown. (g-j) Histogram of the Ising energy ($H$) computed using the OIM and the DIM dynamics for graph #1 and #2, respectively, for 50 independent trials.
• Figure 3: (a) Temporal evolution of $\Delta$ for the DIM dynamics presented in Fig.\ref{['fig:comparison']}; (b) Magnified version clearly showing the onset of bifurcation at $\Delta=0.006$.