arXiv subjects

Stefan Boettcher

Publications and source records attributed to Stefan Boettcher.

At least 19 recordsLinked to original sources

arXiv2026

Cluster-based Message-Passing (CluMP) Optimization for Complex QUBO Problems

Quadratic Unconstrained Boolean Optimization (QUBO) problems are widespread in both industrial applications and scientific studies. A QUBO problem corresponds to the optimization of a system of Ising spins defined on a generally sparse and heterogeneous graph. When the QUBO problem contains conflicting requests, the corresponding Ising system is frustrated, generating a complex energy landscape, which is hard to explore and optimize. Despite extensive algorithmic and hardware developments, finding low-energy configurations in these systems remains challenging (e.g., local-update heuristics typically become trapped in metastable states), especially when the (possibly frustrated) interactions generate extended correlated domains. We introduce CluMP (Cluster-based Message-Passing), an algorithm that performs collective updates on connected clusters of spins using information from Belief Propagation (BP). By controlling the amount of frustration within clusters, CluMP enables BP convergence on large subgraphs and proposes nonlocal rearrangements involving up to hundreds of spins in a single move. We benchmark CluMP against state-of-the-art local-update heuristics on spin-glass models defined on several graph topologies, including random regular graphs and lattice regular graphs in two and three dimensions. Cluster moves consistently bypass local trapping and reach lower energies with fewer effective operations than single-spin dynamics. These results demonstrate that frustration-tolerant cluster updates can be implemented efficiently on sparse graphs. The CluMP framework provides a scalable strategy for large-scale combinatorial optimization and inference problems, where exploiting medium- and long-range correlations is key to navigating complex energy landscapes.

cond-mat.dis-nn↗

arXiv2026

Instability cascades in crumpling mylar sheets follow a log-Poisson statistic

The process of aging following a hard quench into a glassy state is characterized universally, for a wide class of materials, by logarithmic evolution of state variables and a power-law decay of two-time correlation functions that collapse only for the ratio of those times. This stands in stark contrast with relaxation in equilibrium materials, where time-translational invariance holds. It is by now widely recognized that these aging processes, which ever so slowly relax a complex disordered material after a quench, are facilitated by activated events. Yet, theories often cited to describe such a non-equilibrium process can be shown to miss pertinent aspects that are inherent to many experiments. A case in point are recent experiments on crumpling sheets of mylar loaded by a weight whose acoustic emissions are measured while the material buckles. Using extensive simulations to generate long time-series of such buckling events, we show that crumpling is a log-Poisson process activated by increasingly rare record-sized fluctuations in a slowly stiffening material characterized by a logarithmically growing length-scale. Crumpling thus adds to a range of glassy materials exhibiting the log-Poisson property, which can be used to discriminate between theories.

cond-mat.soft↗

arXiv2025

Ground States of the Mean-Field Spin Glass with 3-Spin Couplings

We use heuristic optimization methods in extensive computations to determine with low systematic error ground state configurations of the mean-field $p$-spin glass model with $p=3$. Here, all possible triplets in a system of $N$ Ising spins are connected with a bond. This model has been of recent interest, since it exhibits the ``overlap gap condition'', which should make it prohibitive to find ground states asymptotically with local search methods when compared, for instance, with the $p=2$ case better-known as the Sherrington-Kirkpatrick model (SK). Indeed, it proves more costly to find good approximations for $p=3$ than for SK, even for our heuristic. Compared to SK, the ground-state behavior for $p=3$ is quite distinct also in other ways. For SK, finite-size corrections for large system sizes $N\to\infty$ of both, the ensemble average over ground state energy densities and the width of their distribution, vary anomalously with non-integer exponents. In the $p=3$ case here, the energy density and its distribution appear to scale with $\ln N/N$ and $1/N$ corrections, respectively. The distribution itself is consistent with a Gumbel form. Even more stark is the contrast for the bond-diluted case, where SK has shown previously a notable variation of the anomalous corrections exponent with the bond density, while for $p=3$ no such variation is found here. Hence, for the 3-spin model, all measured corrections scale the same as for the random energy model (REM), corresponding to $p=\infty$. This would suggest that all $p$-spin models with $p\geq3$ exhibit the same ground-state corrections as in REM.

cond-mat.dis-nn↗

arXiv2024

Hysteretic response to different modes of ramping an external field in sparse and dense Ising spin glasses

We consider the hysteretic behavior of Ising spin glasses at $T=0$ for various modes of driving. Previous studies mostly focused on an infinitely slow speed $\dot{H}$ by which the external field $H$ was ramped to trigger avalanches of spin flips by starting with destabilizing a single spin while few have focused on the effect of different driving methods. First, we show that this conventional protocol imposes a system size dependence. Then, we numerically analyze the response of Ising spin glasses at rates $\dot{H}$ that are fixed as well, to elucidate the differences in the response. Specifically, we compare three different modes of ramping ($\dot{H}=c/N$, $\dot{H}=c/\sqrt{N}$, and $\dot{H}=c$ for constant $c$) for two types of spin glass systems of size $N$, representing dense networks by the Sherrington-Kirkpatrick model and sparse networks by the lattice spin glass in $d=3$ dimensions known as the Edwards Anderson model. Depending on the mode of ramping, we find that the response of each system, in form of spin-flip avalanches and other observables, can vary considerably. In particular, in the $N$-independent mode applied to the lattice spin glass, which is closest to experimental reality, we observe a percolation transition with a broad avalanche distribution between phases of localized and system-spanning responses. We explore implications for combinatorial optimization problems pertaining to sparse systems.

cond-mat.dis-nn↗

arXiv2024

Physics of the Edwards-Anderson Spin Glass in Dimensions $d=3,\ldots,8$ from Heuristic Ground State Optimization

We present a collection of simulations of the Edwards-Anderson lattice spin glass at $T=0$ to elucidate the nature of low-energy excitations over a range of dimensions that reach from physically realizable systems to the mean-field limit. Using heuristic methods, we sample ground states of instances to determine their energies while eliciting excitations through manipulating boundary conditions. We exploit the universality of the phase diagram of bond-diluted lattices to make such a study in higher dimensions computationally feasible. As a result, we obtain a verity of accurate exponents for domain wall stiffness and finite-size corrections that allow us to examine their dimensional behavior and their connection with predictions from mean-field theory. We also provide an experimentally testable prediction for the thermal-to-percolative crossover exponent in dilute lattices Ising spin glasses.

cond-mat.dis-nn↗

arXiv2024

Real-space model for activated processes in rejuvenation and memory behavior of glassy systems

We offer an alternative real-space description, based purely on activated processes, for the understanding of relaxation dynamics in hierarchical landscapes. To this end, we use the cluster model, a coarse-grained lattice model of a jammed system, to analyze rejuvenation and memory effects during aging after a hard quench. In this model, neighboring particles on a lattice aggregate through local interactions into clusters that fragment with a probability based on their size. Despite the simplicity of the cluster model, it has been shown to reproduce salient observables of the aging dynamics in colloidal systems, such as those accounting for particle mobility and displacements. Here, we probe the model for more complex quench protocols and show that it exhibits rejuvenation and memory effects similar to those attributed to the complex hierarchical structure of a glassy energy landscape.

cond-mat.soft↗

arXiv2023

Deep reinforced learning heuristic tested on spin-glass ground states: The larger picture

In Changjun Fan et al. [Nature Communications https://doi.org/10.1038/s41467-023-36363-w (2023)], the authors present a deep reinforced learning approach to augment combinatorial optimization heuristics. In particular, they present results for several spin glass ground state problems, for which instances on non-planar networks are generally NP-hard, in comparison with several Monte Carlo based methods, such as simulated annealing (SA) or parallel tempering (PT). Indeed, those results demonstrate that the reinforced learning improves the results over those obtained with SA or PT, or at least allows for reduced runtimes for the heuristics before results of comparable quality have been obtained relative to those other methods. To facilitate the conclusion that their method is ''superior'', the authors pursue two basic strategies: (1) A commercial GUROBI solver is called on to procure a sample of exact ground states as a testbed to compare with, and (2) a head-to-head comparison between the heuristics is given for a sample of larger instances where exact ground states are hard to ascertain. Here, we put these studies into a larger context, showing that the claimed superiority is at best marginal for smaller samples and becomes essentially irrelevant with respect to any sensible approximation of true ground states in the larger samples. For example, this method becomes irrelevant as a means to determine stiffness exponents $θ$ in $d>2$, as mentioned by the authors, where the problem is not only NP-hard but requires the subtraction of two almost equal ground-state energies and systemic errors in each of $\approx 1\%$ found here are unacceptable. This larger picture on the method arises from a straightforward finite-size corrections study over the spin glass ensembles the authors employ, using data that has been available for decades.

cond-mat.dis-nn↗

arXiv2022

Inability of a graph neural network heuristic to outperform greedy algorithms in solving combinatorial optimization problems like Max-Cut

In Nature Machine Intelligence 4, 367 (2022), Schuetz et al provide a scheme to employ graph neural networks (GNN) as a heuristic to solve a variety of classical, NP-hard combinatorial optimization problems. It describes how the network is trained on sample instances and the resulting GNN heuristic is evaluated applying widely used techniques to determine its ability to succeed. Clearly, the idea of harnessing the powerful abilities of such networks to ``learn'' the intricacies of complex, multimodal energy landscapes in such a hands-off approach seems enticing. And based on the observed performance, the heuristic promises to be highly scalable, with a computational cost linear in the input size $n$, although there is likely a significant overhead in the pre-factor due to the GNN itself. However, closer inspection shows that the reported results for this GNN are only minutely better than those for gradient descent and get outperformed by a greedy algorithm, for example, for Max-Cut. The discussion also highlights what I believe are some common misconceptions in the evaluations of heuristics.

cond-mat.dis-nn↗

arXiv2022

Transport and Localization in Quantum Walks on a Random Hierarchy of Barriers

We study transport within a spatially heterogeneous one-dimensional quantum walk with a combination of hierarchical and random barriers. Recent renormalization group calculations for a spatially disordered quantum walk with a regular hierarchy of barriers alone have shown a gradual decrease in transport but no localization for increasing (but finite) barrier sizes. In turn, it is well-known that extensive random disorder in the spatial barriers is sufficient to localize a quantum walk on the line. Here we show that adding only a sparse (sub-extensive) amount of randomness to a hierarchy of barriers is sufficient to induce localization such that transport ceases. Our numerical results suggest the existence of a localization transition for a combination of both, the strength of the regular barrier hierarchy at large enough randomness as well as the increasing randomness at sufficiently strong barriers in the hierarchy.

quant-ph↗

arXiv2022

Density Fluctuations in Granular Piles Traversing the Glass Transition: A Grain-Scale Characterization of the Transition via the Internal Energy

The transition into a glassy state of the ensemble of static, mechanically stable configurations of a tapped granular pile is explored using extensive molecular dynamics simulations. We show that different horizontal sub-regions ("layers") along the height of the pile traverse this transition in a similar manner but at distinct tap intensities. We supplement the conventional approach based purely on properties of the static configurations with investigations of the grain-scale dynamics by which the tap energy is transmitted throughout the pile. We find that the effective energy that particles dissipate is a function of each particle's location in the pile and, moreover, that its value plays a distinctive role in the transformation between configurations. This internal energy provides a "temperature-like" parameter that allows us to align the transition into the glassy state for all layers, as well as different annealing schedules, at a critical value.

cond-mat.soft↗

arXiv2021

Extremal fluctuations driving the relaxation in glassy energy landscapes

Cooperative events requiring anomalously large fluctuations are a defining characteristic for the onset of glassy relaxation across many materials. The importance of such intermittent events has been noted in systems as diverse as superconductors, metallic glasses, gels, colloids, and granular piles. Here, we show that prohibiting the attainment of new record-high energy fluctuations -- by explicitly imposing a ``lid'' on the fluctuation spectrum -- impedes further relaxation in the glassy phase. This lid allows us to directly measure the impact of record events on the evolving system in extensive simulations of aging in such vastly distinct glass formers as spin glasses and tapped granular piles. Interpreting our results in terms of a dynamics of records succeeds in explaining the ubiquity of both, the logarithmic decay of the energy and the memory effects encoded in the scaling of two-time correlation functions of aging systems.

cond-mat.soft↗

arXiv2021

Analysis of landscape hierarchy during coarsening and aging in Ising spin glasses

We use record dynamics (RD), a coarse-grained description of the ubiquitous relaxation phenomenology known as "aging", as a diagnostic tool to find universal features that distinguish between the energy landscapes of Ising spin models and the ferromagnet. According to RD, a non-equilibrium system after a quench relies on fluctuations that randomly generate a sequence of irreversible record-sized events (quakes or avalanches) that allow the system to escape ever-higher barriers of meta-stable states within a complex, hierarchical energy landscape. Once these record events allow the system to overcome such barriers, the system relaxes by tumbling into the following meta-stable state that is marginally more stable. Within this framework, a clear distinction can be drawn between the coarsening dynamics of an Ising ferromagnet and the aging of the spin glass, which are often put in the same category. To that end, we interpolate between the spin glass and ferromagnet by varying the admixture $p$ of ferromagnetic over anti-ferromagnetic bonds from the glassy state (at 50% each) to wherever clear ferromagnetic behavior emerges. The accumulation of record events grows logarithmic with time in the glassy regime, with a sharp transition at a specific admixture into the ferromagnetic regime where such activations saturate quickly. We show this effect both for the Edwards-Anderson model on a cubic lattice as well as the Sherrington-Kirkpatrick (mean-field) spin glass. While this transition coincides with a previously observed zero-temperature equilibrium transition in the former, that transition has not yet been described for the latter.

cond-mat.dis-nn↗

arXiv2020

Universal Features of Annealing and Aging in Compaction of Granular Piles

We explore the compaction dynamics of a granular pile after a hard quench from a liquid into the glassy regime. First, we establish that the otherwise athermal granular pile during tapping exhibits annealing behavior comparable to glassy polymer or colloidal systems. Like those other systems, the pile undergoes a glass transition and "freezes" into different non-equilibrium glassy states at low agitation for different annealing speeds, starting from the same initial equilibrium state at high agitation. Then, we quench the system instantaneously from the highly-agitated state to below the glass transition regime to study the ensuing aging dynamics. In this classical aging protocol, the density increases (i.e., the potential energy of the pile decreases) logarithmically over several decades in time. Instead of system-wide, thermodynamic measures, here we identify the intermittent, irreversible events ("quakes") that actually drive the glassy relaxation process. We find that the event rate decelerates hyperbolically, which explains the observed increase in density when the integrated contribution to the downward displacements is evaluated. We argue that such a hyperbolically decelerating event rate is consistent with a log-Poisson process, also found as a universal feature of aging in many thermal glasses.

cond-mat.soft↗

arXiv2020

Record dynamics of evolving metastable systems: theory and applications

Record Dynamics (RD) deals with complex systems evolving through a sequence of metastable stages. These are macroscopically distinguishable and appear stationary, except for the sudden and rapid changes, called quakes, which induce the transitions from one stage to the next. This phenomenology is well known in physics as "physical aging", but from the vantage point of RD the evolution of a class of systems of physical, biological and cultural origin is rooted in a hierarchically structured configuration space and can therefore be analyzed by similar statistical tools. This colloquium paper strives to present in a coherent fashion methods and ideas that have gradually evolved over time. To this end, it first describes the differences and similarities between RD and two widespread paradigms of complex dynamics, Self Organized Criticality and Continuous Time Random Walks. It then outlines the Poissonian nature of records events in white noise time series, and connects it to the statistics of quakes in metastable hierarchical systems, arguing that the relaxation effects of quakes can generally be described by power laws unrelated to criticality. Several different applications of RD have been developed over the years. Some of these are described, showinghe basic RD hypothesis, the log time homogeneity of quake dynamics, can be empirically verified in a given context. The discussion summarizes the paper and briefly mentions applications not discussed in detail. Finally, the outlook points to possible improvements and to new areas of research where RG could be of use.

cond-mat.stat-mech↗

arXiv2020

Quantum Ultra-Walks: Walks on a Line with Hierarchical Spatial Heterogeneity

We discuss the model of a one-dimensional, discrete-time walk on a line with spatial heterogeneity in the form of a variable set of ultrametric barriers. Inspired by the homogeneous quantum walk on a line, we develop a formalism by which the classical ultrametric random walk as well as the quantum walk can be treated in parallel by using a "coined" walk with internal degrees of freedom. For the random walk, this amounts to a $2^{\rm nd}$-order Markov process with a \emph{stochastic} coin, better known as an (anti-)persistent walk. When this coin varies spatially in the hierarchical manner of "ultradiffusion," it reproduces the well-known results of that model. The exact analysis employed for obtaining the walk dimension $d_{w}$, based on the real-space renormalization group (RG), proceeds virtually identical for the corresponding quantum walk with a $unitary$ coin. However, while the classical walk remains robustly diffusive ($d_{w}=\frac{1}{2}$) for a wide range of barrier heights, unitarity provides for a quantum walk dimension $d_{w}$ that varies continuously, for even the smallest amount of heterogeneity, from ballistic spreading ($d_{w}=1$) in the homogeneous limit to confinement ($d_{w}=\infty$) for diverging barriers. Yet for any $d_{w}<\infty$ the quantum ultra-walk never appears to localize.

quant-ph↗

arXiv2020

Ground State Properties of the Diluted Sherrington-Kirkpatrick Spin Glass

We present a numerical study of ground states of the dilute versions of the Sherrington-Kirkpatrick (SK) mean-field spin glass. In contrast to so-called "sparse" mean-field spin glasses that have been studied widely on random networks of finite (average or regular) degree, the networks studied here are randomly bond-diluted to an overall density $p$, such that the average degree diverges as $\sim pN$ with the system size $N$. Ground-state energies are obtained with high accuracy for random instances over a wide range of fixed $p$. Since this is a NP-hard combinatorial problem, we employ the Extremal Optimization heuristic to that end. We find that the exponent describing the finite-size corrections, $ω$, varies continuously with $p$, a somewhat surprising result, as one would not expect that gradual bond-dilution would change the $T=0$ universality class of a statistical model. For $p\to1$, the familiar result of $ω(p=1)\approx\frac{2}{3}$ for SK is obtained.

cond-mat.dis-nn↗

arXiv2019

Analysis of the Relation between Quadratic Unconstrained Binary Optimization (QUBO) and the Spin Glass Ground-State Problem

We analyze the transformation of QUBO from its conventional Boolean presentation into an equivalent spin glass problem with coupled $\pm1$ spin variables exposed to a site-dependent external field. We find that in a widely used testbed for QUBO these fields tend to be rather large compared to the typical coupling and many spins in each optimal configurations simply align with the fields irrespective of their constraints. Thereby, the testbed instances tend to exhibit large redundancies - seemingly independent variables which contribute little to the hardness of the problem, however. We demonstrate various consequences of this insight, for QUBO solvers as well as for heuristics developed for finding spin glass ground states. To this end, we implement the Extremal Optimization (EO) heuristic, in a new adaptation for the QUBO problem. We also propose a novel way to assess the quality of heuristics for increasing problem sizes based on asymptotic scaling.

cond-mat.dis-nn↗

arXiv2018

Aging is a log-Poisson Process, not a Renewal Process

Aging is a ubiquitous relaxation dynamic in disordered materials. It ensues after a rapid quench from an equilibrium "fluid" state into a non-equilibrium, history-dependent jammed state. We propose a physically motivated description that contrasts sharply with a continuous-time random walk (CTRW) with broadly distributed trapping times commonly used to fit aging data. A renewal process like CTRW proves irreconcilable with the log-Poisson statistic exhibited, for example, by jammed colloids as well as by disordered magnets. A log-Poisson process is characteristic of the intermittent and decelerating dynamics of jammed matter usually activated by record-breaking fluctuations ("quakes"). We show that such a record dynamics (RD) provides a universal model for aging, physically grounded in generic features of free-energy landscapes of disordered systems.

cond-mat.soft↗