There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Physical Review B

The computational brain: Patricia S. Churchland and Terrence J. Sejnowski (MIT Press, Cambridge, MA, 1992); xi, 544 pages, $39.95

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Electronic Transport In Mesoscopic Systems

Journal of Physics Condensed Matter: Preface

Conventional computers operate deterministically using strings of zeros and ones called bits to represent information in binary code. Despite the evolution of conventional computers into sophisticated machines, there are many classes of problems that they cannot efficiently address, including inference, invertible logic, sampling and optimization, leading to considerable interest in alternative computing schemes. Quantum computing, which uses qubits to represent a superposition of 0 and 1, is expected to perform these tasks efficiently1–3. However, decoherence and the current requirement for cryogenic operation4, as well as the limited many-body interactions that can be implemented, pose considerable challenges. Probabilistic computing1,5–7 is another unconventional computation scheme that shares similar concepts with quantum computing but is not limited by the above challenges. The key role is played by a probabilistic bit (a p-bit)—a robust, classical entity fluctuating in time between 0 and 1, which interacts with other p-bits in the same system using principles inspired by neural networks8. Here we present a proof-of-concept experiment for probabilistic computing using spintronics technology, and demonstrate integer factorization, an illustrative example of the optimization class of problems addressed by adiabatic9 and gated2 quantum computing. Nanoscale magnetic tunnel junctions showing stochastic behaviour are developed by modifying market-ready magnetoresistive random-access memory technology10,11 and are used to implement three-terminal p-bits that operate at room temperature. The p-bits are electrically connected to form a functional asynchronous network, to which a modified adiabatic quantum computing algorithm that implements three- and four-body interactions is applied. Factorization of integers up to 945 is demonstrated with this rudimentary asynchronous probabilistic computer using eight correlated p-bits, and the results show good agreement with theoretical predictions, thus providing a potentially scalable hardware approach to the difficult problems of optimization and sampling. A probabilistic computer utilizing probabilistic bits, or p-bits, is implemented with stochastic nanomagnetic devices in a neural-network-inspired electrical circuit operating at room temperature and demonstrates integer factorization up to 945.

Integer factorization using stochastic magnetic tunnel junctions

This paper draws attention to a hardware system which can be engineered so that its intrinsic physics is described by the generalized Ising model and can encode the solution to many important NP-hard problems as its ground state. The basic constituents are stochastic nanomagnets which switch randomly between the ±1 Ising states and can be monitored continuously with standard electronics. Their mutual interactions can be short or long range, and their strengths can be reconfigured as needed to solve specific problems and to anneal the system at room temperature. The natural laws of statistical mechanics guide the network of stochastic nanomagnets at GHz speeds through the collective states with an emphasis on the low energy states that represent optimal solutions. As proof-of-concept, we present simulation results for standard NP-complete examples including a 16-city traveling salesman problem using experimentally benchmarked models for spin-transfer torque driven stochastic nanomagnets.

/pdf/intrinsic-optimization-using-stochastic-nanomagnets-1pk2ube63u.pdf

Intrinsic optimization using stochastic nanomagnets

Digital electronics are based on deterministic units called bits that can have one of two values, 0 and 1. New theoretical work suggests that circuits built out of probabilistic units that fluctuate randomly in value between 0 and 1 can be used to perform multiple functions: A multiplier, for example, can also function as a factorizer.

https://link.aps.org/pdf/10.1103/PhysRevX.7.031014

Stochastic p -Bits for Invertible Logic

This paper draws attention to a hardware system which can be engineered so that its intrinsic physics is described by the generalized Ising model and can encode the solution to many important NP-hard problems as its ground state. The basic constituents are stochastic nanomagnets which switch randomly between the $\pm 1$ Ising states and can be monitored continuously with standard electronics. Their mutual interactions can be short or long range, and their strengths can be reconfigured as needed to solve specific problems and to anneal the system at room temperature. The natural laws of statistical mechanics guide the network of stochastic nanomagnets at GHz speeds through the collective states with an emphasis on the low energy states that represent optimal solutions. As proof-of-concept, we present simulation results for standard NP-complete examples including a 16-city traveling salesman problem using experimentally benchmarked models for spin-transfer torque driven stochastic nanomagnets.

/pdf/intrinsic-optimization-using-stochastic-nanomagnets-nbjn6fdhe9.pdf

Magnetic tunnel junctions (MTJs) utilizing unstable magnets with low barriers have been shown to be well-suited for the implementation of random number generators (RNGs). It has recently been shown that completely new applications involving optimization, inference, and invertible Boolean logic would be enabled if many RNGs can be interconnected to form large scale correlated networks. However, this requires a new device, namely, a three-terminal tunable RNG or a p-bit, whose input terminal can be used to pin its output to 0 or 1. In this letter, we show that a voltage driven p-bit can be implemented simply by incorporating existing RNGs into a transistor circuit using experimentally demonstrated 2-terminal MTJs, without requiring a new device. Using established SPICE models, we show that this proposed p-bit can be interconnected to build correlated p-circuits to implement useful functionalities including a representative example of an invertible AND gate that “factors” the output of an AND gate into consistent input combinations.

Kerem Y. Camsari

Papers

Integer factorization using stochastic magnetic tunnel junctions

Intrinsic optimization using stochastic nanomagnets

Stochastic p -Bits for Invertible Logic

Intrinsic optimization using stochastic nanomagnets

Implementing p-bits With Embedded MTJ