arXiv subjects

Nikhil Shukla

Publications and source records attributed to Nikhil Shukla.

At least 19 recordsLinked to original sources

arXiv2026

Astroid Spinodal Boundary in Phase-Based Ising Machines

Oscillator Ising machines (OIMs) and dynamical Ising machines (DIMs) encode binary spins in phase states stabilized by second-harmonic injection (SHI). In a coupled network, the competition between SHI and the instantaneous local network field reshapes each oscillator's conditional energy landscape. We show that this competition drives a transition between monostable and bistable regimes through an astroid spinodal boundary. Near this boundary, the barrier scales as $\Delta E_i\propto\mu_i^{3/2}$ at generic smooth points and as $\Delta E_i\propto\mu_i^{2}$ at the longitudinal cusp. OIMs and DIMs obey the same spinodal geometry, with their conditional landscapes related by a reversal of the transverse field. Finally, the first-harmonic conditional landscape is mathematically equivalent, up to an additive constant, to the Stoner--Wohlfarth energy of a uniaxial magnetic particle.

physics.comp-ph↗

arXiv2026

Learning from Noise: Effective-Rank Collapse and Out-of-Distribution Rejection in Restricted Boltzmann Machines

Restricted Boltzmann machines (RBMs) represent data by shaping an energy landscape over visible and hidden configurations, but their discriminative use is fragile under out-of-distribution (OOD) inputs: samples outside the training distribution can be absorbed into one of the learned class basins rather than rejected. Here, we analyze this failure mode through the spectrum of the induced visible--visible interaction $J=WW^{T}$, where $W$ is the visible--hidden weight matrix. Relative to a Marchenko--Pastur random-matrix reference, conventional training spreads spectral weight into many weak, bulk-compatible directions, increasing the effective rank of $J$. When auxiliary random binary images are assigned to a rejection label during training, the learned interaction undergoes effective-rank collapse: weak bulk-like modes are depleted, spectral weight concentrates into fewer dominant eigendirections, and the effective rank of $J$ approaches that of the empirical data covariance matrix. The resulting RBM rejects structured OOD image datasets while preserving MNIST classification accuracy, showing that random auxiliary exposure can reshape both the interaction spectrum and the free-energy landscape of an energy-based classifier.

cs.LG↗

arXiv2026

Leveraging Population Dynamics to Steer Efficient Search in Large-Scale Combinatorial Optimization

Combinatorial optimization problems pose substantial computational challenges because their feasible solution spaces grow exponentially with problem size. This paper presents a GPU-accelerated augmented Population Annealing Monte Carlo (PAMC) framework for large-scale graph-partitioning problems, with emphasis on Max-Cut and Max-K-Cut. The proposed framework extends conventional PAMC by coupling population-based resampling with two stagnation-driven mechanisms: adaptive temperature control and energy-preserving nonlocal cluster moves. By using population-level optimization history as feedback, these mechanisms regulate the balance between exploration and refinement by reheating stalled populations and enabling collective transitions across locally confined regions of the solution space. Experiments on G-set benchmark instances show that the augmented PAMC framework achieves competitive or lower time-to-solution than reported state-of-the-art baselines on several large Max-Cut instances, while matching or improving solution quality under comparable runtime budgets. The solver also discovers a new best-known solution for the G63 Max-Cut instance and scales to a fully connected 100,000-spin Ising instance. For Max-3-Cut, the same framework establishes new best-known solutions on 36 G-set instances, demonstrating its applicability beyond binary Ising formulations. These results indicate that feedback-controlled population dynamics provide an effective and scalable strategy for steering stochastic search in large-scale combinatorial optimization.

physics.comp-ph↗

arXiv2026

Breakdown of Gradient-Flow Dynamics in Oscillator Ising Machines from Harmonic Misalignment

Oscillator Ising machines (OIMs) are often viewed as physical systems that perform gradient descent on an energy landscape encoding Ising solutions. Here, we show that this interpretation is not generic and breaks down in a broad class of oscillator implementations. We establish that gradient-flow dynamics require a harmonic-by-harmonic quadrature relation between the oscillator waveform and its phase response. Deviations from this condition, which we term harmonic misalignment, introduce even components in the pairwise interaction function, leading to non-conservative phase dynamics and precluding a gradient-flow description. We introduce a normalized metric for this non-gradient contribution and evaluate it across representative oscillator models relevant to OIMs. This metric reveals substantial non-gradient contributions in ring oscillators and across other hardware-realistic oscillator models. These findings identify harmonic misalignment as a fundamental mechanism for the breakdown of energy-based dynamics in OIMs and motivate nonequilibrium analysis and algorithms that explicitly account for and potentially exploit non-gradient behavior.

physics.comp-ph↗

arXiv2026

Cosm: Collective Switched Motion for Fast and Accurate Sparse Ising Optimization

We introduce Collective Switched Motion (Cosm), a dynamical system-based heuristic algorithm. Cosm combines locally interacting continuous circular variables with novel global coordination rules that facilitate collective dynamics. Pairwise interactions occur sequentially over a set of conflict-free edge partitions, resulting in an interaction network that switches periodically. Unlike conventional gradient-based approaches, Cosm employs structured, non-smooth switching dynamics with finite-magnitude interactions that sustain collective fluctuations and promote exploration beyond local minima. A correlated perturbation mechanism further promotes coordinated cluster motion in the circular phase space. On the three largest Ising problems from the Gset suite, which have 10,000-20,000 variables and represent 2D spin glasses, Cosm attains the optimal solutions (verified with an exact solver) heuristically for the first time. On two large bounded-degree non-lattice graph instances, Cosm reduces the state-of-the-art times-to-target from hundreds of hours to 36-303 s. Results on benchmark problems with tuned hardness suggest favorable scaling relative to previously characterized dynamical solvers. These results suggest that Cosm's synthesis of local interactions, structured switching dynamics, and global coordination provides an effective computational framework for sparse optimization.

cs.CE↗

arXiv2026

How Physical Dynamics Shape the Properties of Ising Machines: Evaluating Oscillators vs. Bistable Latches as Ising Spins

Ising machines exploit the natural dynamics of physical systems to minimize the Ising Hamiltonian and thereby address computationally hard combinatorial optimization problems. This paradigm has motivated a range of physical implementations. In the electronic domain, coupled networks of oscillators and bistable latches have emerged as two prominent realizations of Ising machines and are the focus of the present work. Despite this common abstraction, we demonstrate that differences in the underlying physical dynamics of oscillators and latches lead to fundamentally different stability properties of the resulting dynamical systems. Specifically, we show analytically that in Bistable Latch Ising Machines (BLIMs) all discrete Ising configurations possess identical linear stability, whereas in Oscillator Ising Machines (OIMs) the Jacobian spectrum depends explicitly on the spin configuration, enabling selective destabilization of higher-energy states. We further corroborate this analysis using finite-noise perturbation experiments initialized near prescribed Ising configurations. These results highlight how the characteristics of the device nonlinearity directly shape the local dynamical properties of Ising machine implementations.

physics.comp-ph↗

arXiv2026

Mind the Gap: Where Analog Ising Machines Cease to Minimize the Ising Hamiltonian

The design of nonlinear dynamical systems whose gradient flows minimize the Ising Hamiltonian has emerged as a compelling paradigm for realizing Ising machines, forming the foundation of architectures including coherent Ising machines, simulated bifurcation machines, oscillator-based Ising machines, and dynamical Ising machines. Here, we identify a fundamental structural feature shared by these systems a functional parameter gap defined by the separation between the destabilization of the trivial state and the stabilization of Ising-encoded states. We demonstrate that this separation creates a finite parameter interval in which convergence to an Ising-encoded solution is no longer functionally guaranteed, and the resulting evolution is dictated by the spectral structure of the Jacobian at bifurcation. Subsequently, by introducing a hybrid dynamical framework that reshapes the bifurcation topology, we establish a principled pathway for modulating this parameter gap. The parameter gap thus emerges as a unifying structural principle for the analysis, design and optimization of analog Ising machines.

physics.comp-ph↗

arXiv2026

At the Top of the Mountain, the World can Look Boltzmann-Like: Sampling Dynamics of Noisy Double-Well Systems

The success of the transistor as the cornerstone of digital computation motivates analogous efforts to identify an equivalent hardware primitive, the probabilistic bit or p-bit, for the emerging paradigm of probabilistic computing. Here, we uncover a fundamental ubiquity in the stochastic dynamics of double well energy systems when initialized near the barrier top. Using a topological framework grounded in Morse theory and singularity theory, we make use of the result that all smooth, even double well potentials reduce near the saddle point to a canonical quartic normal form. Within this regime, the interplay of noise, synaptic bias, and potential curvature produces a topologically robust short time evolution characterized by a tanh like response. This enables Boltzmann like sampling that is largely independent of the detailed shape of the potential, apart from its effective temperature scaling. Analytical derivations and numerical simulations across multiple representative systems corroborate this behavior. Our work provides a unifying foundation for assessing and engineering a broad class of physical platforms, including oscillators, bistable latches, and magnetic devices, as p-bits operating within a synchronous framework for stochastic sampling and probabilistic computation.

physics.comp-ph↗

arXiv2026

Bridging the Analog and the Probabilistic Computing Divide: Configuring Oscillator Ising Machines as P-bit Engines

Oscillator Ising Machines (OIMs) and probabilistic bit (p-bit)-based computing platforms have emerged as promising paradigms for tackling complex combinatorial optimization problems. Although traditionally viewed as distinct approaches, this work presents a theoretically grounded framework for configuring OIMs as p-bit engines. We demonstrate that this functionality can be enabled through a novel interplay between first- and second harmonic injection to the oscillators. Our work identifies new synergies between the two methods and broadens the scope of applications for OIMs. We further show that the proposed approach can be applied to other analog dynamical systems, such as the Dynamical Ising Machine.

physics.comp-ph↗

arXiv2025

Analyzing Parametric Oscillator Ising Machines through the Kuramoto Lens

Networks of coupled nonlinear oscillators are emerging as powerful physical platforms for implementing Ising machines. Yet the relationship between parametric-oscillator implementations and traditional oscillator-based Ising machines remains underexplored. In this work, we develop a Kuramoto-style, canonical phase description of parametric oscillator Ising machines by starting from the Stuart-Landau oscillator model -- the canonical normal form near a Hopf bifurcation, and a natural reduced description for many parametric oscillator implementations such as the degenerate optical parametric oscillator (DOPO) among others. The resulting phase dynamics combine the usual phase-difference coupling observed in the standard Kuramoto model along with an intrinsic phase sum term that is generated when conjugate coupling is considered. Moreover, our formulation helps explain why explicit second-harmonic driving is unnecessary in parametric oscillators and also reveals how quasi-steady amplitude heterogeneity scales the original strength of the spin interaction with potentially adverse impacts on the solution quality. Our work helps develop a unifying view of the oscillator-based approach to designing Ising machines.

eess.SY↗

arXiv2025

New Best-Known Max-Cut Solution for the G63 Instance in the G-Set Benchmark

For over two decades, the G-set benchmark has remained a cornerstone challenge for combinatorial optimization solvers. Remarkably, it continues to yield new best-known solutions even to the present day. Here, we report a new best-known Max-Cut of 27,047 for the 7000-node G63 instance-one of the two instances in the benchmark with the largest number of edges. This result is achieved using an optimized Population Annealing Monte Carlo framework, augmented with adaptive control of stochasticity and the periodic introduction of non-local moves, and accelerated on a GPU platform.

math-ph↗

arXiv2025

Spin Freezing in Oscillator Ising Machines: When Second Harmonic Injection Impedes Computation

Second harmonic injection (SHI) has emerged as a critical mechanism in enabling networks of coupled oscillators to function as Oscillator Ising Machines (OIMs), capable of minimizing the Ising Hamiltonian. While SHI facilitates phase binarization essential for mapping oscillator phases to spin states, we demonstrate that it can also induce a previously unreported phenomenon -- spin freezing -- where oscillator spins are unable to transition between spin states, even when such a transition can reduce the Ising energy. This freezing effect can impair the analog dynamics of the OIM, preventing it from reaching lower-energy spin configurations. Through theoretical analysis and numerical simulations, we show that the onset of spin freezing is highly sensitive to the initial phase configuration of the oscillators. Contrary to conventional practice, which favors random initialization, we find that initializing all oscillators at specific phase values ($ϕ= π$ or $ϕ= \fracπ{2}$) delays the onset of spin freezing and consistently yields higher-quality solutions. These findings point to the need to carefully engineer the SHI for optimal performance.

physics.comp-ph↗

arXiv2024

Designing a K-state P-bit Engine

Probabilistic bit (p-bit)-based compute engines utilize the unique capability of a p-bit to probabilistically switch between two states to solve computationally challenging problems. However, when solving problems that require more than two states (e.g., problems such as Max-3-Cut, verifying if a graph is K-partite (K>2) etc.), additional pre-processing steps such as graph reduction are required to make the problem compatible with a two-state p-bit platform. Moreover, this not only increases the problem size by entailing the use of auxiliary variables but can also degrade the solution quality. In this work, we develop a unique framework for implementing a K-state (K>2) p-bit engine. Furthermore, from an implementation standpoint, we show that such a K-state p-bit engine can be implemented using N traditional (2-state) p-bits, and one multi-state p-bit -- a novel concept proposed here. Augmenting traditional p-bit platforms, our approach enables us to solve an archetypal combinatoric problem class requiring multiple states, namely Max-K-Cut (K=3, 4 shown here), without using any additional auxiliary variables. Thus, our work fundamentally advances the functional capability of p-bit engines, enabling them to solve a broader class of computationally challenging problems more efficiently.

cs.ET↗

arXiv2023

CMOS-based Single-Cycle In-Memory XOR/XNOR

Big data applications are on the rise, and so is the number of data centers. The ever-increasing massive data pool needs to be periodically backed up in a secure environment. Moreover, a massive amount of securely backed-up data is required for training binary convolutional neural networks for image classification. XOR and XNOR operations are essential for large-scale data copy verification, encryption, and classification algorithms. The disproportionate speed of existing compute and memory units makes the von Neumann architecture inefficient to perform these Boolean operations. Compute-in-memory (CiM) has proved to be an optimum approach for such bulk computations. The existing CiM-based XOR/XNOR techniques either require multiple cycles for computing or add to the complexity of the fabrication process. Here, we propose a CMOS-based hardware topology for single-cycle in-memory XOR/XNOR operations. Our design provides at least 2 times improvement in the latency compared with other existing CMOS-compatible solutions. We verify the proposed system through circuit/system-level simulations and evaluate its robustness using a 5000-point Monte Carlo variation analysis. This all-CMOS design paves the way for practical implementation of CiM XOR/XNOR at scaled technology nodes.

cs.AR↗

arXiv2023

A Note on Analyzing the Stability of Oscillator Ising Machines

The rich non-linear dynamics of the coupled oscillators (under second harmonic injection) can be leveraged to solve computationally hard problems in combinatorial optimization such as finding the ground state of the Ising Hamiltonian. While prior work on the stability of the so-called Oscillator Ising Machines (OIMs) has used the linearization method, in this letter, we present a complementary method to analyze stability using the second order derivative test of the energy / cost function. We establish the equivalence between the two methods, thus augmenting the tool kit for the design and implementation of OIMs.

math.OC↗

arXiv2023

Stability of Oscillator Ising Machines: Not All Solutions Are Created Equal

Nonlinear dynamical systems such as coupled oscillators are being actively investigated as Ising machines for solving computationally hard problems in combinatorial optimization. Prior works have established the equivalence between the global minima of the Lyapunov function (commonly referred to as the energy function) describing the coupled oscillator system and the ground state of the Ising Hamiltonian. However, the properties of the oscillator Ising machine (OIM) from a nonlinear control viewpoint, such as the stability of the OIM solutions remains unexplored. Therefore, in this work, using nonlinear control-theoretic analysis, we (i) Identify the conditions required to ensure the functionality of the coupled oscillators as an Ising machine; (ii) Show that all globally optimal phase configurations may not always be stable, resulting in some configurations being more favored over others, and thus, creating a biased OIM; (c) Elucidate the impact of the stability of locally optimal phase configurations on the quality of the solution computed by the system. Our work, fostered through the unique convergence between nonlinear control theory and analog systems for computing, provides a new toolbox for the design and implementation of dynamical system-based computing platforms.

math.DS↗

arXiv2022

Constructing Dynamical Systems to Model Higher Order Ising Spin Interactions and their Application in Solving Combinatorial Optimization Problems

The Ising model provides a natural mapping for many computationally hard combinatorial optimization problems (COPs). Consequently, dynamical system-inspired computing models and hardware platforms that minimize the Ising Hamiltonian, have recently been proposed as a potential candidate for solving COPs, with the promise of significant performance benefit. However, the Ising model, and consequently, the corresponding dynamical system-based computational models primarily consider quadratic interactions among the nodes. Computational models considering higher order interactions among Ising spins remain largely unexplored. Therefore, in this work, we propose dynamical-system-based computational models to consider higher order (>2) interactions among the Ising spins, which subsequently, enables us to propose computational models to directly solve many COPs that entail such higher order interactions (COPs on hypergraphs). Specifically, we demonstrate our approach by developing dynamical systems to compute the solution for the Boolean NAE-K-SAT (K is greater than 3) problem as well as solve the Max-K-Cut of a hypergraph. Our work advances the potential of the physics-inspired 'toolbox' for solving COPs.

math.DS↗

arXiv2022

Formulating Oscillator-Inspired Dynamical Systems to Solve Boolean Satisfiability

Dynamical systems can offer a novel non-Boolean approach to computing. Specifically, the natural minimization of energy in the system is a valuable property for minimizing the objective functions of combinatorial optimization problems, many of which are still challenging to solve using conventional digital solvers. In this work, we formulate two oscillator-inspired dynamical systems to solve quintessential computationally intractable problems in Boolean satisfiability (SAT). The system dynamics are engineered such that they facilitate solutions to two different flavors of the SAT problem. We formulate the first dynamical system to compute the solution to the 3-SAT problem, while for the second system, we show that its dynamics map to the solution of the Max-NAE-3-SAT problem. Our work advances understanding of how this physics-inspired approach can be used to address challenging problems in computing.

math.DS↗