Papers

We show that cavity quantum electrodynamics (QED) devices can realize dissipative Sachdev-Ye-Kitaev (SYK) universality, a paradigmatic setting for quantum chaos in open many-body systems. Ultracold fermions with disordered, all-to-all cavity-mediated interactions provide two complementary routes: atomic spontaneous emission in a single-mode cavity and photon leakage from a multimode cavity. Strikingly, both converge to the same non-Hermitian random-matrix universality despite originating from integrable and chaotic closed-system limits, respectively. In the single-mode cavity, dissipation creates quantum chaos from an integrable Hamiltonian. We trace this convergence to a tunable growth in dissipative rank, controlled, respectively, by the Lamb-Dicke parameter and the cavity-mode spacing. The integrability-to-chaos crossover is accompanied by a dynamical crossover from long-lived prethermal memory to rapid thermalization, visible in single-atom-resolved densities.

The Kuramoto model provides a paradigmatic framework for studying synchronization of interacting oscillators, and has been generalized to arbitrary dimensions to describe swarms, flocks and multi-dimensional opinion dynamics. Yet, existing formulations neglect inertia, a key mechanism known to enhance information propagation and collective responsiveness. Here, we introduce and analyze an inertial Kuramoto model in arbitrary dimensions. We show that inertia fundamentally alters the nature of the synchronization transition, inducing a crossover from continuous to discontinuous behavior, with the onset of hysteresis depending explicitly on both inertia and dimensionality parity. Our analytical theory is supported by extensive numerical simulations. These results establish inertia as a crucial ingredient of high-dimensional collective dynamics and reveal a novel universal structure in synchronization phenomena.

Random walks with long-range jumps can drive superdiffusive transport, replacing ordinary diffusion with an effective long-range kinetic operator. Such superdiffusive kinetics is also central to critical phenomena, notably the self-avoiding walk with long-range jump statistics, or Lévy-SAW. This work investigates how the critical behavior is affected when the long-range connectivity itself becomes random. We study self-avoiding walks (SAWs) on a one-dimensional long-range random ring graph, where bonds are independently generated with Bernoulli probability $\sim|i-j|^{-(1+\sigma)}$. We term this walk Sparse-SAW. The same random bonds are responsible for both long-range superdiffusive transport and quenched disorder, with both simultaneously controlled by the single parameter $\sigma$, placing the problem beyond the conventional Harris and Weinrib-Halperin frameworks. Through large-scale Monte Carlo simulations and a Gaussian-truncated field theory, we show that Sparse-SAW belongs to the same universality class as the clean superdiffusive Lévy-SAW. The random bonds generate short-range uncorrelated and long-range correlated mass disorder while simultaneously producing the long-range kinetic operator. Under coarse-graining, the latter dominates, restoring the clean critical behavior. Our study suggests that the full non-Gaussian Bernoulli statistics may lead to disorder physics beyond the conventional theory of quenched disorder, while establishing random graphs as an efficient platform for extracting the critical exponents of the clean superdiffusive Lévy-SAW universality class.

We study the class of transversal field models with pair interactions, including the quantum Curie-Weiss and the transversal field Ising model, as well as quantum spin glasses based on the Sherrington-Kirkpatrick (SK) and Edwards-Anderson models. Our main result is a general correlation inequality based on a Gaussian convolution estimate, which is derived from an extension of the Ding-Song-Sun inequality and an application of Brascamp-Lieb convexity techniques. The results apply for sufficiently strong transversal fields and, for quantum spin glasses, yield a high-field regime without replica order. Combined with the previously established low-field replica-symmetry-breaking regime, this gives distinct low- and high-field overlap phases. As a corollary of the logarithmic Sobolev inequalities for the underlying functional integrals derived here, we show the concentration of the self- and replica-overlap, and determine the high-field asymptotics of the pressure in the quantum SK model.

Training large language models is costly. How low a loss the same compute can ultimately reach depends on how each step's gradient is converted into a weight update; the rule that performs this conversion is the optimizer. From SGD and AdamW to the recent Muon, effective update rules have mostly been shaped by engineering intuition and then selected on benchmarks. Muon semi-orthogonalizes the momentum matrix before applying the update and has kept breaking records on public training benchmarks; yet why the semi-orthogonalized direction works, and over how long a history the momentum should average, are two questions at present answered mainly by experience. Here we treat the weight matrix during training as a responsive medium with memory and build a physical model for it, in which both questions find answers: the semi-orthogonalized direction is the maximally dissipative response under an output-side safety budget, which explains why it works; momentum is the internal stress accumulated by the medium; how long it should average is set by the relaxation of this stress, and a real medium relaxes on more than one timescale, the simplest form being one fast and one slow. On this basis we propose the Bi-Maxwell optimizer. The framework further yields a testable consequence: gradient directions change fast early in training and more slowly later, so the optimal memory length should grow with training stage; step-by-step measurements of a proxy for it by a read-only probe across 8 independent training trajectories are consistent with this consequence. Replacing the memory kernel alone, from a single timescale to two, brings training to the target loss in noticeably fewer steps on a public large-language-model optimizer benchmark.

Machine-learning interatomic potentials enable accurate and efficient atomistic simulations, yet their reliability for out-of-distribution configurations far beyond the training domain remains a significant challenge. Here, we introduce a semiparametric interatomic potential based on a generalization of the Abell–Tersoff bond-order potential, incorporating a chemically informed functional form and explicit high-order many-body correlations to address this challenge. The model is trained and evaluated on various datasets of silicon, carbon, water, and small molecules, achieving interpolation accuracy comparable to existing MLIP models while exhibiting improved extrapolation to unseen configurations, including those at high pressures and temperatures. These results provide insights into the design of specific inductive biases for reliable extrapolation in interatomic potentials. inductive biases that can be used to control extrapolation in machine-learning interatomic potentials.

Cooperation is ubiquitous in both natural and human societies, yet its evolutionary basis remains a major challenge. A long-standing puzzle is whether having more information leads to better decision-making and thus a higher level of cooperation. To address this question, we adopt a recently developed reinforcement learning framework in which individuals learn through trial and error to maximize cumulative rewards - a paradigm that has successfully explained diverse emergent patterns in human behaviors. Specifically, we equip a structured population with the Q-learning algorithm and systematically vary the size of the interactive neighborhood, which serves as a proxy for perceived information. Interestingly, we observe a non-monotonic relationship between cooperation prevalence and neighborhood size in both two-dimensional square lattices and Barabasi-Albert scale-free networks. This inverted U-shaped dependence reveals that an optimal amount of information exists, yielding the highest level of cooperation. Mechanistic analyses show that a moderate neighborhood size enables individuals to strike an optimal balance between information sufficiency and decision-making tractability. This balance allows them to detect reciprocal opportunities while avoiding the deterioration of decision quality due to information overload. Our findings challenge everyday intuition, suggesting that a proper amount of information - not more - is optimal for the emergence of cooperation.

In recent years, both academia and industry have focused on the development of computational architectures inspired by the distributed, adaptive, and event-driven characteristics of biological neural systems, with the aim of reducing the computational cost associated with conventional training approaches [1]. However, a major challenge is the lack of general models and design guidelines for emerging computational systems and hardware. This work introduces a general model based on an input-dependent stochastic weight network, referred to as a substrate. The substrate weights evolve through input-triggered stochastic updates, with correlations between weight coefficients described by a matrix-valued covariance kernel. The proposed framework is implemented using quadratic polynomial weight functions, where the input amplitude controls the magnitude of the stochastic perturbation and a substrate-dependent distance determines the correlation structure. Numerical simulations show that correlations in the stochastic weight evolution significantly affect the system response, suggesting a potential mechanism for neuromorphic-inspired computation without conventional weight training. The aim of this work is to provide a general formulation of the model and identify its main properties and characteristics. 1 H. Jaeger, Towards a generalized theory comprising digital, neuromorphic and unconventional computing, Neuromorphic Comput. Eng., vol. 1, no. 1, p. 012002, Sep. 2021

Finite datasets can share the same size and low-order statistics while differing strongly in structural complexity. We connect this dataset complexity to loss-landscape geometry by pairing local label mixing across neighborhood scales with local entropy around trained neural-network solutions. Adapted from the Franz–Parisi construction in spin-glass theory, local entropy measures the effective volume of low-loss, solution-like parameter configurations at each distance from a reference. We estimate it in finite networks using adaptive sequential Monte Carlo. In a controlled synthetic sweep, greater dataset complexity produces a larger decrease in local entropy near the reference. Farther away, its radial derivative becomes weak and nearly common across conditions. Dataset complexity therefore changes where the effective solution volume contracts, rather than making it decrease uniformly faster. Experiments on real image data show the same qualitative trend, with label randomization further amplifying the effect. These results show that dataset structure shapes how low-loss neighborhoods are organized across finite distances from trained solutions.

We investigate localization phenomena and exact mobility edges in a quasiperiodic Aharonov-Bohm chain, where the 1D canonical diagonal and off-diagonal Aubry-André-Harper models are laterally coupled to an auxiliary sublattice threaded by a synthetic magnetic flux. By analytically computing the Lyapunov exponent via Avila's global theory of one-frequency Schrödinger operators, we derive exact expressions for mobility edges. In the diagonal limit, the coupling to the auxiliary sublattice generates hyperbolic mobility edges that exhibit a sign-changing divergence and can be continuously tuned by the magnetic flux. In the off-diagonal regime, the interference between quasiperiodic hopping and indirect tunneling through the auxiliary sublattice gives rise to exact anomalous mobility edges that separate critical and extended states. Critical states and anomalous mobility edges emerge even in the absence of incommensurately distributed zeros in the hopping modulation, a behavior distinct from that of conventional off-diagonal models. Our results establish a rigorous theoretical framework for engineering controllable mobility edges, with the synthetic flux providing a tunable experimental knob that paves the way for realizations in platforms such as superconducting quantum circuits and photonic waveguides.

We investigate analytically and numerically a one-dimensional periodically driven free-fermionic system subjected to monitoring of the local particle density. Based on the analytical approach that describes the long-wavelength physics of the time-dependent Hamiltonian in the field-theoretical language using the nonlinear sigma-model (NLSM), we reveal that driving does not alter the universality class of the problem. As a consequence, the system retains the area-law behavior in the thermodynamic limit, with an intermediate diffusive regime giving rise to logarithmic growth of entanglement entropy for a small monitoring rate. At the same time, driving leads to a renormalization of the bare coupling constant of the NLSM, which controls the space-time “conductivity” in the diffusive regime. We derive the analytic form of this renormalization, which becomes particularly strong in the case of a “maximally symmetric” drive and sufficiently short driving period. In addition, we employ the Wiener-Hopf method to investigate the ballistic-diffusive crossover. These analytical predictions are corroborated by numerical simulations of the von-Neumann entanglement entropy and the density correlation function. Our numerical results clearly demonstrate that, with an increase in the system size, there are successive crossovers from ballistic to diffusive behavior and ultimately to localization. Furthermore, in the diffusive regime, we observe weak-localization corrections that are in agreement with the analytical predictions of the NLSM. Overall, our results provide a unified analytical and numerical framework for understanding the effects of monitoring in time-modulated fermionic systems, paving a way for broader investigations of driven quantum matter.

It has long been known that well-trained neural networks can be compressed very strongly without affecting their performance, an important phenomenon that remains poorly understood. We prove a uniform compressibility theorem for deep multilayer perceptrons with analytic activations. For a deep, wide fixed teacher network, there exists a narrow (same depth) network that approximately represents the same function as the original. The reachable compressed width is strikingly independent of the original width, but is $O((\log(1/\varepsilon))^{d_{in}})$, where $\varepsilon$ is the error budget and $d_{in}$ is the effective input dimension. Our construction involves a novel derivative-matching technique which is aware of the low-dimensional input, and a layer-wise reweighting that preserves the input-output mapping.

Polyethylene (PE) is one of the most commonly used synthetic polymers. While the synthesis and processing protocols for PE are well established, precise experimental assignment of microscopic structures at atomistic resolution (i.e., the position of each atom) remains largely limited to highly crystalline systems. This gap is often addressed via computer simulations using empirical interatomic potentials, which use approximate but efficient descriptions of interatomic interactions to reach the length and time scales needed to describe macromolecules. These empirical potentials typically perform well for bulk and/or collective properties but face challenges with chemical realism for complex systems, e.g., during reactive processes. In this work, we address this challenge by combining the computational efficiency of a deep potential (DP) machine-learning force field and the chemical realism of first-principles van der Waals (vdW) corrected hybrid density functional theory (DFT) enabled by a SeA high-throughput framework. Using this approach, we study the structure and dynamics of PE oligomers and polymers in an ethylene solvent under common high-pressure (supercritical) radical polymerization conditions. We found that the local solvation environment of radical-containing PE oligomers converges for chain lengths greater than (n~6), suggesting extensibility of our oligomer-trained MLFF to significantly longer polymers. We then confirmed the extensibility of these models to long PE chains by characterizing the molecular weight scaling of single-chain structure and dynamics, which showed classic good solvent behavior. Our PE MLFF retained a consistent level of fidelity and stability across a wide range of thermodynamic state points and chain lengths, at full atomistic resolution, therefore paving the way towards first-principles-based polymer structure and property prediction.

Under mild assumptions, any quantum mechanical (QM) model or quantum field theory (QFT) admits a representation in terms of an ensemble of neural networks with countably many random parameters. We investigate the features of NN-QM and NN-FT models with finitely many parameters, such as a feedforward network of width $N < \infty$. We find that, generically, such models must violate one of the properties of conventional Euclidean QFTs, such as reflection positivity or cluster decomposition. We present several complementary ways of understanding which features can and cannot be preserved at finite $N$, both in QM and in QFT.

Long timescales in brain dynamics give rise to power law correlations measured in experiments. A simple linear recurrent neural network model can reproduce this power law behavior, but the recurrent interaction strength needs to be fine tuned in order to sit at the edge of stability. We show that by adding a dynamical recurrent feedback gain to the simple linear model removes the need for fine tuning, because the gain self-organizes so that the spectrum of the effective recurrent matrix is always gapless with the top eigenvalue landing on the stability edge for any recurrent strength. As a consequence, long timescales and scale free correlations arise generically, as we demonstrate both analytically and numerically.

Modern quantum experiments can probe many-body wavefunctions through projective measurements, providing snapshots of individual many-body configurations. A fundamental question is whether the intrinsic structure of their measurement distributions can reveal quantum dynamics beyond predefined observables. Here, we address this question at the many-body ergodicity-breaking transition using two complementary characteristics of nonequilibrium wavefunction snapshots: their binary intrinsic dimension (BID) and the topology of wavefunction networks constructed from Hamming distances. We show that the critical point is characterized by an extensive but submaximal BID, and scale-free network connectivity with highly connected hubs. These signatures distinguish the ergodic, critical, and nonergodic regimes, and provide experimentally accessible probes of ergodicity breaking from projective measurement snapshots.

Markov chain Monte Carlo algorithms can be viewed as feedback devices that compare a proposed move with the target distribution and then accept or reject it. In this paper the Maxwell demon is identified with the acceptance module: it measures a proposed edge, stores the outcome in the accept/reject bit, and uses that bit to shape the probability current. The decision bit carries a genuine Shannon mutual information about the proposal, whereas only its directional part is converted into a cooling information flow. The relative-entropy relaxation rate obeys $v(t)=\dot{\mathcal I}_{\rm cool}(t)+\dot S(t)$, which separates useful cooling from housekeeping circulation in nonreversible chains.

The dramatic slowing down of structural relaxation in supercooled liquids is accompanied by the emergence of dynamic heterogeneity. A monotonically increasing dynamical correlation length, measured at the $\alpha$-timescale, is one of the remarkable features of this phenomenon. Here we show that this picture is incomplete: the dynamical correlation length measured in the $\beta$-relaxation regime exhibits a striking non-monotonic temperature dependence, reaching a maximum near the mode-coupling crossover temperature $T_c$ and decreasing upon further cooling, even as local dynamical fluctuations continue to intensify. This behavior suggests a crossover from spatially extended, maximally cooperative motion near $T_c$ to increasingly compact and localized relaxation events below it. We demonstrate that this evolution is quantitatively captured by stochastic beta-relaxation theory, an extension of mode-coupling theory beyond mean-field that explicitly predicts an avoided dynamical transition in finite dimensions. Our results provide the first direct spatial evidence in favor of the avoided-transition picture of the mode-coupling crossover, and establish the peak of the $\beta$-regime correlation length as a robust indicator of the mode-coupling crossover.

The Trachenko-Zaccone equation provides a compact nonlinear dynamical framework for describing non-Debye relaxation in disordered condensed matter. Originally developed to rationalize stretched- and compressed-exponential relaxation in liquids and glasses from the dynamics of interacting local relaxation events, the same equation has subsequently appeared in broader contexts, including polymer relaxation and nonlinear models of global population dynamics. This review retraces the conceptual development of the equation, with particular emphasis on its physical origin in Kostya Trachenko's feed-forward interaction mechanism, its mathematical structure, and its possible generalizations. We also include personal recollections of the work with Kostya Trachenko at Queen Mary University of London in August 2019, during which the equation emerged in essentially its present form. After reviewing applications to stress relaxation, glassy materials, polymer relaxation and population dynamics, we discuss future directions including time-dependent feedback parameters, coupled order parameters, heterogeneous and spatially resolved formulations, flux terms, network versions and stochastic extensions. The central theme of the review is that the Trachenko-Zaccone equation should be viewed not only as a model of glassy relaxation, but as a general nonlinear feedback equation with potential applications across complex systems.

We determine the aging phase diagram of mixed spherical $(p+s)$-spin glasses quenched from random configurations to zero temperature. Solving the asymptotic dynamical equations of Cugliandolo and Kurchan, we find transitions between aging states with one, two, and continuously many effective temperatures, including phases in which discrete and continuous hierarchies coexist. We obtain analytic expressions for the energies reached asymptotically by the dynamics. In models combining pairwise and higher-order interactions, and sufficiently separated interaction orders ($s>3$), a fully continuous phase exists, where the gradient descent relaxation converges to the algorithmic energy lower bound. Instead, in mixed models with only high-order interactions ($p\ge3$) the energy relaxation remains strictly above it. Numerical integration of the dynamical equations is consistent with the predicted two-step and mixed discrete–continuous aging states.

Motivated by a previous Ising study, we identify a ${\it higher}$ Nishimori line in the learning phase diagram of the $2D$ $q$-state Potts model $(2 < q\leq 4)$ under bond-energy measurements. This ${\it higher}$ Nishimori line meets the critical temperature line of the Potts model, in a ${\it higher}$ Nishimori critical point – a tricritical point at finite inference strength that separates a paramagnetic, a ferromagnetic and a 'spin-glass' phase. With analytical tools, we discuss the general structure of the rich phase diagram, which contains two unstable and three stable fixed points, and obtain a number of exact results for universal quantities, including the decay exponent of the Edwards-Anderson correlator, using a Gaussian measurement protocol which allows for exact calculations. Using extensive numerical tools, we confirm these statements for a generic, discrete $q$-state measurement protocol and determine precise numerical estimates for the location of higher and ordinary Nishimori critical points as well as RG flows between the various fixed points. We also discuss the Casimir effective central charges of the critical points in the learning phase diagram, and their monotonic ${\it decrease}$ along measurement-induced RG flows, as established non-perturbatively by the c-effective theorem and its extensions, and contrast it to the monotonic increase along the corresponding RG flows in the random-bond Potts model. Finally, we discuss a general argument based on ${\it Elitzur's \; theorem}$ that establishes stability of the ordinary Nishimori critical points in their respective learning phase diagrams. Equivalently, our results describe a monitored deformed $\mathbb{Z}_q$ toric code where the tricritical ${\it higher}$ Nishimori point is an 'information' critical point that separates stable quantum, classical, and no memory phases.

What is the relation between subsystem quantum complexity and the emergence of computationally useful structure? We address this by studying a family of minimally tunable postvariational quantum circuits, and show how spectral nonflatness and metrological response directly control the ensemble-typical information processing power. This unveils an intermediate "learning phase" that precedes the onset of quantum chaos, characterized by pronounced nonflatness and sensitivity of readout states. The optimal information processing capacity improves with system size, while deep scrambling suppresses observable response. The results reveal how such features of random quantum dynamics can be viewed as computational resources for scalable nonlinear computation.

Accurate digital quantum simulation at long times is limited by the accumulation of errors inherent to approximate simulation. Here we introduce RL-Trotter, a reinforcement-learning framework that treats unavoidable approximation errors as resources for error correction rather than merely imperfections to suppress. We show that low-dimensional information from conservation laws, such as the energy and energy variance, provides a sufficient learning signal to guide the agent, which learns to adapt a single scalar—the next Trotter step size—without access to the target wave function. By optimizing the entire long-time evolution rather than individual steps, RL-Trotter discovers self-correcting sequences in which later errors compensate for those accumulated earlier, increasing the accuracy of the long-time dynamics. The learned policies are intrinsically robust to measurement noise, substantially reducing measurement overhead. They also generalize to previously unseen, physically similar initial states and transfer from small, classically simulable systems to systems an order of magnitude larger. This enables a practical protocol based on classical pretraining followed by direct deployment or limited fine-tuning on quantum hardware. Our results establish a broader perspective for quantum algorithms: errors in approximate evolution can be orchestrated into resources for accurate and resource-efficient quantum dynamics.

Reputation is widely recognized as a key mechanism for sustaining cooperation. However, most existing game-theoretic models treat reputation primarily as an external factor that modulates payoffs, interaction structures, or strategy update rules. In many social contexts, though, reputation operates primarily as information – it shapes how individuals interpret their own experiences and assess the behavior of others. To bridge this gap, we propose a spatial prisoner's dilemma game grounded in the reinforcement learning paradigm, in which agents equipped with Q-learning integrate both individual and social information via a locally defined reputation metric to guide their decisions. Our results reveal that reputation-modulated learning significantly promotes the emergence of cooperative behavior, and we observe a discontinuous phase transition from full cooperation to full defection as the temptation increases. Cooperation spreads through the nucleation of cooperative clusters, whereas the disintegration of these clusters drives the system into an absorbing state of complete defection. Overall, this study demonstrates that reputation facilitates cooperation not only by providing direct incentives but also by reshaping the social information landscape that agents rely on for learning and adaptation.

Dissipation is usually regarded as a source of decoherence that suppresses quantum interference and localization. Here we show that suitably engineered dissipation can instead be used to select localized or extended states in a strictly non-disordered one-dimensional lattice. The underlying clean lattice has spatially inhomogeneous hopping and supports both extended bulk states and localized boundary states, including an algebraically localized bound state in the continuum. We introduce a nonlocal bond jump operator with a tunable relative phase and show that this phase selectively favors eigenstates with different spatial phase correlations. As a result, the long-time density matrix can be steered toward sectors dominated by localized or extended Hamiltonian eigenstates without changing any Hamiltonian parameter. The microscopic origin of the selection is quantified by the fraction of site pairs separated by a distance $l$ that are phase matched with the dissipative channel. We further characterize the dissipative quench through the quantum fidelity and show that the selected character of the steady state can persist after the dissipation is removed. Our results establish phase-selective bond dissipation as a route to controllable state preparation and transport manipulation in non-disordered lattices.

We formulate a theory of pseudogap formation generated by dynamic finite nesting vector ${\bf Q}$ fluctuations with characteristic oscillation frequency $\omega_0$, and damping $\gamma$. Starting from a Hamiltonian describing electrons coupled to a classical Gaussian random field, we introduce the double - series representations of the single - particle Green's function within an Abelian (commuting) approximation to the exact SU(2) time evolution of the pseudogap problem. The resulting propagator naturally acquires a generalized Bogoliubov structure in which every stochastic scattering history is characterized by an effective dynamic gap, leading to a coherent superposition of dynamically broadened sidebands with complex Poisson weights. A central result of the theory is the emergence of a dynamically generated decoherence scale $\Gamma_{\rm eff}$ governing the crossover between two qualitatively different pseudogap regimes. For $\omega_0>\Gamma_{\rm eff}$ the fluctuating field is resolved coherently and the double-series representation provides a controlled description of dynamic sideband formation. Conversely, when $\Gamma_{\rm eff}\gtrsim\omega_0$, coherence is progressively lost and the theory crosses over to the quasistatic fluctuating - gap regime described by the exact continued-fraction solution. The coherent and quasistatic descriptions are therefore interpreted as two complementary asymptotic limits of the same microscopic dynamic pseudogap model. The detailed results of numerical calculations for electron spectral density and density of states supporting our approach are presented for different sets of model parameters confirming the general picture of this crossover.

How many memories can a local learning rule protect, and what fixes the number? Physical learning rules train resistive networks through local measurements, and the standard rules conserve a mass-like function of the conductances, a law whose general form was posed as an open problem with the expectation that no useful solution theory exists. We answer it, in the direction the expectation ran against, and the answer is a capacity theory. The Tellegen identity behind the conserved mass localizes: the feedback state is pinned to zero at the inputs, so every sector of the circuit that the input electrodes separate carries its own private conserved mass, by an argument consuming only Kirchhoff's law and that boundary condition, hence valid for arbitrary nonlinear branch laws. The number of independent sector masses is a topological property of the circuit, bounded by the number of output electrodes. An untrainable element can destroy the mass of its own sector and of no other, which says where fixed nonlinear components may be placed; per-edge learning rates select which functionals are conserved and never how many; and adjoint coupled learning, which clamps its outputs before measuring, drains every output-carrying sector at a rate set by the squared output multipliers. In linear circuits we then classify the identities: exact rational computation over all 502 circuits on up to five vertices with two inputs and one or two outputs, and hundreds of larger ones, yields a closed-form count, proved on that family and conjectured in general, with the sector statement a theorem at every circuit size. The anticipated series-parallel polynomial laws appear as exactly the series-class cubic differences, and the number the opening question asks for is a budget: one designed mass per sector, the differences the topology donates, and one broadcast scalar for each protected functional beyond.

We modify the NN/QFT duality [1] to incorporate the layerwise permutation symmetry of the network, resulting in a $(0\!+\!1)$-dimensional lattice gauge theory, in which each layer of $N$ neurons acts as an $N$-component lattice site, and the weight matrices play the role of gauge fields living on the links. In this framework, we compute the tree-level neuron-neuron propagator which describes the evolution of layer variance in the network, and develop the Feynman diagram machinery to compute interactions in the perturbative expansion in $1/N$. In particular, we obtain a recursive expression for all corrections to the exact propagator at $O(1)$, representing statistical fluctuations in the ensemble of networks, including infinitely-many loop diagrams mediating the interactions from previous layers. We also present a preliminary analysis of neuron scattering amplitudes that contribute order-by-order in $1/N$, which provides a field-theoretic framework for studying higher-point correlations, and by extension information propagation, in deep networks. We remark on some interesting directions for future work at the intersection of neural networks and quantum field theory.

Machine-learned exchange–correlation (XC) functionals offer a route to improve Kohn–Sham density-functional theory without incurring the cost of explicitly correlated electronic-structure methods. Their use in production simulation codes, however, requires a well-defined mapping between the learned model and the host-code density representation. We formulate and implement a Skala-1.1 interface in CP2K through the external GauXC library. CP2K supplies the geometry, Gaussian basis, spin-resolved atomic-orbital density matrix, and communicator, while GauXC evaluates the XC energy, atomic-orbital potential matrix, and available nuclear derivatives. The interface accepts both all-electron and valence-only density matrices. The latter may arise from separable dual-space pseudopotentials or molecular effective-core potentials. Implementation errors are isolated from functional differences by comparing the Perdew–Burke–Ernzerhof (PBE) functional evaluated through GauXC with native CP2K PBE. The resulting interface gives consistent energies, forces validated against finite-difference total-energy checks, and force-based molecular-virial diagnostics for representative molecular cases. The dietGMTKN55 benchmark suite is evaluated with an all-electron Gaussian augmented plane-wave treatment for elements up to bromine and def2 effective-core potentials for the heavier elements. The resulting aggregate mean absolute deviation of 1.255 kcal/mol is within 0.012 kcal/mol of the corresponding Skala reference value of 1.243 kcal/mol. This work establishes a validated molecular implementation of Skala in CP2K through GauXC.

Hard combinatorial optimization problems, many of which are NP-hard, present fundamental algorithmic challenges. Average-case analysis on random instances has emerged as a powerful framework for understanding typical algorithmic performance beyond worst-case guarantees. A substantial body of work has established negative results: for sufficiently hard instances (often controlled by the underlying graph connectivity/constraints density), no known polynomial-time algorithm can significantly outperform naive heuristics in the double asymptotic limit where both problem size and constraints density tend to infinity. We revisit this picture by studying the finite-size behavior of some optimization algorithms across easy, intermediate, and hard regimes. Through rigorous analysis of large-graph asymptotics combined with numerical experiments on canonical problems (maximum independent set and maximum $K$-SAT), we demonstrate that while algorithms do eventually converge to theoretically predicted bounds, this convergence can be remarkably slow. In the intermediate regime where instances are already highly constrained, local algorithms achieve solutions substantially better than their predicted performance in the high-constraint-density limit. This gap between finite-regime and asymptotic behavior has important practical implications: sophisticated algorithmic design remains crucial even when asymptotic theory predicts inevitable failure.