The slowing down of Moore's law has driven the development of unconventionalcomputing paradigms, such as specialized Ising machines tailored to solvecombinatorial optimization problems. In this paper, we show a new applicationdomain for probabilistic bit (p-bit) based Ising machines by training deepgenerative AI models with them. Using sparse, asynchronous, and massivelyparallel Ising machines we train deep Boltzmann networks in a hybridprobabilistic-classical computing setup. We use the full MNIST and FashionMNIST (FMNIST) dataset without any downsampling and a reduced version ofCIFAR-10 dataset in hardware-aware network topologies implemented in moderatelysized Field Programmable Gate Arrays (FPGA). For MNIST, our machine using only4,264 nodes (p-bits) and about 30,000 parameters achieves the sameclassification accuracy (90%) as an optimized software-based restrictedBoltzmann Machine (RBM) with approximately 3.25 million parameters. Similarresults follow for FMNIST and CIFAR-10. Additionally, the sparse deep Boltzmannnetwork can generate new handwritten digits and fashion products, a task the3.25 million parameter RBM fails at despite achieving the same accuracy. Ourhybrid computer takes a measured 50 to 64 billion probabilistic flips persecond, which is at least an order of magnitude faster than superficiallysimilar Graphics and Tensor Processing Unit (GPU/TPU) based implementations.The massively parallel architecture can comfortably perform the contrastivedivergence algorithm (CD-n) with up to n = 10 million sweeps per update, beyondthe capabilities of existing software implementations. These resultsdemonstrate the potential of using Ising machines for traditionallyhard-to-train deep generative Boltzmann networks, with further possibleimprovement in nanodevice-based realizations.
Ising machines, hardware accelerators for combinatorial optimization andprobabilistic sampling problems, have gained significant interest recently. Akey element is stochasticity, which enables a wide exploration ofconfigurations, thereby helping avoid local minima. Here, we refine thepreviously proposed concept of coupled chaotic bits (c-bits) that operatewithout explicit stochasticity. We show that augmenting chaotic bits withstochasticity enhances performance in combinatorial optimization, achievingalgorithmic scaling comparable to probabilistic bits (p-bits). We firstdemonstrate that c-bits follow the quantum Boltzmann law in a 1D transversefield Ising model. We then show that c-bits exhibit critical dynamics similarto stochastic p-bits in 2D Ising and 3D spin glass models, with promisingpotential to solve challenging optimization problems. Finally, we propose anoise-augmented version of coupled c-bits via the adaptive parallel temperingalgorithm (APT). Our noise-augmented c-bit algorithm outperforms fullydeterministic c-bits running versions of the simulated annealing algorithm.Other analog Ising machines with coupled oscillators could draw inspirationfrom the proposed algorithm. Running replicas at constant temperatureeliminates the need for global modulation of coupling strengths. Mixingstochasticity with deterministic c-bits creates a powerful hybrid computingscheme that can bring benefits in scaled, asynchronous, and massively parallelhardware implementations.
An emerging paradigm in modern electronics is that of CMOS + $\sf X$requiring the integration of standard CMOS technology with novel materials andtechnologies denoted by $\sf X$. In this context, a crucial challenge is todevelop accurate circuit models for $\sf X$ that are compatible with standardmodels for CMOS-based circuits and systems. In this perspective, we presentphysics-based, experimentally benchmarked modular circuit models that can beused to evaluate a class of CMOS + $\sf X$ systems, where $\sf X$ denotesmagnetic and spintronic materials and phenomena. This class of materials isparticularly challenging because they go beyond conventional charge-basedphenomena and involve the spin degree of freedom which involves non-trivialquantum effects. Starting from density matrices $-$ the central quantity inquantum transport $-$ using well-defined approximations, it is possible toobtain spin-circuits that generalize ordinary circuit theory to 4-componentcurrents and voltages (1 for charge and 3 for spin). With step-by-step examplesthat progressively become more complex, we illustrate how the spin-circuitapproach can be used to start from the physics of magnetism and spintronics toenable accurate system-level evaluations. We believe the core approach can beextended to include other quantum degrees of freedom like valley andpseudospins starting from corresponding density matrices.
Abstract In this paper, we propose two random matrix based extended target tracking models, which apply to the trajectory‐aligned and drifting target motions. The trajectory‐aligned model is specifically designed to handle targets moving along the direction of their extent orientations, while the drift model is tailored to targets whose trajectories deviate from their orientations in time. We utilise the well‐known variational Bayes method to perform inference and obtain posterior densities via computationally efficient, analytical, iterative steps. Through comprehensive experiments conducted on simulated and real data, our methods have demonstrated superior performance compared to previous approaches in scenarios involving both drifting and trajectory‐aligned targets. These results highlight the efficacy of our proposed models in accurately tracking targets and estimating their extent.
We aim to explore the properties of the Be/X-ray binary system MXB 0656-072from a timing analysis perspective through an investigation of the RXTE/PCA andFermi/GBM data during its 2007-2008 type I outbursts. We applied two newtechniques, for the first time, along with the conventional Deeter method toproduce higher-resolution power density spectra (PDS) of the torquefluctuations. We also investigated the spin frequency evolution of the sourceby utilising a pulse timing technique. The PDSs show a red noise pattern, witha steepness of $\Gamma \sim -2$ and a saturation timescale of $\sim$150 d,indicating that MXB 0656-072 is a disc-fed source. With the obtained long termspin frequency evolution, we reveal the torque-luminosity correlation of MXB0656-072 for the first time. We also demonstrate that the frequency evolutionis largely consistent with the Ghosh-Lamb model. In the RXTE/PCA observations,the pulsed emission disappears below $\sim$5$\times 10^{35}$ erg s$^{-1}$,while the profiles remain stable above this value in our analysis time frame.We show that the magnetic field strength deduced from the torque model iscompatible with the field strength of the pulsar derived from the cyclotronresonance scattering feature. Utilising the new distance of MXB 0656-072measured by Gaia, we show that the spectral transition of MXB 0656-072 occursat a luminosity that matches the expected theoretical transition from thesubcritical to supercritical accretion regime.
Stochastic magnetic tunnel junctions (s-MTJ) is a promising component ofprobabilistic bit (p-bit), which plays a pivotal role in probabilisticcomputers. For a standard cell structure of the p-bit, s-MTJ is desired to beinsensitive to voltage across the junction over several hundred millivolts. Inconventional s-MTJs with a reference layer having a fixed magnetizationdirection, however, the stochastic output significantly varies with the voltagedue to spin-transfer torque (STT) acting on the stochastic free layer. In thiswork, we study a s-MTJ with a "double-free-layer" design theoretically proposedearlier, in which the fixed reference layer of the conventional structure isreplaced by another stochastic free layer, effectively mitigating the influenceof STT on the stochastic output. We show that the key device propertycharacterized by the ratio of relaxation times between the high- andlow-resistance states is one to two orders of magnitude less sensitive to biasvoltage variations compared to conventional s-MTJs when the top and bottom freelayers are designed to possess the same effective thickness. This work opens apathway for reliable, nanosecond-operation, high-output, and scalablespintronics-based p-bits.