Physical Review Applied

Implementing artificial neural networks in electronic hardware has been pursued for decades, but suffers drawbacks such as device mismatch and circuit complexity, hence requiring sophisticated fabrication facilities. This study exploits the concept of memristors to implement a neuromorphic circuit of extreme simplicity. Both the neuron and synaptic circuits quantitatively realize conventional mathematical models of theoretical neuroscience and are implemented with off-the-shelf analog electronic components. This hardware model provides an affordable and easily available platform to implement spiking neural networks for basic research and practical applications.

Improving the performance of parallel computing usually comes at the cost of complexity and high energy consumption. This study combines multiple in-memory computing solvers via a parallel-tempering framework, and shows an increase in the speed and energy efficiency of solving binary optimization problems with negligible energy overhead. This approach is expected to have an impact on engineering solutions to Boolean satisfiability problems, Ising machines, and other binary optimization problems with applications in fields such as circuit design and supply-chain management, among others.

Magnetic tunnel junctions (MTJs) are promising building blocks for energy-efficient computing in neuromorphic and optimization applications, but implementing logic functions in coupled MTJs remains a significant challenge. This study offers an approach to building a universal reversible Toffoli gate using interacting macrospins—representing MTJ free layers—evolving under Landau-Lifshitz-Gilbert dynamics. With tuning and thermal annealing, the system reliably converges on correct logic outputs, revealing strategies for embedding logic into magnetic hardware. This approach could enable more complex spintronic computational architectures founded on MTJ-based Boltzmann machines.

Networks of spintronic nano-oscillators promise to process time series in a fast and energy-efficient way. However, realizations leveraging the transient dynamics of spintronic oscillators have been limited to training-free or single-layer networks. Through numerical simulations, the authors show how to train a multilayer dynamical spintronic network using standard machine-learning tools and derive design guidelines. These results are a key step toward using deep dynamical networks in applications such as smart sensors, personal assistants, and medical devices.

Resistive networks that train themselves using local learning rules such as equilibrium propagation show promise as energy-efficient alternatives to neural networks. Their computational capabilities remain unclear, though, as they solve circuit equations rather than standard neural-network equations. This study demonstrates mathematically that a deep resistive network built from (ideal) ohmic resistors, diodes, voltage sources, and voltage amplifiers can approximate to arbitrary accuracy any neural network based on the rectified-linear-unit activation function. This insight is expected to inform the design of self-learning resistor networks capable of universal function approximation.

In the ever-evolving landscape of computational science, tensor networks have emerged as a versatile toolset to simulate both quantum and classical many-body systems. This study investigates their applicability to complex optimization problems, where quantum annealing devices have generated significant interest. A challenge in applying tensor networks here is the high connectivity of the devices, which this work effectively leverages by utilizing sparse structures in construction, plus hardware acceleration. The authors quantify the limitations of their deterministic approach, and find that in certain scenarios it might outperform quantum annealers or randomized classical solvers.

Adaptive devices that can display multiple resistance states could enable compact, energy-efficient hardware for neuromorphic computing, by sharing intelligence with software via co-design. This study combines machine learning with a VO${}_2$ device and its volatile phase-relaxation dynamics to realize decision trees within a single physical device. These results show a fresh way to utilize distinct phases in electronically complex crystals for emerging AI hardware.

Conventional digital systems struggle with the complexity of NP-complete problems due to sequential processing and energy inefficiency, and this motivates alternative computing paradigms inspired by physical dynamics. This study presents a CMOS network of injection-locked ring oscillators that solve Boolean satisfiability problems by mapping them onto Ising models, to realize a fully invertible one-bit full adder in hardware. The approach leverages the analogy between Ising spin interactions and the synchronization of oscillator phases to binary states via injection locking.

Ising machines offer hardware acceleration for combinatorial optimization, artificial intelligence, and quantum simulation, but their reliance on dense graphs restricts large-scale deployment. To address this limitation, the authors introduce a sparsification algorithm that distributes each node’s connections across multiple copies, enabling constant-frequency operation in ASIC designs and FPGA prototypes. Evaluation of runtime overhead during optimization tasks reveals a trade-off between hardware feasibility and execution time; notably, this overhead vanishes for inherently sparse problems such as integer factorization.

Oscillator Ising machines (OIMs) are nonlinear dynamical systems that offer a physics-inspired approach to minimizing the Ising Hamiltonian. Performance, however, is often sensitive to the alignment of the system’s intrinsic dynamics with the structure of the input graph. This work presents the dynamical Ising machine (DIM), a complementary system operating in a fundamentally different phase-space geometry and exploring the solution landscape differently from an OIM. The hybrid approach of running OIMs and DIMs in parallel leverages diversity in dynamical behavior, enhancing both the effectiveness and robustness of the analog approach to solving combinatorial optimization.

One promising approach to “beyond von Neumann” computing architectures is to use networks of inherently stochastic units called probabilistic bits (p-bits). Here the authors exploit enthalpic competition among charge, orbital, lattice, and spin degrees of freedom plus natural phase inhomogeneity in a strongly correlated oxide to design a cascade of energy states, between which the system can switch probabilistically. Substrate-induced tunability of the energy landscape yields two distinct modes of voltage-controlled stochastic operation: clocked binary switching between two current states in one sample, and unclocked multibit switching between multiple current regimes in another.

The authors demonstrate an approach for solving an NP-hard problem, based on coupled-oscillator networks implemented with charge-density-wave-condensate devices. Prototype hardware based on $1T$-TaS${}_2$ enables room-temperature operation of the network. The oscillator operation relies on hysteresis in I-V characteristics and bistability triggered by applied electrical bias. The nature of the transitions between the charge-density-wave phases creates the potential for low-power operation and compatibility with conventional silicon technology.

Neuromorphic computing promises transformative advances for high-speed, energy-efficient artificial intelligence. This study reports how circuits based on resonant tunneling diode (RTD) neurons can encode information in both the spike rate and spike latency, directly emulating strategies found in biological neurons. Through experiments and numerical studies, the authors show how RTD design parameters influence the spiking dynamics, identifying practical routes toward subnanosecond spike-rate encoding in future neuromorphic hardware.

Domain-wall (DW) devices have garnered interest for diverse applications including memory, logic, and neuromorphic primitives; thus fast, accurate device models are imperative. Existing models of DW motion are suboptimal for large systems: They either devour computational resources, or oversimplify the physics. The authors propose a DW model inspired by the phenomenological similarities between the motion of a DW and that of a classical object subject to forces like friction or drag. This model predicts essentially the same DW motion as do micromagnetic simulations, but 4000× as fast. It is also faster than collective-coordinate models, and much more accurate than hyper-reduced models.

Wave dynamics offer intrinsic parallelism that remains largely untapped in current wave-based computing systems. This work proposes a framework for a dynamical wave-propagating network to tackle combinatorial optimization. By empowering both nodes and edges to actively process signals through frequency mixing and programmable time delays, this technique exploits parallelism across frequency, space, and time. The approach is validated on canonical benchmarks—including number partitioning, the 0/1 knapsack problem, and the traveling-salesman problem—while rigorously addressing practical constraints such as pseudopolynomial complexity and energy density.

Analog in-memory computing based on crossbar arrays offers a path to energy-efficient AI hardware, but has been limited by reliance on bulky, power-hungry analog-to-digital converters. This study introduces stochastic nanomagnets driven by spin-orbit torque as intrinsic analog-to-digital interfaces, enabling compact, fast, low-power digitization while maintaining high computational accuracy. These results suggest a promising direction for energy-efficient AI hardware accelerators and significant advances in next-generation machine-learning hardware.

A single magnetic tunnel junction (MTJ) can operate as an entropy source for probabilistic bits (p-bits), thanks to tunable stochasticity, CMOS compatibility, and room-temperature operation. The average probability measured from the random signals generated by MTJs can be tuned through e.g. spin torques or voltage-controlled exchange coupling, and furthermore such mechanisms can be combined. This article reviews experimental and theoretical work on all of the biasing mechanisms that have been proposed for MTJ-based p-bits, and provides an overview of the advantages and disadvantages of each biasing mechanism.

Oscillator-based Ising machines are currently explored for their potential to solve complex optimization problems. A key challenge lies in developing theoretically sound models that capture both phase and amplitude dynamics of a given system. This study uses Wirtinger calculus to develop a rigorous mathematical framework with a complex-valued oscillator representation, introducing a real-valued energy function and corresponding dynamics that faithfully represent the system’s behavior. This provides a stronger theoretical foundation and practical design principles for engineering next-generation oscillator-based Ising machines.

Synaptic spintronic devices based on domain-wall motion driven by spin-orbit torque are appealing for neuromorphic computing applications. Unlike in other technologies, though, here the long-term stability of multiple consecutive synaptic states is rare. To address this issue, the authors exploit the high thermal stability of magnetization in a heavy-metal/ferromagnetic-metal stack with graded thickness of the heavy metal. In micrometer-scale devices made from the stack, they demonstrate the stability of approximately 30 synaptic states for up to 1200 seconds, with some states stable for up to 50,000 seconds. These results pave the way for spintronic neuromorphic computing technologies.

Probabilistic Ising machines (PIMs) show promise in solving optimization problems. However, the binary nature of probabilistic bits (p-bits) does not permit the natural mapping of more than two state variables, which are common in real-world applications. To sidestep the potential increase in time to solution for these problems, the authors investigate the concept of a d-dimensional probabilistic bit (p-dit). Three different implementations of p-dit-based computers show large improvements over traditional PIMs, showcasing their adaptability.

Ising machines realized on field-programmable gate arrays are used to solve combinatorial optimization problems. Performance is limited by the speed of the digital clocked arithmetic that is part of the algorithm, so in this study the authors simplify the Ising machine by replacing all arithmetic with a set of look-up tables, while retaining the polynomial scaling of chip resources with problem size. The result is reduction of both resource usage and time to solution by more than an order of magnitude, compared to previous approaches. Future implementations of Ising machines may be inspired by this simplified design.

Modern neural networks require enormous server infrastructure, as parameter count and memory demands grow with data richness. In contrast, biological brains learn and process information using limited memory and power through extensive reuse of neural representations, encoding shared features across related concepts (such as horses and zebras) while adding only a small number of synaptic connections to capture variations. This study implements a mathematical model of the neocortex that economizes on neuronal usage for image classification, and proposes a magnetic hardware platform to realize key aspects of its functionality.

Stochastic magnetic tunnel junctions (MTJs) are promising building blocks for neuromorphic and probabilistic computing, but conventional approaches rely on thermally unstable superparamagnetic devices with limited reliability and tunability. In this work, thermally stable perpendicular MTJs are electrically driven to produce random telegraph noise using nanosecond spin-torque pulses, the response being well described by a simple Poisson process. This approach enables broad, continuous tuning of both fluctuation rate and probability bias in a single device, pointing to a practical route for combining memory elements with programmable stochastic functionality on a single hardware platform.

Physics-based memcomputing is of interest for solving hard combinatorial-optimization problems in computer science, engineering, and physics by embedding a problem directly into the dynamics of a nonlinear system with memory. Few studies, however, have addressed how the phase-space structure controls performance and scalability. This work uses systematic phase-space engineering and simulations of memcomputing machines to identify the dynamical mechanisms that enable efficient solution-finding. Its insights into how memory-driven collective behavior influences phase-space geometry could help in designing more reliable and scalable physics-inspired devices for hard computational problems.

The traveling-salesman problem has important applications in logistics and path optimization. For autonomous devices like drones, edge computing provides key advantages compared to cloud computing, but has strict energy-consumption and memory limitations. Ising machines would be suitable here, but they lack compatibility with generalized variants of route-optimization problems and struggle with constraint-based problems. The authors present strategies to overcome these limitations, impacting future engineering solutions that employ Ising machines for edge-computing applications.

Stochastic Ising machines (sIMs) are promising accelerators for optimization and sampling in computational problems that can be formulated as an Ising model. The authors investigate the computational advantage of sIMs for simulating quantum magnets with neural-network quantum states, a very powerful method for probabilistic quantum simulation. Based on the autocorrelation time, the team predicts sampling advantage without requiring deployment on hardware. For massively parallel hardware sIMs a speed-up factor of 100 to 10000 is projected, suggesting that sIMs may drastically increase the scale at which probabilistic quantum simulation is possible.

$XY$ (planar-spin) Hamiltonians arise in phase synchronization and retrieval and analog formulations of hard optimization, which motivates fast, low-power physical solvers. However, gain-based $XY$ systems that encode each spin with a single complex field can become trapped in metastable states. The authors introduce an annealer that represents each spin with two coupled complex components, and uses a graph-independent locking term that exploits the extra degree of freedom to bypass barriers. For challenging graph families, this higher-dimensional annealing improves ground-state recovery compared to one-component approaches, supporting more reliable photonic and analog $XY$ optimization.