Spin Glass Dynamics on Complex Hardware Topologies: A Bond-Correlated Percolation Approach

TL;DR

This work addresses how frustration and quenched disorder shape non-exponential relaxation of spin glasses on quantum-annealing topologies. Using FKCK cluster percolation and finite-size scaling, the authors relate the percolation temperature $T_p$ to a Potts-like transition and analyze spin-glass crossover temperatures $T_{SG}$ and $T_C$ across Chimera, Pegasus, Zephyr, and 3D graphs. They show that network topology strongly modulates barrier distributions and relaxation scales, with higher connectivity yielding larger $T_p$ and $T_C$ and a progression from stretched-exponential to simple exponential relaxation as temperature rises above $T_p$. The results provide quantitative benchmarks for QA architectures and offer a framework to evaluate how topology, disorder, and frustration govern relaxation dynamics in complex energy landscapes.

Abstract

Understanding how frustration and disorder shape relaxation in complex systems is a central problem in statistical physics and quantum annealing. Spin-glass models provide a natural framework to explore this connection, as their energy landscapes are governed by competing interactions and constrained topologies. We investigate the non-exponential relaxation behavior of spin glasses on network architectures relevant to quantum annealing hardware -- such as finite size Chimera, Pegasus, and Zephyr graphs -- where embedding constraints and finite connectivity strongly modulate the distribution of barriers and metastable states. This slow relaxation arises from the combined effects of frustration and disorder, which persist even beyond the conventional spin-glass transition. Within the Fortuin-Kasteleyn-Coniglio-Klein (FKCK) cluster formalism, the appearance of unfrustrated cluster regions gives rise to multiple relaxation scales, as distinct domains follow different dynamical pathways across a rugged energy landscape. This framework enables a more comprehensive characterization of spin-glass energy landscapes and offers valuable insight into how topological constraints and disorder jointly govern relaxation dynamics, providing quantitative benchmarks for evaluating the performance and limitations of quantum annealing architectures.

Figures (4)

• Figure 1: Illustrative energy landscapes (left) and their corresponding spin–spin autocorrelations (right). Below the spin glass transition, the energy landscape becomes extremely rugged, with many deep valleys separated by high barriers; ergodicity is broken and spin autocorrelations decay slowly. Below the Potts (percolation) transition, frustration persists, while below the Griffiths temperature, disorder induces rare correlated regions—both mechanisms give rise to stretched-exponential relaxation. Above these phases, the system is paramagnetic, exhibiting exponential decay and a simple energy landscape.
• Figure 2: A) Spanning probability $P_{\infty}(T)$ for the largest system size of each lattice: Blue – 3D ($N=1000$), Orange – Chimera ($N=1568$), Green – Pegasus ($N=1664$), Pink – Zephyr ($N=1248$). B) Average path length for all system sizes studied in each network. The results are well described by a finite-size scaling analysis, which allows us to estimate the percolation temperature $T_p$ in the thermodynamic limit (see Table \ref{['tab:combined_parameters']}). The distances are weighted by the coupling strengths $J_{ij}$, but this weighting does not affect the position of the APL peak, only its absolute value. For the largest system sizes, the APL is fitted according to Eq. \ref{['eq:APL_model']}.
• Figure 3: Log–log plots of the spin–spin autocorrelation function (Eq. \ref{['eq:correlation']}) for different system sizes on a 3D network. A)$T=0.8$, spin-glass phase. B)$T=2.2$, non-exponential relaxation regime. C)$T=5.0$, paramagnetic phase.
• Figure 4: Spin–spin autocorrelations (Eq. \ref{['eq:correlation']}) for different network topologies: A) 3D lattice with $N = 1000$ and $T_p = 2.84$, B) Chimera network with $N = 1568$ and $T_p = 2.27$, C) Pegasus network with $N = 1664$ and $T_p = 7.18$, and D Zephyr network with $N = 1248$ and $T_p = 9.78$. Here, $t$ corresponds to a single-spin Monte Carlo flip attempt rather than a full-lattice update, in order to obtain higher temporal resolution. The reference power-law curves were obtained from fits to the lowest-temperature data, yielding exponents $x_{\text{3D}} = 0.067$, $x_{\text{Chimera}} = 0.104$, $x_{\text{Pegasus}} = 0.050$, and $x_{\text{Zephyr}} = 0.040$.