arXiv subjects

Natalia G. Berloff

Publications and source records attributed to Natalia G. Berloff.

At least 19 recordsLinked to original sources

arXiv2026

Exact Spin Elimination for Quadratic and k-Local Ising Optimization

Ising solvers have a finite spin budget. Quadratization uses auxiliary spins to replace higher-order interactions by pairwise ones. We show that removing spins while allowing more complex interactions can fit larger problems within the same spin budget and improve optimization. Walsh elimination minimizes over one spin at a time, writes the resulting interaction as a sum of spin products in the Walsh basis, and stores a rule for recovering the removed spin. Explicit resource limits control which eliminations are accepted. In a held-out test with the protocol frozen before instance generation and one annealing schedule shared by both formulations, mean observed success rises from 10.4% to 97.5% on sparse quadratic spin glasses and from 16.9% to 87.5% on three-body problems. Observed success increases on every instance with a verified reference minimum, and pooled time-to-solution estimates decrease by factors of about 34 and 11, respectively, with preprocessing included. For zero-field pairwise models on connected simple cubic graphs with more than four vertices, we prove that a fixed spin budget accommodates 50% larger inputs while retaining pairwise interactions, zero fields, residual degree at most six, and no increase in coupling count. These results establish exact spin elimination as a controlled trade between active-spin capacity and interaction complexity.

quant-ph↗

arXiv2026

Dynamic distortion from discretized coupling feedback in soft-spin Ising machines

Although Ising machines (IMs) have grown in popularity due to their potential as physics-based accelerators for combinatorial optimization problems (COPs), most literature focuses on empirically-obtained benchmarks with low universality. In this work, we propose dynamic feedback distortion as a universal metric to quantify the effect of coupling field discretization in time-multiplexed IMs. Starting from the transfer functions of realistic hardware, we analytically show how this distortion can be mitigated using a global bias (or choosing the correct discrete coupling element) and injecting coloured noise. Our numerical results show that our mitigation protocols are very effective, especially in complicated energy landscapes (namely tunable Möbius ladders and 2D tiling problems); where an analogue reduction of the distortion leads to a two--three order of magnitude increase in the success rate of complex COPs.

physics.app-ph↗

arXiv2026

Higher-Order Kuramoto Oscillator Network for Dense Associative Memory

Networks of phase oscillators can serve as dense associative memories when they incorporate genuine many-body coupling beyond the classical Kuramoto model's pairwise interaction. Here we introduce a generalized Hebbian Kuramoto model that combines a conventional two-body, first-Fourier-harmonic coupling with a genuine four-body phase interaction, inspired by dense Hopfield memory theory. For identical oscillators with Langevin noise, equilibrium mean-field theory yields a phase diagram with a tricritical point when the four-body coupling is three times the pairwise coupling, where continuous retrieval onset gives way to a discontinuous, hysteretic transition. We separately analyze a deterministic model with Lorentzian frequency disorder using the Ott--Antonsen ansatz. In that model the onset changes from supercritical to subcritical when the four-body coupling is six times the pairwise coupling, and the two descriptions agree only at the linear instability threshold. In the four-body-dominated equilibrium regime, stored-pattern and incoherent states coexist. We determine the bistable region and the free-energy barriers for transitions in both directions, and show that the noise-induced memory-loss time grows exponentially with the number of oscillators, with a rate set by the relevant barrier. At finite memory load, a phenomenological cavity signal--to--noise closure models pattern crosstalk, while simulations measure finite-size strong-cue recovery and retention thresholds. Nominal power-law fits over the simulated size range have exponents above one in several four-body-dominated cases, with the largest fitted exponent in the purely four-body zero-temperature data; these fits are not asymptotic capacity laws.

nlin.AO↗

arXiv2026

Programmable k-local Ising interactions and shallow optical Kolmogorov--Arnold networks through repeated data encounters

Photonic processors are naturally suited to linear transformations, but independently programmable higher-order interactions usually require nonlinear media or a reduction to pairwise models. We introduce a repeated-encounter architecture that combines linear optical propagation with square-law detection to evaluate sparse and structured $k$-local Ising objectives. Each hyperedge is routed to a resolved channel, returned through the spin-dependent mask, and reconstructed from calibrated encounter-order signals. Within the stated architecture class, a $k$-spin Walsh term requires at least $\lceil k/2\rceil$ data encounters. A reciprocal rank-one recollection attains this bound for every finite order, allowing hyperedge identity, interaction order, and signed coupling to be programmed independently without quadratization ancillas or material optical nonlinearities. We test the $R=2$, $k=4$ member in a finite discrete-Fourier model of an ideal folded $4f$ relay. Reciprocal recollection produces the four-body response, whereas a fixed-patch control does not; configuration-level calibration measures the leakage caused by finite windows. Under signed-amplitude encoding, the same encounter hierarchy spans polynomial edge functions for shallow optical Kolmogorov--Arnold networks. The arbitrary-$k$ result is analytic; the finite Fourier calculation tests its two-encounter member and does not replace experimental validation.

physics.optics↗

arXiv2026

Mirror-mediated long-range coupling and robust phase locking of spatially separated exciton-polariton condensates

Lattice arrays are valuable simulators for complex mathematical problems, but physical systems typically allow only short-range coupling. We demonstrate a method for independently tunable, long-range interactions between polariton condensates in two-dimensional lattices by using vertical emission and external imaging to couple arbitrary sites. Two geometrically isolated condensates are phase-locked without planar coupling, verified via phase-resolved interferometry. Analytical modeling reveals mechanisms for robust coherence. The mirror-based scheme, free of cameras or modulators, offers a pure, high-bandwidth analogt element. Extension to dense graphs via segmented micro-mirrors is limited only by imaging optics, enabling scalable, energy-efficient polaritonic hardware for neuromorphic computation.

physics.optics↗

arXiv2026

Attractor-Keyed Memory

Physical selectors (lasers choosing a mode, Ising machines settling on a ground state, condensates occupying a spin state) produce high-dimensional signatures at the moment of decision: full field amplitudes, multimode interference patterns, or scattering responses. These signatures are richer than the winner's index, yet they are routinely discarded. We show that when the signatures are repeatable across trials (stereotyped) and linearly independent across routes, a single linear decoder compiled from calibration data maps them to arbitrary payloads, merging selection and memory access into one event and eliminating the fetch that dominates latency and energy in sparse routing architectures. The construction requires one SVD of measured device responses, which certifies capability and bounds worst-case error for any downstream payload before the task is chosen. Runtime error separates into two independently diagnosable channels, decoding fidelity (controlled by dictionary conditioning $σ_{\min}(Φ)$) and routing reliability (controlled by the margin-to-noise ratio $Δ/T_{\mathrm{eff}}$), each with a distinct physical origin and targeted remedy. We derive the full error decomposition, give Ising-machine selector constructions, and validate the predicted scalings on synthetic speckle-signature simulations across three measurement modalities. No hardware demonstration exists; we provide a falsifiable four-step experimental protocol specifying what a first experiment must measure. Whether real device signatures satisfy stereotypy is the central open question.

physics.optics↗

arXiv2026

Soft vector spins with dimensional annealing for combinatorial optimization

Recently, purpose-built analog hardware that can efficiently minimize the Ising energy and thereby solve a variety of combinatorial optimization problems has been receiving widespread attention. In this work, we show how multidimensional, vectorial degrees of freedom, that are either naturally present or can be artificially created in such hardware, could strengthen the capability to find optimal solutions to optimization problems. In order to achieve this, we introduce a simple model of soft vector spins that should be implementable on a variety of analog hardware platforms as well as three different dimensional annealing methods which harness the enlarged phase space of the vectorial degrees of freedom to minimize the Ising energy. We perform simulations on different benchmark problems and show that for all dimensional annealing methods we tested, vectorial degrees of freedom improve upon one-dimensional degrees of freedom when it comes to finding the ground state of the Ising model. In particular, we find that this advantage becomes most pronounced for $d \gtrsim 3$ dimensional degrees of freedom, with diminishing returns as the dimension is increased further. Our results could inspire new analog optimization hardware and algorithms that explicitly incorporate the advantage of vectorial degrees of freedom.

cond-mat.dis-nn↗

arXiv2025

Coupling Light with Matter for Identifying Dominant Subnetworks

We introduce DOMINO, a light-matter computing platform that exploits the full complex amplitude of coupled condensate networks to solve maximum-weight clique problems and reveal hidden indirect correlations in large graphs. By embedding network structure directly into a gain-controlled polaritonic (or photonic) oscillator array, DOMINO performs analog optimization, directly solving the maximum-weight clique problem via the gain-controlled minimisation, through a physically enforced global-intensity constraint, allowing the system to converge rapidly to dominant subnetworks while simultaneously extracting phase, encoded co- and counter-regulation patterns. This gain-based mechanism unlocks capabilities inaccessible to conventional Ising-type simulators: all degrees of freedom (amplitude and phase) participate in the computation, dramatically expanding the class of problems that can be efficiently encoded. Our approach is inherently ultrafast, energy-efficient, and naturally robust to noise, requiring no digital post-processing. Applied to real gene-gene coexpression data, DOMINO reliably identifies biologically meaningful transcription-regulator modules and exposes latent regulatory relationships. Because the method applies generically to any weighted network, it establishes a scalable physical route to solving high-value graph-analytic tasks across biology, finance, social systems, and engineered networks.

cond-mat.dis-nn↗

arXiv2025

A Unified Analytic Framework for Microlensing Caustics: Geode Solutions and Hyper--Catalan Signatures

We give a preparation-invariant analytic description of image formation near microlensing caustics. After a local Weierstrass preparation at any multiple image (order $d\ge2$), the lens mapping reduces to a single geode variable $m$ satisfying $m=U\,φ(m)$, where $U$ is a prepared source coordinate and $φ$ is an image-side kernel. The coefficients of $m(U)$ obey closed Hyper-Catalan (HC) recurrences, allowing termwise derivatives and truncation control from the characteristic system. We also use the same form for a short HC predictor-corrector: evaluate the series within its certified radius and apply a brief Newton polish near the boundary. We define an HC signature (first nonzero kernel coefficients) and an HC spectrum (branch points and analyticity radius $ρ_U$), which quantify sparsity, stiffness, and safe evaluation domains. The construction covers folds and cusps of any global degree. On a binary fold and cusp, an artificial decic with a resonant unit, and two triple-lens cusps of a challenging geometry, HC seeds plus a few Newton steps recover the exact images to machine precision within the certified domain and maintain continuity under continuation. The resulting single-series templates (with $(\mathrm{Sig}_R,ρ_U)$ metadata) are ready for photometric and astrometric modeling.

astro-ph.IM↗

arXiv2025

Solving Distance-Based Optimization Problems Using Optical Hardware

We present a practical approach to solving distance-based optimization problems using optical computing hardware. The objective is to minimize an energy function defined as the weighted sum of squared differences between measured distances and the squared Euclidean distances of point coordinates. By representing coordinates as complex numbers, we map the optimization problem onto optical fields, enabling its solution through either the canonical transformation (CT) method or the gain-based bifurcation (GBB) method. To further enhance the performance of the CT method, we introduce two techniques: asynchronous update and steepened gradient. Both the CT and GBB methods can effectively solve the distance-based problem and are adaptable to various optical hardware platforms. Our optical implementation is inspired by recent progress in analog optical computing for combinatorial optimization, highlighting its promise for efficient and scalable problem-solving in high-dimensional settings.

physics.optics↗

arXiv2025

Instantons and Rarefaction Pulses as Pathways to Global Phase Coherence in Gain-Based Optical Networks

We investigate how to reliably remove unwanted global phase windings in gain-based optical oscillator networks, thereby ensuring convergence to the true synchronized configuration corresponding to the XY Hamiltonian's global minimum. Focusing on one-dimensional rings and two-dimensional toroidal lattices, we show that two key strategies greatly enhance the probability of reaching the defect-free state. First, operating at a low effective injection rate just above threshold, exploits the amplitude degree of freedom, allowing the system to form transient zero-amplitude holes: instantons in one dimension, or vortex-antivortex rarefaction pulses in two-dimensional space, that enable phase slips. Second, preparing the initial condition with amplitude or phase inhomogeneities can directly seed these amplitude collapses and prevent the system from getting trapped in higher-energy states with nonzero winding. Using the Stuart--Landau and Ginzburg--Landau equations as models for fast-reservoir lasers, we derive analytic and numerical evidence that even relatively minor amplitude dips can trigger global unwinding in 1D. We further demonstrate that the pump's slow annealing favours these amplitude-driven events, leading to improved success in finding the globally coherent ground state. These findings highlight the critical role of amplitude freedom in analogue solvers for XY optimization problems, showing how local amplitude suppression provides a direct route to ejecting topological defects.

physics.optics↗

arXiv2025

Phase Retrieval via Gain-Based Photonic XY-Hamiltonian Optimization

Phase-retrieval from coded diffraction patterns (CDP) is important to X-ray crystallography, diffraction tomography and astronomical imaging, yet remains a hard, non-convex inverse problem. We show that CDP recovery can be reformulated exactly as the minimisation of a continuous-variable XY Hamiltonian and solved by gain-based photonic networks. The coupled-mode equations we exploit are the natural mean-field dynamics of exciton-polariton condensate lattices, coupled-laser arrays and driven photon Bose-Einstein condensates, while other hardware such as the spatial photonic Ising machine can implement the same update rule through high-speed digital feedback, preserving full optical parallelism. Numerical experiments on images, two- and three-dimensional vortices and unstructured complex data demonstrate that the gain-based solver consistently outperforms the state-of-the-art Relaxed-Reflect-Reflect (RRR) algorithm in the medium-noise regime (signal-to-noise ratios 10--40 dB) and retains this advantage as problem size scales. Because the physical platform performs the continuous optimisation, our approach promises fast, energy-efficient phase retrieval on readily available photonic hardware. uch as two- and three-dimensional vortices, and unstructured random data. Moreover, the solver's accuracy remains high as problem sizes increase, underscoring its scalability.

physics.optics↗

arXiv2025

Ising Hamiltonian Minimization: Gain-Based Computing with Manifold Reduction of Soft-Spins vs Quantum Annealing

We investigate the minimization of Ising Hamiltonians, comparing the performance of gain-based computing paradigms based on the dynamics of semi-classical soft-spin models with quantum annealing. We systematically analyze how the energy landscape for the circulant couplings of a Mobius graph evolves with increased annealing parameters. Our findings indicate that these semi-classical models face challenges due to a widening dimensionality landscape. To counteract this issue, we introduce the `manifold reduction' method, which restricts the soft-spin amplitudes to a defined phase space region. Concurrently, quantum annealing demonstrates a natural capability to navigate the Ising Hamiltonian's energy landscape due to its operation within the comprehensive Hilbert space. Our study indicates that physics-inspired or physics-enhanced optimizers will likely benefit from combining classical and quantum annealing techniques.

quant-ph↗

arXiv2024

A Fully Analog Pipeline for Portfolio Optimization

Portfolio optimization is a ubiquitous problem in financial mathematics that relies on accurate estimates of covariance matrices for asset returns. However, estimates of pairwise covariance could be better and calculating time-sensitive optimal portfolios is energy-intensive for digital computers. We present an energy-efficient, fast, and fully analog pipeline for solving portfolio optimization problems that overcomes these limitations. The analog paradigm leverages the fundamental principles of physics to recover accurate optimal portfolios in a two-step process. Firstly, we utilize equilibrium propagation, an analog alternative to backpropagation, to train linear autoencoder neural networks to calculate low-rank covariance matrices. Then, analog continuous Hopfield networks output the minimum variance portfolio for a given desired expected return. The entire efficient frontier may then be recovered, and an optimal portfolio selected based on risk appetite.

q-fin.PM↗

arXiv2024

Complex Vector Gain-Based Annealer for Minimizing XY Hamiltonians

This paper presents the Complex Vector Gain-Based Annealer (CoVeGA), an analog computing platform designed to overcome energy barriers in XY Hamiltonians through a higher-dimensional representation. Traditional gain-based solvers utilizing optical or photonic hardware typically represent each XY spin with a single complex field. These solvers often struggle with large energy barriers in complex landscapes, leading to relaxation into excited states. CoVeGA addresses these limitations by employing two complex fields to represent each XY spin and dynamically evolving the energy landscape through time-dependent annealing. Operating in a higher-dimensional space, CoVeGA bridges energy barriers in this expanded space during the continuous phase evolution, thus avoiding entrapment in local minima. We introduce several graph structures that pose challenges for XY minimization and use them to benchmark CoVeGA against single-dimension XY solvers, highlighting the benefits of higher-dimensional operation.

cond-mat.dis-nn↗

arXiv2024

Fully Programmable Spatial Photonic Ising Machine by Focal Plane Division

Ising machines are an emerging class of hardware that promises ultrafast and energy-efficient solutions to NP-hard combinatorial optimization problems. Spatial photonic Ising machines (SPIMs) exploit optical computing in free space to accelerate the computation, showcasing parallelism, scalability, and low power consumption. However, current SPIMs can implement only a restricted class of problems. This partial programmability is a critical limitation that hampers their benchmark. Achieving full programmability of the device while preserving its scalability is an open challenge. Here, we report a fully programmable SPIM achieved through a novel operation method based on the division of the focal plane. In our scheme, a general Ising problem is decomposed into a set of Mattis Hamiltonians, whose energies are simultaneously computed optically by measuring the intensity on different regions of the camera sensor. Exploiting this concept, we experimentally demonstrate the computation with high success probability of ground-state solutions of up to 32-spin Ising models on unweighted maximum cut graphs with and without ferromagnetic bias. Simulations of the hardware prove a favorable scaling of the accuracy with the number of spins. Our fully programmable SPIM enables the implementation of many quadratic unconstrained binary optimization problems, further establishing SPIMs as a leading paradigm in non von Neumann hardware.

physics.optics↗

arXiv2024

Training of Physical Neural Networks

Physical neural networks (PNNs) are a class of neural-like networks that leverage the properties of physical systems to perform computation. While PNNs are so far a niche research area with small-scale laboratory demonstrations, they are arguably one of the most underappreciated important opportunities in modern AI. Could we train AI models 1000x larger than current ones? Could we do this and also have them perform inference locally and privately on edge devices, such as smartphones or sensors? Research over the past few years has shown that the answer to all these questions is likely "yes, with enough research": PNNs could one day radically change what is possible and practical for AI systems. To do this will however require rethinking both how AI models work, and how they are trained - primarily by considering the problems through the constraints of the underlying hardware physics. To train PNNs at large scale, many methods including backpropagation-based and backpropagation-free approaches are now being explored. These methods have various trade-offs, and so far no method has been shown to scale to the same scale and performance as the backpropagation algorithm widely used in deep learning today. However, this is rapidly changing, and a diverse ecosystem of training techniques provides clues for how PNNs may one day be utilized to create both more efficient realizations of current-scale AI models, and to enable unprecedented-scale models.

physics.app-ph↗

arXiv2024

Efficient Computation Using Spatial-Photonic Ising Machines: Utilizing Low-Rank and Circulant Matrix Constraints

We explore the potential of spatial-photonic Ising machines (SPIMs) to address computationally intensive Ising problems that employ low-rank and circulant coupling matrices. Our results indicate that the performance of SPIMs is critically affected by the rank and precision of the coupling matrices. By developing and assessing advanced decomposition techniques, we expand the range of problems SPIMs can solve, overcoming the limitations of traditional Mattis-type matrices. Our approach accommodates a diverse array of coupling matrices, including those with inherently low ranks, applicable to complex NP-complete problems. We explore the practical benefits of low-rank approximation in optimization tasks, particularly in financial optimization, to demonstrate the real-world applications of SPIMs. Finally, we evaluate the computational limitations imposed by SPIM hardware precision and suggest strategies to optimize the performance of these systems within these constraints.

physics.comp-ph↗