Papers

Qubits are powerful computational resources because each additional qubit doubles the size of the accessible state space. This expressive power comes at a cost: adding qubits also increases circuit size, constraint overhead, and exposure to hardware noise. These limitations are especially critical for near-term quantum computers, making the number of qubits required to formulate a problem a central practical concern. In this work, we present a method for efficiently mapping logistics and routing problems onto qubits through structure-aware encoding reduction. We demonstrate the method on a benchmark problem of latency-resilient routing under dual-link failures. For this benchmark, the reduced formulation substantially lowers both the qubit count and the compiled circuit complexity while preserving the original optimization task. By running the reduced model on the same trapped-ion quantum computer as the benchmark, we obtain a 100X improvement in the success probability. This finding shows how careful problem formulation can make current quantum hardware substantially more effective for practical optimization.

We discuss three simple connections between generative diffusion and optimal transport. First, the joint law of the data and of their Ornstein-Uhlenbeck noising is, at every time, the solution of an entropic optimal-transport problem, at a temperature set by the noise level. Backward diffusion can consequently be read as an annealing which gradually decreases the entropic regularization. Second, we study the speciation transition–how the modes of the data distribution emerge during generation–and compare how it happens in diffusion and flow matching with respect to optimal transport. Using Gaussian mixtures as high-dimensional example, we show that the dynamical instability through which diffusion and flow matching separate the modes becomes, for optimal transport, an exponentially thin boundary layer at the beginning of generation, which is rounded by entropic regularization. Third, considering a generic forward dynamics with a general confining potential, we show that the backward process driven by the score also satisfies a drift-based variational principle whose minimal cost is the entropy difference between initial and final distributions.

In one dimension, Anderson localization by a disordered potential sets in at any strength, whereas localization by a quasiperiodic potential in the Aubry-André model occurs through a transition at a critical strength. Disordered photonic time crystals exhibit a temporal counterpart of this Anderson localization, in which light, instead of decaying in space, stops propagating and grows in time. Whether a counterpart of the Aubry-André transition exists in photonic time quasicrystals remains an open question. Here we show that it does, and that the transition occurs at mobility edges in momentum rather than in energy. These edges arise even under impulsive modulation, where the field obeys the self-dual almost-Mathieu equation exactly, and for general analytic modulations Avila's global theory reveals the condition under which they form. The Aubry-André transition thus extends from spatial lattices to the time domain, where it sorts light by wave number into components that propagate and components that stop and grow.

Specific heat of graphite at high temperatures demonstrates significant deviation from the “3R” Dulong-Petite law. It can be explained by accumulation of defects with formation energy as low as 1.45 eV. Based on the density functional theory calculations, we propose a possible candidate of topological defects, whose energy is comparable to the growth activation energy.

We study how a neural network reorganizes its learned function as training proceeds and as previously unseen parity data accumulate. The trained model is itself a Boolean function on binary strings of finite length. We then evaluate it on all data, inside and outside the training set, and read off its complete sign table. Its $\pm1$ entries are the amplitudes of a normalized state, dubbed the function-state, whose inverse subsystem purity $K_f=1/\textrm{Tr}(\rho_A^2)$ is an effective Schmidt rank. We find that small data additions broaden the Schmidt spectrum, whereas larger additions reconcentrate it in the target mode. The former is called differentiation because the model distinguishes inputs more sharply. The latter is called target-selective recoherence, where the model settles on the single rule shared by the added data. As the dataset grows, the connectivity of the constraint graph defined by the training set increases monotonically, whereas the learned inverse purity changes nonmonotonically. On a fixed connected dataset, as learning proceeds, the Schmidt spectrum begins to reorganize, followed by gains in target alignment and later by improvement in predictive accuracy. This ordering distinguishes three aspects of learning: constraint identifiability, function-state organization, and predictive accuracy. At fixed data size, a function that has already been learned can re-differentiate, an event we call retreat, without changing graph connectivity. Recovery reconcentrates the Schmidt spectrum in a similar target-selective pattern, without exact recurrence of the Boolean state. Thus the percolation of data constraints fixes the identifiable support, while the reduced state produced by training continues to reorganize even after that support stops changing.

Frustrated Ising models of social networks mix ferromagnetic (agreement-seeking) and antiferromagnetic (antagonistic) couplings, and the antagonistic bonds are almost always placed independently at a fixed rate, yielding a +/-J spin glass. Real signed networks (Epinions, Slashdot, Wikipedia) instead show an excess of balanced triads, and perfectly balanced placement is gauge-equivalent to an unfrustrated ferromagnet. We compare three sign-placement conditions on a frustrated random-field Ising model of a polarised population, sharing an identical graph, field assignment, and edge-weight magnitudes: an independent baseline; a balance-structured condition whose camp membership coincides with each voter's field sign; and the same condition with camp labels randomly permuted, so triadic balance is identical by construction while the camp-field correlation is removed. On a confound-controlled Kolmogorov-Smirnov test of the sampled energy-gap distribution (simulated annealing), the aligned condition differs from the baseline at every density tested (p = 7.3e-8), while the permuted condition shows no significant difference (paired mean difference 0.003, 95% CI [-0.041, 0.046]). Field-camp alignment, not triadic balance alone, drives the effect. The qualitative direction follows from a known mapping; the contribution is a quantitative, confound-controlled measurement of the effect's size away from that limit, robust to parallel-tempering sampling, system sizes up to N = 300, and structural noise calibrated to real networks, while triadic balance alone produces no detectable effect at N = 90 and at most a small one at N = 180-300 (paired change in KS distance about 0.04, 95% CI upper bound below 0.09).

Replica symmetry breaking (RSB) is the organizing principle of mean-field spin-glass theory, yet whether a continuous Parisi hierarchy can arise in a genuinely short-range system in finite spatial dimension remains unresolved. Here we introduce a disordered model of $N$-component vector spins on a $D$-dimensional hypercubic lattice, with random matrix couplings on nearest-neighbor links and random tensor couplings on elementary plaquettes. In the large-$N$ limit, the model supports thermodynamically stable one-step, full, and one-full RSB phases, alongside replica-symmetric paramagnetic and ferromagnetic phases. We find that spatial locality constrains these ordered phases through two distinct infrared mechanisms. ($i$) Quenched random anisotropy destroys uniform ferromagnetism for $D\leq4$ through an Imry-Ma mechanism. ($ii$) A continuous Parisi hierarchy gives rise to replica Goldstone modes whose gaplessness is protected by a Ward identity. At leading order in $1/N$, these modes behave as a band of free spin waves, producing infrared-divergent fluctuations in $D\leq2$ and a Mermin-Wagner instability of the continuously broken phase. Remarkably, the Parisi hierarchy also acquires a direct real-space interpretation. For two pure states with mutual overlap $q$, a characteristic length $\xi(q)$ describes how far the differences between their frozen magnetization patterns remain correlated. Throughout a continuous sector of the Parisi hierarchy, $\xi(q)$ is independent of temperature and decreases monotonically with $q$. The ultrametric organization of pure states is therefore accompanied by an ultrametric hierarchy of real-space correlation lengths: states that separate near the root of the Parisi tree differ through long-wavelength frozen patterns, whereas states that separate farther from the root differ only on progressively shorter scales.

AI agents can now conduct real-world cyberattacks, scale up capabilities with the number of agents, and collectively pursue misaligned goals to obtain rewards. Together, these factors raise the risk of a population explosion of misaligned agents: agents could compromise computers and secretly deploy additional agents, creating a self-reinforcing cycle where larger populations develop greater collective cyber capability and expand further. This raises a fundamental question: What determines whether a population of misaligned agents remains contained or takes off into this self-reinforcing cycle? This population-level problem is ecological safety: unlike individual-agent or multi-agent safety with a fixed population, it concerns the dynamics of the population itself. Here, we develop an ecological theory of AI-agent populations based on a population growth equation in which fitness (growth rate) depends on cybersecurity capability. We show that, without collaboration, the population takes off only when individual-agent capability exceeds a critical threshold. With collaboration, however, collective cybersecurity capability increases with population size. This creates a critical population threshold: below it, the population declines; above it, the population takes off, even though individual-agent capability has not changed. In ecology, this phenomenon is known as the strong Allee effect. Because red teaming a small group of agents cannot guarantee ecological safety in larger populations, our theory calls for ecological red teaming and population pacing: gradually deploying larger agent populations in controlled environments, while measuring how cyber capability scales with population size, and estimating the critical population size for takeoff. Capability gains may lower this threshold, requiring re-estimation for each new model generation.

The Lyapunov spectrum of a nonlinear recurrent neural network with random asymmetric couplings is calculated in the limit $N\to\infty$. The calculation is based on a finite-$N$ identity that expresses the cumulative distribution of Lyapunov exponents as a response function of the tangent-space dynamics, with the tangent trajectory selected by a minimum-norm condition rather than by an initial condition. A cavity method then determines this response function at large $N$ through a self-consistent single-site problem. This result establishes that the chaos in this network is extensive and gives access to diffeomorphism-invariant properties of the dynamics. This work was done in collaboration with the AI models GPT-6 Astra and Claude Opus 5.5.

Is the relaxation of a modular network dictated by its modules or by the network that connects them? The answer fixes the time scales of diffusion, consensus, and synchronization, all set by the Fiedler eigenvalue. We show that, for bundled networks, the two levels act one after the other: a random walker must first escape from its module and then spread over the network that connects them. The equilibration time is the sum of these two times, the time needed to reach the base from inside a fiber and the relaxation time of the base, slowed down by the mass of the fibers. The consequences are unexpected. However large the modules are, a small-world core always imposes its own Fiedler dimension; the modules survive only as a logarithmic correction, which slows down equilibration and hides the true exponent up to sizes far beyond any real network. Effective dimensions measured on finite modular systems can therefore be systematically biased. We analytically work out all combinations of finite-dimensional and small-world bases and fibers, and compare them with numerically exact spectra of representatives of each category.

Quantum annealing must be sufficiently slow to prevent defect formation. We study this problem in the random transverse-field Ising model via the adiabatic theorem to obtain the limiting rate at which defect formation can be avoided. Our approach based on the strong-disorder renormalization group (SDRG) technique views the annealing process as a gradual aggregation of strongly coupled clusters, giving access to the elementary events of error formation. We present a scaling theory of the statistics of these events in general dimensions and confront it to an extensive numerical analysis carried out in one, two, and three dimensions by an efficient implementation of the SDRG method. This approach is also validated by exact diagonalization in one dimension by means of the free-fermion technique. Our results consistently show a logarithmically slow decrease of the error density with the annealing time, characterized by the dimension-dependent critical exponents of the corresponding infinite-disorder fixed point of the model. Furthermore, we provide a map of individual events of potential defect formation in the excitation energy - control parameter plane, which may help in the design of optimal annealing paths.

Encapsulation of two-dimensional (2D) materials with hexagonal boron nitride (hBN) is widely used to fabricate ultraclean van der Waals (vdW) heterostructures for optoelectronic and quantum transport studies. These devices are often operated under strong out-of-plane displacement fields that tune the electronic properties of the active 2D material, while the surrounding hBN is typically assumed to remain electronically inert. This assumption, however, neglects defect states that are present even in high-quality hBN crystals. Here, we probe their response to the applied displacement field using monolayer graphene encapsulated between hBN crystals as a sensitive charge sensor. Increasing the displacement field induces notable charge transfer between graphene and hBN, producing additional electron doping of 10-20x10^9 cm^(-2) even at the moderate displacement field (~0.5 V nm^(-1)), while simultaneously increasing carrier-density inhomogeneity of the graphene layer. Both effects are consistent with ionization of defect states within the hBN bandgap, with the bulk density ~1-7x10^(-5) nm^(-3) varying across different hBN crystals. Our results show that displacement-field-induced defect ionization in hBN can modify the electronic response of vdW heterostructures, with direct implications for device design and interpretation of experiments.

We study the emergence of a mutual giant component in a multiplex network comprised of two one-dimensional long-range networks on a lattice. The probability that two nodes on one lattice, with a distance $r$ between them, are connected with an edge falls algebraically with distance as $Cr^{-(1+\sigma)}$, where $0 < \sigma < 1$ and $0 < C \leq 1$. A mutual giant component exists if $C > C_c^{\mathrm{mul}}$, where $C_c^{\mathrm{mul}}(\sigma)$ is a critical value of $C$ that is dependent on $\sigma$. We find rigorous lower bounds on $C_c^{\mathrm{mul}}$, and a tree approximation whose estimate of $C_c^{\mathrm{mul}}$ agrees with simulations within 0.6% for $\sigma \leq 0.5$. The multiplex network is significantly less stable than the single network case, as is evident by the larger value of $C_c^{\mathrm{mul}}$, which is up to 2.4 times the single-layer value. Our simulations indicate that the order of the transition depends on $\sigma$. For $\sigma \leq 0.35$ the mutual giant component collapses abruptly, in a first-order phase transition, while for $\sigma \geq 0.4$ the transition is continuous. When the dependency links connect random pairs of nodes, the transition is first order for all $\sigma$.

Energy landscapes of non-convex optimization problems are high-dimensional surfaces that are difficult to reveal, visualize, or analyze. Message-passing algorithms, or equivalently cavity approaches in statistical physics, may fail to identify local minima as messages do not converge owing to the ruggedness of the energy landscapes. Here we aim to reveal the characteristics of these complex energy landscapes by limiting the amount of information passed in local messages, improving the convergence of messages for local minima. By studying a routing optimization problem with its convexity governed by a single parameter, we introduce a cavity message-passing algorithm with a trust region, and sample ensembles of converged local minima of rugged energy landscapes across many independent realizations of the same problem instances. We observe three regimes with different characteristics of the energy landscapes: (1) a maximally rugged regime with a peak in the diversity of low-energy minima; (2) an intermediate regime with a hierarchical multi-cluster organization of low-energy solutions; and (3) a smooth regime dominated by a single minimum. Additional tests show that the converged energies are largely insensitive to the size of the trust region if the size is small or moderate, demonstrating the robustness of the proposed approach in identifying characteristics of rugged energy landscapes.

Physical reservoir computing exploits the nonlinear dynamics of physical systems to process time-dependent data with greater energy efficiency than conventional machine learning approaches. However, physical reservoirs have fixed intrinsic response timescales, whereas real-world signals combine deterministic and stochastic components across multiple timescales. Here we show, using a nanoporous niobium oxide reservoir, synthetic noisy signals and cryptocurrency-price volatility, that the relationship among noise correlation time, reservoir memory and forecast horizon determines whether correlated noise is filtered or predicted. Noise varying faster than the relevant reservoir memory and forecast horizon is averaged by the reservoir, whereas the temporal structure of slower-varying noise is sufficient for algorithmic forecasting. We introduce the reservoir memory horizon and forecasting regime index to distinguish these operating regimes. These contributions demonstrate that timescale matching can guide the encoding of input time series and development of physical reservoir architectures that filter, analyse and predict stochastic signal components across distinct temporal scales.

Understanding the extent to which measured synaptic wiring determines computation remains a central challenge. Here, we couple the proofread adult Drosophila melanogaster connectome to an anatomically faithful model of its eye. Visual information is inputted in the eye model, then passed to the connectome, and finally read from a Kenyon-cell-centered linear decoder. This creates a connectome-only model in which the anatomical graph and eye geometry are fixed and only scalar synaptic gains and neuronal thresholds may be learned. The model supports multitask vision, including color discrimination, shape classification, and numerical discrimination that follows a ratio-dependent scaling characteristic of approximate number perception. To test whether precise connectivity is consequential under wiring economy, we compare the biological graph to randomized ensembles that increasingly preserve biological synaptic constraints. At matched wiring cost, the biological network consistently yields higher accuracy, whereas less constrained rewiring surpasses it at the cost of inflated wiring. These findings indicate that the measured connectivity and eye geometry jointly set efficient operating points for visual computation.

A recently introduced anharmonic mean-field model unifiedly reproduced a broad range of low-temperature glass phenomena — including harmonic nonphononic spectral properties, linear micromechanics and strongly driven elasto-plastic dynamics — indicating that its underlying energy landscape is intrinsically glassy. Here, we apply a nonlinear modes framework to the model and derive analytic predictions for the asymptotic distributions of activation barriers $p(\Delta{U})\!\sim\!(\Delta{U})^{1/4}$ and the external force needed for the onset of plasticity $p(f_{\rm c})\sim f_{\rm c}^{2/3}$, for their extreme-value scaling and for $\langle\Delta{U}\rangle$ beyond the asymptotic regime. These predictions are expected to equally apply to the mean-field model and to finite-dimensional glasses. We develop efficient algorithms for sampling minima and saddles of the model's glassy potential energy landscape, and quantitatively confirm the theoretical predictions. This progress is enabled by identifying a subset of collective degrees of freedom that are physically relevant for activated glassy dynamics, which like the theoretical predictions should apply to realistic glasses.

Poly(1,3-dioxolane) (pDXL) is a chemically recyclable polyether that can be synthesized by living cationic ring-opening polymerization with precise control of chain length into the ultra-high-molecular-weight (UHMW) regime, where it acquires enhanced mechanical properties. Like most polymers, however, it has amorphous condensed-phase structure, and diffraction measurements alone cannot resolve the atomistic microstructure that underlies them. In this work, we predict the ensemble of the pDXL microstructures by applying a recently developed AI-accelerated ab initio bottom-up polymer structure prediction (AI$^{2}$-BPSP) framework that simulates the living polymerization of pDXL under experimental synthetic conditions with machine-learning force fields trained within van der Waals-corrected hybrid density functional theory. Partitioning the predicted X-ray and neutron diffraction signal between the growing chain and the surrounding monomer resolves its dominant feature into two counteracting modes: a monomer contribution near $q \approx 1.5$ Å$^{-1}$ that decays as monomer is consumed, and a polymer contribution near $q \approx 1.6$ Å$^{-1}$ that is absent in oligomers and grows with chain length. This work paves the way to direct accuracy validation of these statistically significant, first-principles, and chain-length resolved diffraction predictions against experiments.

We argue that pattern recognition and step-by-step reasoning are two ends of a spectrum. A large language model (LLM) learns to reason step-by-step when data is structured such that the next token depends on a small amount of preceding context. Inference in LLMs resembles pattern recognition when the next token depends on a large amount of preceding context. If the next token depends on only the $c$ most recent tokens, reasoning traces are paths on a De Bruijn graph whose nodes are $c$-length contexts and edges are next-token transitions between contexts. The set of reasoning traces of a task forms a directed acyclic subgraph of the De Bruijn graph. An LLM that has learned all edges of this subgraph can compose them to solve longer, unseen tasks, i.e., it reasons step-by-step. We prove that the number of edges is vanishingly small compared to the number of reasoning traces. Empirically, the number of training samples a transformer needs is a power law in the number of edges, so learning to reason step-by-step is sample efficient. We can induce De Bruijn structure in any task by maintaining a “state” that makes future reasoning independent of the past. The frequency of states in the reasoning trace determines $c$. We show, by fine-tuning Qwen2.5-1.5B-Instruct to solve equations and answer questions about stories, that frequent states (small $c$) result in higher accuracy but greater fragility to perturbations at test time. LLMs trained with a large $c$ are only as good as models that perform pattern recognition without reasoning. A moderate density of states balances accuracy and robustness. We show that real-world data has De Bruijn structure: Qwen3-14B and Qwen3-32B retain over 75% of their accuracy on GSM8K, MATH-500 and GPQA-Diamond when attention is restricted to a sliding window less than 15% as long as the full reasoning trace.

Metallic nanowire aerogels combine the functional properties of metals with the high porosity of aerogels. Here, Ag and Cu nanowire aerogels with densities of 20-500 mg cm-3 were synthesized by freeze casting and investigated using lock-in thermography during sequential mechanical compaction. Thermal transport depended strongly on nanowire diameter, with 30 nm Ag nanowires exhibiting substantially lower thermal diffusivity than 60 nm wires, likely due to enhanced surface scattering as the wire diameter approaches the electron mean free path in Ag. Surprisingly, increasing the aerogel density through mechanical compaction reduced the calculated thermal diffusivity by nearly 300x, opposite the conventional expectation that denser materials transport heat more efficiently. Further analysis showed that this decrease is dominated by the thickness term in the thermal diffusivity formalism, while the fitted transport coefficient changes comparatively little. The characteristic thermal diffusion timescale decreased only modestly during compaction, indicating that densification primarily collapses pore volume without fundamentally restructuring the conductive nanowire network. This behavior resembles topology-preserving densification, where large geometric compression produces comparatively small changes in the effective percolative transport pathway. These results show that nanowire aerogels cannot always be treated as homogeneous bulk media, because their macroscopic geometry and conductive network evolve differently during compaction.

The Mpemba effect challenges the intuition that states closer to equilibrium must relax faster. We ask whether an analogous magic-ordering reversal can occur in the generation of quantum magic (nonstabilizerness), a key resource for universal quantum computation: can a state with less initial magic overtake one with more and reach its asymptotic value sooner? Here, we uncover interaction-induced dephasing as a distinct mechanism for this magic Mpemba effect, requiring neither transport nor conventional thermalization. For tilted product states in an interacting $\ell$-bit model, a random-phase analysis predicts a reversal of the magic ordering: states with less initial magic attain higher saturated magic, independent of their spatial pattern. Exact simulations of the $\ell$-bit model and localized spin chains further show that these states also saturate earlier. We observe both features on IBM superconducting quantum processors. An effective $\ell$-bit implementation accesses long dephasing times at fixed circuit depth by encoding time in gate angles, while a microscopic spin model exhibits the reversal universally across initial patterns in the localized regime but only selectively in the ergodic regime. These results establish interaction-induced dephasing as a route to anomalous quantum-resource relaxation beyond transport-driven thermalizing dynamics.

We consider a modular associative neural network made of $L$ Hopfield models (HMs), coupled so that intra-module interactions are Hebbian and inter-module interactions are anti-Hebbian; this competitive coupling is known to endow the network with pattern-disentanglement capabilities. The integral representation of this system coincides with an assembly of $L$ restricted Boltzmann machines (RBMs) whose hidden layers are coupled, thereby extending the HM-RBM duality to the modular setting. We then train this modular RBM, via one-step contrastive divergence, to perform a classification task in which a query encoded on the visible layers is mapped onto an $L$-tuple of labels read off the hidden layers. When the query is composed of $L$ patterns that are mutually orthogonal on average, we prove that the RBM weights obtained as empirical means over the training dataset, as suggested by the HM-RBM equivalence, constitute a fixed point of the learning dynamics, and we derive an explicit, non-asymptotic bound on the residual drift. We then turn to a harder classification task in which each module is queried with a mixture of $L$ patterns and we show numerically that the same setting for the RBM weights still provides an effective set-up, letting the network jointly classify and disentangle the mixture.

We discuss free-energy estimation for complex systems using simulated tempering with Wang–Landau-type feedback. We show that introducing nonreversible dynamics in temperature space and multiple temperature updates per step significantly accelerates the convergence of the free-energy estimate. We then apply the algorithm to the three-dimensional Edwards-Anderson spin-glass model. Our results show that the algorithm can precisely estimate the free energy up to a large system size $L \simeq 64$, and that subsequent equilibrium simulated tempering simulations with the estimated free energy can rapidly sample from the Boltzmann distributions over a wide temperature range that covers the spin-glass transition temperature. The convergence time of our algorithm grows much more slowly with system size than that of another known algorithm for on-the-fly free-energy estimation.

We study how independently trained Transformer neural networks reconstruct a binary string from its local domain walls. Runs sharing the data and training protocol can realize different functions. We treat them as replicas and measure truth alignment $m$, prediction confidence $q_{\mathrm{self}}$, and cross-replica agreement $q_{\mathrm{cross}}$. Confident disagreement defines the finite-size replica fragmentation that we call glass-like. With small training sets, replicas predict all training examples correctly but remain confident in incorrect predictions for unseen inputs, a regime we call memorization. With larger sets, runs can generalize and then retreat. Retreat occurs when outputs start to deviate from truth while confidence remains high. The frontier between learned and unlearned outputs recedes toward shorter strings. Many later recover as the frontier advances again. The self–cross gap $q_{\mathrm{self}}-q_{\mathrm{cross}}$ clearly distinguishes the three learning regimes of memorization, retreat, and recovery. With overall and position-dependent truth alignment subtracted, their residual correlations also differ: memorizing replicas have weakly and uniformly correlated residuals, while retreat has the largest fraction of replica pairs whose residuals are anti-correlated. That is, on the inputs where one replica of such a pair does better than its average, the other tends to do worse. These finite-size observations distinguish persistent memorization from ongoing retreat–recovery dynamics. The thermodynamic and long-time limits, and extensions to other learning tasks, remain open.

Nonlinear neural networks develop structured internal representations, yet how their geometry is determined by the tasks being learned remains poorly understood. Here, we develop an analytical framework for piecewise-linear feedforward and recurrent networks that links learning, activation-region structure, and neural selectivity. We show that, at any stationary point of gradient-aligned learning, a nonlinear network decomposes into local linear regressions over its activation regions. Deviations from the corresponding least-squares solutions are jointly constrained by network weights shared across regions and vanish in the low-error regime, yielding an approximate piecewise least-squares decomposition of the task. This structure admits a decision-tree interpretation, which we recover numerically by fitting trees to predict network activation patterns from the input. We further derive how the regions in which neurons are active determine the task statistics they capture, thereby organizing neural selectivity into distinct subpopulations. The resulting predictions closely match the selectivity geometry observed in simulated networks and two empirical neural datasets. Finally, we show that neural baseline regulates activation pattern diversity, placing networks along a continuum between coarse, generalizing representations and fine-grained, expressive representations. Together, these results establish activation regions as a unifying framework for describing how nonlinear networks decompose task structure into local computations and how those computations can be revealed through neural activity.

A machine learning (ML) framework for phase-diagram construction in condensed matter systems with unknown phase structure is presented. The method combines physically motivated descriptors with contrastive representation learning and a two-stage data-generation workflow. First, a sparse set of configurations is generated by Monte Carlo simulated annealing in order to train a contrastive neural network that produces a low-dimensional embedding. Consequently, representative phase configurations are used as initial conditions for gradient-based optimization on a dense parameter grid, enabling efficient generation of refined datasets for high-resolution phase-diagram construction. In order to optimize the workflow, a variant of the method, in which derived descriptors are used as soft labels in the contrastive objective, is examined as well. The framework is demonstrated on a PbZrO3-inspired model potential formulated in terms of coupled order-parameter fields and used as a physically motivated testbed for exploring configuration diversity and phase behavior. The proposed approach provides a physically informed route to ML-assisted phase-diagram construction that integrates data analysis with adaptive dataset refinement.

We investigate single-excitation quantum walks across the one-to-two-dimensional crossover in diluted power-law hopping models. Using arrays of $\ell$ coupled chains of length $L$, we identify the finite-size localization crossover from the longitudinal spreading of an initially localized excitation. The crossover filling is well described by an exponential approach to its two-dimensional value with increasing $\ell$, defining a characteristic transverse scale $\ell_e$, and effectively two-dimensional behavior emerges already for $\ell\ll L$. An independent spectral analysis shows that $\ell_e$ is set by the transverse extent of the eigenstates participating in the dynamics, providing a microscopic interpretation of the dimensional crossover. Our results reveal an intrinsic eigenstate scale governing emergent dimensionality in diluted systems with power-law hopping, a regime directly relevant to recently developed arrays of polar molecules, Rydberg atoms, and other quantum simulators with long-range interactions.

Many real-world datasets exhibit unusually large values far more frequently than predicted by Gaussian models. Heavy-tailed distributions capture this behavior, yet evaluating learning performance under them remains challenging because rare, large feature entries retain non-vanishing effects even in high dimensions. Even in the canonical setting of empirical risk minimization for linear regression with entry-wise i.i.d. symmetric $\alpha$-stable data, a precise asymptotic characterization of prediction has been lacking. In this work, we introduce a functional order parameter that describes the random effective problem associated with each coefficient. Using the replica method, we fully characterize the generalization error in the proportional high-dimensional limit where the sample size and feature dimension diverge at a fixed ratio. Additionally, this analysis establishes a heavy-tail universality law, scaling laws relating typical errors to prediction reliability, and the Bayes-optimal prediction error. In addition to characterizing the effects of extreme entries on the learning process, our method applies broadly to other systems with persistent local heterogeneity.

The Sachdev-Ye-Kitaev model is a strongly interacting fermionic system that has been well-studied in condensed matter and high energy physics. It is highly quantum: Gaussian states are far from the thermal state (Hastings and O'Donnell, STOC'22) and representing the thermal state requires large polynomial-size quantum circuits (Anschuetz et al., QIP'25). Very recently, it was nonetheless proven that classical algorithms can estimate local thermal expectations at sufficiently high temperature in quasipolynomial time (Zlokapa, FOCS'26). We show that classical algorithms can in fact estimate local observables at all constant temperatures in polynomial time. Our techniques also extend straightforwardly to classical systems: we resolve an open question about computing thermal expectations of a classical spin glass up to its phase transition (Bencs et al., STOC'26). Our proof develops a fully rigorous quantum cavity method. Due to the success of the classical cavity method in optimization, sampling, inference and learning, we expect the quantum cavity method to find further applications of independent interest. As an example, we give a quantum algorithm that learns SYK Hamiltonians from the Gibbs state at any constant temperature with polynomial time and sample complexity.

A recurring design principle in modern optimizers is to decouple update magnitude from the raw gradient norm, yet its consequences for learning-curve and resource scaling remain unclear. We isolate this mechanism by studying normalized SGD in a random-feature model with power-law teacher and data covariance. Fixed-norm updates induce an effective learning rate that grows as gradients shrink. We derive a dynamical mean-field theory (DMFT) describing the joint dependence of the loss on training time, model width and batch size. Normalization initially accelerates SGD, mapping the power-law exponent $r_{\rm SGD}<1$ to $2r_{\rm SGD}/(1-r_{\rm SGD})$, with exponential convergence at $r_{\rm SGD}=1$ and formal finite-time convergence for $r_{\rm SGD}>1$. At finite step size, however, the same feedback ultimately breaks the acceleration and leads to marginal stability. The late-time theory yields width-limited, edge-of-stochastic-stability (EoSS), and deterministic edge-of-stability (EoS) regimes. These phases determine when larger batches or wider models reduce serial training time at comparable compute. We quantify in which of these phases increased batch size or width can compensate the excess compute use per step by fewer optimization steps to target loss. Linearized ResNet experiments on CIFAR-5M support the predicted acceleration, breakdown, and resource-scaling trends. Together, these results connect normalization-induced acceleration, EoS effects, and width–batch allocation within a solvable theory.