Despite their appeal as physics-inspired, energy-based and generative nature, general Boltzmann Machines (BM) are considered intractable to train. This belief led to simplified models of BMs with restricted intralayer connections or layer-by-layer training of deep BMs. Recent developments in domain-specific hardware -- specifically probabilistic computers (p-computer) with probabilistic bits (p-bit) -- may change established wisdom on the tractability of deep BMs. In this paper, we show that deep and unrestricted BMs can be trained using p-computers generating hundreds of billions of Markov Chain Monte Carlo (MCMC) samples per second, on sparse networks developed originally for use in D-Wave's annealers. To maximize the efficiency of learning the p-computer, we introduce two families of Mean-Field Theory assisted learning algorithms, or xMFTs (x = Naive and Hierarchical). The xMFTs are used to estimate the averages and correlations during the positive phase of the contrastive divergence (CD) algorithm and our custom-designed p-computer is used to estimate the averages and correlations in the negative phase. A custom Field-Programmable-Gate Array (FPGA) emulation of the p-computer architecture takes up to 45 billion flips per second, allowing the implementation of CD-$n$ where $n$ can be of the order of millions, unlike RBMs where $n$ is typically 1 or 2. Experiments on the full MNIST dataset with the combined algorithm show that the positive phase can be efficiently computed by xMFTs without much degradation when the negative phase is computed by the p-computer. Our algorithm can be used in other scalable Ising machines and its variants can be used to train BMs, previously thought to be intractable.

The slowing down of Moore's Law has led to a crisis as the computing workloads of Artificial Intelligence (AI) algorithms continue skyrocketing. There is an urgent need for scalable and energy-efficient hardware catering to the unique requirements of AI algorithms and applications. In this environment, probabilistic computing with p-bits emerged as a scalable, domain-specific, and energy-efficient computing paradigm, particularly useful for probabilistic applications and algorithms. In particular, spintronic devices such as stochastic magnetic tunnel junctions (sMTJ) show great promise in designing integrated p-computers. Here, we examine how a scalable probabilistic computer with such magnetic p-bits can be useful for an emerging field combining machine learning and quantum physics.

Quantum issues may now be solvable without the use of a quantum computer. New research by engineers at Purdue University and Tohoku University in Japan, has unveiled the researchers' new hardware -- a probabilistic computer -- which can solve quantum problems without using any quantum computers. The probabilistic computer with its new hardware has fundamental units -- known as p-bits -- which can perform a calculation that usually only a quantum computer could solve. The study was published this week in Nature. The new device built by the engineers will be the basis for building probabilistic computers.

We study the empirical meaning of randomness with respect to a family of probability distributions $P_\theta$, where $\theta$ is a real parameter, using algorithmic randomness theory. In the case when for a computable probability distribution $P_\theta$ an effectively strongly consistent estimate exists, we show that the Levin's a priory semicomputable semimeasure of the set of all $P_\theta$-random sequences is positive if and only if the parameter $\theta$ is a computable real number. The different methods for generating ``meaningful'' $P_\theta$-random sequences with noncomputable $\theta$ are discussed.