The Physics of Local Optimization in Complex Disordered Systems

TL;DR

The paper investigates how local optimization can approximate global ground states in disordered spin systems by analyzing a Local Single Bond Solver (LSBS) and a contraction-based scheme guided by subsystem critical thresholds $J_{\mathrm{c},ij}^{\rm sub}$ to stitch local solutions into a global GS for Edwards-Anderson and Sherrington-Kirkpatrick models. The framework centers on the Hamiltonian $H = - \sum_{\langle ij \rangle} J_{ij} \sigma_i \sigma_j$, with $\sigma_i = \pm 1$, and demonstrates that LSBS predictions are accurate away from criticality due to locality, while near critical thresholds they become fragile due to zero-energy droplets and avalanche-like excitations. Key findings include algebraic decay of the LSBS error with subsystem size, with exponents $\kappa \approx 0.685$ in 2D and $\kappa \approx 0.18$ in 3D, and an exponential decay form when the system approaches the disordered ferromagnetic phase; for the SK model the local hardness also exhibits system-size dependence, reflecting nonlocal effects. The results propose a local-hardness descriptor beyond P vs NP classifications, enable scalable GS estimation via contraction of well-determined bonds, and suggest broader applicability to NP-hard problems likeMinimum Vertex Cover and to diverse physical and computational systems where local actions drive global behavior.

Abstract

Limited resources motivate decomposing large-scale problems into smaller,``local" subsystems and stitching together the so-found solutions. We explore the physics underlying this approach and discuss the concept of ``local hardness", i.e., the complexity of predicting local properties of the solution from local information, for the ground-state problem of both P- and NP-hard spin-glasses and related frustrated spin systems. Depending on the model considered, we observe varying scaling behaviors in how errors associated with local predictions decay as a function of the size of the solved subsystem. These errors are intimately connected to global critical threshold instabilities, characterized by gapless, avalanche-like excitations that follow scale-invariant size distributions. Away from criticality, local solvers quickly achieve high accuracy, aligning closely with the results of the computationally much more expensive global minimization. We leverage these findings to introduce a heuristic contraction-based algorithm for globally studying spin-glass ground states. The local solvers further display sharp imprints of the phase transition from the spin-glass to the ferromagnetic phase as the distribution of spin-glass couplings is shifted, as well as characteristic differences for the infinite-range model, implying the existence of specific classes of local hardness. Our findings shed light on how Nature may operate solely through local actions at her disposal.

Figures (8)

• Figure 1: Computational setup for determining the relative spin orientation $\sigma_{i_{0}}\sigma_{j_{0}}$ across the central bond (shown in red). The correct value of $\sigma_{i_{0}}\sigma_{j_{0}}$ is that within the GS of the entire system (comprised of black, light blue, and red bonds). The LSBS computes $\sigma_{i_{0}}\sigma_{j_{0}}$ within the GS of the local subsystem (light blue and red bonds).
• Figure 2: The disorder-averaged LSBS error rate $\mathcal{E}_{ij}$ as a function of the subsystem size $L_{\rm sub}$ (resp. $N_{\rm sub}$) for (a) the square-lattice (2D) EA model, (b) the cubic lattice (3D) EA model, and (c) the SK model. For each plot, the results for varying system sizes $L$ ($N$) are shown. For (a), both cases $\overline{J}=0$ and $\overline{J}=1.2$ are shown, whereas in (b) and (c) $\overline{J} = 0$. Note the nearly perfect collapse of different system-size data, indicating that the error of the LSBS becomes asymptotically independent of system size for (a) and (b), while this is only the case if one scales $N_{\rm sub}$ with $N$ for the SK model in (c). For the lattice systems in (a)--(b) and $\overline{J}=0$, the error rate is fit by Eq. (\ref{['eq:solver']}), cf. the straight dashed lines.
• Figure 3: LSBS solutions for nearest-neighbor bonds $\langle ij\rangle$ in an $L=128$ system. An $L_{\rm sub}=40$ subsystem is chosen to be centered about any such bond $\langle i j \rangle$ (near the boundaries, the subsystem becomes correspondingly smaller). $J_{\mathrm{c},ij}^{\rm sub}$ is the critical threshold of bond $\langle ij\rangle$. $\color{red} \times$ denotes an error wherein the local solver does not match the global solution, while $\checkmark$ refers to correct LSBS predictions. Errors arise more readily for smaller $|J_{\mathrm{c},ij}^{\rm sub}-J_{ij}|$. It is noteworthy that correct LSBS predictions also appear for small values of $|J_{\mathrm{c},ij}^{\rm sub}- J_{ij} |$. Zoomed in fragment: individual bond $|J_{\mathrm{c},ij}^{\rm sub}-J_{ij}|$ values are overlayed on color coded correct/incorrect LSBS predictions for these bonds.
• Figure 4: Error rate $\mathcal{E}_{ij}$ as a function of $|J_{\mathrm{c},ij}^{\rm sub}-J_{ij}|$ for (a) a 2D (square lattice), $L=1024$ system with $L_{\rm sub}=8$, $16$, $32$, $64$ and (b) a 3D (cubic) system of $L=12$ with $L_{\rm sub}=4$, $10$. Point percentages denote cumulative probabilities (i.e., fraction of instances) for values of $|J_{\mathrm{c},ij}^{\rm sub}-J_{ij}|$ lower than their abscissa.
• Figure 5: Average change of critical threshold values ($[ |\Delta {J_{\mathrm{c},ij}}|]$) of bonds a distance of $r$ along a Cartesian direction from $-\infty$ to $+\infty$ in (a) 2D ($L=128$) and (b) 3D ($L=12$) systems. Dashed: Eq. (\ref{['eq:deltajc']}) with parameters in Table \ref{['tab:fit_params']}. As is more evident in 2D, when $r= {\cal{O}}(L)$, the average $[ |\Delta J_{\mathrm{c},ij}|]$ is bounded from above by these ultra-local forms with the fitted $\ell_{c}$ being of the order of the lattice constant.