Self-contained relaxation-based dynamical Ising machines

TL;DR

This work addresses the rounding bottleneck in relaxation-based Ising machines for the max-cut problem by introducing the V2 model, a core relaxation whose objective is $\mathcal{C}_{\text{V2}}(\boldsymbol{\xi}) = \tfrac{1}{4} \sum_{m,n} A_{m,n} \lvert \xi_m - \xi_n \rvert_{[-2,2]}$. The key result is that from any non-binary initial state, the V2 dynamics converge to a state that trivially rounds to a binary configuration with a cut not worse than the best possible rounding of the initial state; moreover, if the initial state is a small perturbation of a binary solution, the terminal state yields a cut not smaller than the original binary state, and random agitations drive convergence to the maximum cut almost surely. The authors demonstrate these properties both theoretically (via the clustering structure of critical points and rounding invariance) and empirically (comparing V2 against 1-opt and a coherent Ising machine on random 3-regular graphs), showing that the V2 model can serve as the self-contained final stage of relaxation-based Ising machines. This advances practical optimization by enabling self-contained, dynamical rounding and robust performance on large NP-hard instances. The work thus broadens the applicability of relaxation-based Ising architectures to a wider class of combinatorial problems.

Abstract

Dynamical Ising machines are based on continuous dynamical systems evolving from a generic initial state to a state strongly related to the ground state of the classical Ising model on a graph. Reaching the ground state is equivalent to finding the maximum (weighted) cut of the graph, which presents the Ising machines as an alternative way to solving and investigating NP-complete problems. Among the dynamical models, relaxation-based models are distinguished by their relations with guarantees of performance achieved in time scaling polynomially with the problem size. However, the terminal states of such machines are essentially non-binary, necessitating special post-processing relying on disparate computing. We show that an Ising machine implementing a special continuous dynamical system (called the V${}_2$ model) solves the rounding problem dynamically. We prove that the V${}_2$ model, starting from an arbitrary non-binary state, terminates in a state that trivially rounds to a binary state with the cut at least as big as obtained by optimal rounding of the initial state. Besides showing that relaxation-based dynamical Ising machines can be made self-contained, this result presents a non-Boolean realization of solving a non-trivial information processing task on Ising machines. Moreover, we prove that if the initial state of the V${}_2$-machine is a random limited amplitude perturbation of a binary state, the machine progresses to a state with at least as high cut as that of the initial binary state. Since the probability of improving the cut is finite, this shows that the V${}_2$-machine with random agitations converges to a maximum cut state almost surely.

Figures (7)

• Figure 1: Comparison of core functions of several relaxation-based Ising machines: $\Phi_{\mathrm{I}}$ is the discrete function of the binary Ising model, $\Phi_{\mathrm{SDP}}(\boldsymbol{\xi})$, $\Phi_{\mathrm{Tr}}(\boldsymbol{\xi})$, and $\Phi_{\mathrm{GW}}(\boldsymbol{\xi})$ are the core functions of the rank-$2$ SDP relaxation, the triangular model from Refs. shuklaScalable2022shuklaCustom2023, and the V${}_2$ model investigated in the present paper.
• Figure 2: A relaxation-based heterogeneous Ising machine returning a binary state. First, a dynamical system based on a relaxation (for instance, rank-2 SDP) is initiated by a generic state. Then, the dynamical core is switched to the V${}_2$ model, which delivers the optimal rounding of the state obtained during the first stage and performs basic post-processing by small perturbations of the terminal state (see Theorem \ref{['thm:mdii-non-worsening']}).
• Figure 3: The topology of the representation $\xi = \sigma + X \mod P$. The bold circles represent intervals $(-1,1]$, and the thin lines indicate the transitions at the boundaries of these intervals.
• Figure 4: The characteristic clustered form of critical points of $\mathcal{C}_{{\text{V}_2}}(\boldsymbol{\xi})$ with bold points indicating $\xi_n$ at nodes in sets forming a partition $\mathcal{V} = \mathcal{V}^{(1)} \cup \mathcal{V}^{(2)} \cup \mathcal{V}^{(3)}$. Each cluster is made of two components, $\mathcal{V}^{(p)} = \mathcal{V}^{(p)}_- \cup \mathcal{V}^{(p)}_+$, corresponding to the respective values of the binary component in the $(\sigma, X)$-representation and occupying the opposite points on the circle. Varying the positions of $\mathcal{V}^{(p)}$ on the circle yields different critical points corresponding to the same critical value.
• Figure 5: The dependence of the normalized cut, $c_D = \left( \mathcal{C} / M - 1 /2 \right) \sqrt{D} / P_*$, obtained by the V${}_2$-machine on the number of agitations for random $(D = 3)$-regular graphs. For each number of agitations, $c_D\left( \boldsymbol{\sigma}^{(h)} \right)$ was evaluated for $100$ random initial binary states $\boldsymbol{\sigma}^{(0)}$. The solid line and the error bars show the mean, and the maximal and minimal values of $c_D\left( \boldsymbol{\sigma}^{(h)} \right)$.

Theorems & Definitions (13)

• Theorem 1
• proof : Proof of Theorem \ref{['thm:extremal-manifolds']}
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof