Fan-out and Fan-in properties of superconducting neuromorphic circuits

Sze V., Chen Y., Yang T., Emer J.S.

Shainline J.M.

Much of the information processing performed by a biological neuron occurs in the dendritic tree. For artificial neural systems using light for communication, it is advantageous to convert signals to the electronic domain at synaptic terminals, so dendritic computation can be performed with electrical circuits. Here, we present circuits based on Josephson junctions and mutual inductors that act as dendrites, processing signals from synapses receiving single-photon communication events with superconducting detectors. We show simulations of circuits performing basic temporal filtering, logical operations, and nonlinear transfer functions. We further show how the synaptic signal from a single photon can fan out locally in the electronic domain to enable the dendrites of the receiving neuron to process a photonic synapse event or pulse train in multiple different ways simultaneously. Such a technique makes efficient use of photons, energy, space and information.

Hopkins P.F., Brevik J.A., Castellanos-Beltran M., Donnelly C.A., Flowers-Jacobs N.E., Fox A.E., Olaya D., Dresselhaus P.D., Benz S.P.

We review our development of Josephson-junction-based programmable reference sources to synthesize spectrally pure waveforms with quantum-based voltage accuracy for characterizing and improving future communication devices and systems. The goal is to provide sources that together span the entire radio frequency band, equivalent to seven orders of magnitude in frequency (20 kHz to 300 GHz). To synthesize waveforms in the microwave frequency band, we are extending the existing Josephson arbitrary waveform synthesizer (JAWS) technology used for ac voltage metrology at audio frequencies (

Shainline J.M., Buckley S.M., McCaughan A.N., Chiles J.T., Jafari Salim A., Castellanos-Beltran M., Donnelly C.A., Schneider M.L., Mirin R.P., Nam S.W.

Superconducting optoelectronic hardware has been proposed for large-scale neural computing. In this work, we expand upon the circuit and network designs previously introduced. We investigate circuits using superconducting single-photon detectors and Josephson junctions to perform signal reception, synaptic weighting, and integration. Designs are presented for synapses and neurons that perform integration of rate-coded signals as well as detect coincidence events for temporal coding. A neuron with a single integration loop can receive input from thousands of synaptic connections, and many such loops can be employed for dendritic processing. We show that a synaptic weight can be modified via a superconducting flux-storage loop inductively coupled to the current bias of the synapse. Synapses with hundreds of stable states are designed. Spike-timing-dependent plasticity can be implemented using two photons to strengthen and two photons to weaken the synaptic weight via Hebbian-type learning rules. In addition to the synaptic receiver and plasticity circuits, we describe an amplifier chain that converts the current pulse generated when a neuron reaches threshold to a voltage pulse sufficient to produce light from a semiconductor diode. This light is the signal used to communicate between neurons in the network. We analyze the performance of the elements in the amplifier chain to calculate the energy consumption per photon created. The speed of the amplification sequence allows neuronal firing up to at least 20 MHz, independent of connectivity. We consider these neurons in network configurations to investigate near-term technological potential and long-term physical limitations. By modeling the physical size of superconducting optoelectronic neurons, we calculate the area of these networks. A system with 8100 neurons and 330 430 total synapses will fit on a 1 × 1 cm 2 die. Systems of millions of neurons with hundreds of millions of synapses will fit on a 300 mm wafer. For multiwafer assemblies, communication at light speed enables a neuronal pool the size of a large data center ( 10 5 m 2) comprised of trillions of neurons with coherent oscillations at 1 MHz.Superconducting optoelectronic hardware has been proposed for large-scale neural computing. In this work, we expand upon the circuit and network designs previously introduced. We investigate circuits using superconducting single-photon detectors and Josephson junctions to perform signal reception, synaptic weighting, and integration. Designs are presented for synapses and neurons that perform integration of rate-coded signals as well as detect coincidence events for temporal coding. A neuron with a single integration loop can receive input from thousands of synaptic connections, and many such loops can be employed for dendritic processing. We show that a synaptic weight can be modified via a superconducting flux-storage loop inductively coupled to the current bias of the synapse. Synapses with hundreds of stable states are designed. Spike-timing-dependent plasticity can be implemented using two photons to strengthen and two photons to weaken the synaptic weight via Hebbian-type learning rules. In addition...

Leonard E., Beck M.A., Nelson J., Christensen B.G., Thorbeck T., Howington C., Opremcak A., Pechenezhskiy I.V., Dodge K., Dupuis N.P., Hutchings M.D., Ku J., Schlenker F., Suttle J., Wilen C., et. al.

High-fidelity gate operations are essential to the realization of a fault-tolerant quantum computer. In addition, the physical resources required to implement gates must scale efficiently with system size. A longstanding goal of the superconducting qubit community is the tight integration of a superconducting quantum circuit with a proximal classical cryogenic control system. Here we implement coherent control of a superconducting transmon qubit using a Single Flux Quantum (SFQ) pulse driver cofabricated on the qubit chip. The pulse driver delivers trains of quantized flux pulses to the qubit through a weak capacitive coupling; coherent rotations of the qubit state are realized when the pulse-to-pulse timing is matched to a multiple of the qubit oscillation period. We measure the fidelity of SFQ-based gates to be ~95% using interleaved randomized benchmarking. Gate fidelities are limited by quasiparticle generation in the dissipative SFQ driver. We characterize the dissipative and dispersive contributions of the quasiparticle admittance and discuss mitigation strategies to suppress quasiparticle poisoning. These results open the door to integration of large-scale superconducting qubit arrays with SFQ control elements for low-latency feedback and stabilization.

Cheng R., Goteti U.S., Hamilton M.C.

Superconducting circuits that operate by propagation of small voltage or current pulses, corresponding to propagation of single flux or charge quantum, are naturally suited for implementing spiking neuron circuits. Quantum phase-slip junctions (QPSJs) are 1-D superconducting nanowires that have been identified as exact duals to Josephson junctions, based on charge-flux duality in Maxwell’s equations. In this paper, a superconducting quantized-charge circuit element, formed using quantum phase-slip junctions, is investigated for use in high-speed, low-energy superconducting spiking neuron circuits. By means of a SPICE model developed for QPSJs, operation of this superconducting circuit to produce and transport quantized charge pulses, in the form of current pulses, is demonstrated. The resulting quantized-charge-based operation emulates spiking neuron circuits for brain-inspired neuromorphic applications. Additionally, to further demonstrate the operation of QPSJ-based neuron circuits, a QPSJ-based integrate and fire neuron circuit is introduced, along with simulation results using WRSPICE. Estimates for operating speed and power dissipation are provided and compared to Josephson junction and CMOS-based spiking neuron circuits. Current challenges are also briefly mentioned.

Soloviev I.I., Schegolev A.E., Klenov N.V., Bakurskiy S.V., Kupriyanov M.Y., Tereshonok M.V., Shadrin A.V., Stolyarov V.S., Golubov A.A.

We consider adiabatic superconducting cells operating as an artificial neuron and synapse of a multilayer perceptron (MLP). Their compact circuits contain just one and two Josephson junctions, respectively. While the signal is represented as magnetic flux, the proposed cells are inherently nonlinear and close-to-linear magnetic flux transformers. The neuron is capable of providing the one-shot calculation of sigmoid and hyperbolic tangent activation functions most commonly used in MLP. The synapse features both positive and negative signal transfer coefficients in the range ∼ ( − 0.5 , 0.5 ). We briefly discuss implementation issues and further steps toward the multilayer adiabatic superconducting artificial neural network, which promises to be a compact and the most energy-efficient implementation of MLP.We consider adiabatic superconducting cells operating as an artificial neuron and synapse of a multilayer perceptron (MLP). Their compact circuits contain just one and two Josephson junctions, respectively. While the signal is represented as magnetic flux, the proposed cells are inherently nonlinear and close-to-linear magnetic flux transformers. The neuron is capable of providing the one-shot calculation of sigmoid and hyperbolic tangent activation functions most commonly used in MLP. The synapse features both positive and negative signal transfer coefficients in the range ∼ ( − 0.5 , 0.5 ). We briefly discuss implementation issues and further steps toward the multilayer adiabatic superconducting artificial neural network, which promises to be a compact and the most energy-efficient implementation of MLP.

Schneider M.L., Donnelly C.A., Russek S.E., Baek B., Pufall M.R., Hopkins P.F., Dresselhaus P.D., Benz S.P., Rippard W.H.

Clustered ferromagnetic Josephson junctions form ultralow energy synaptic elements. Neuromorphic computing promises to markedly improve the efficiency of certain computational tasks, such as perception and decision-making. Although software and specialized hardware implementations of neural networks have made tremendous accomplishments, both implementations are still many orders of magnitude less energy efficient than the human brain. We demonstrate a new form of artificial synapse based on dynamically reconfigurable superconducting Josephson junctions with magnetic nanoclusters in the barrier. The spiking energy per pulse varies with the magnetic configuration, but in our demonstration devices, the spiking energy is always less than 1 aJ. This compares very favorably with the roughly 10 fJ per synaptic event in the human brain. Each artificial synapse is composed of a Si barrier containing Mn nanoclusters with superconducting Nb electrodes. The critical current of each synapse junction, which is analogous to the synaptic weight, can be tuned using input voltage spikes that change the spin alignment of Mn nanoclusters. We demonstrate synaptic weight training with electrical pulses as small as 3 aJ. Further, the Josephson plasma frequencies of the devices, which determine the dynamical time scales, all exceed 100 GHz. These new artificial synapses provide a significant step toward a neuromorphic platform that is faster, more energy-efficient, and thus can attain far greater complexity than has been demonstrated with other technologies.

Segall K., LeGro M., Kaplan S., Svitelskiy O., Khadka S., Crotty P., Schult D.

Conventional digital computation is rapidly approaching physical limits for speed and energy dissipation. Here we fabricate and test a simple neuromorphic circuit that models neuronal somas, axons, and synapses with superconducting Josephson junctions. The circuit models two mutually coupled excitatory neurons. In some regions of parameter space the neurons are desynchronized. In others, the Josephson neurons synchronize in one of two states, in-phase or antiphase. An experimental alteration of the delay and strength of the connecting synapses can toggle the system back and forth in a phase-flip bifurcation. Firing synchronization states are calculated >70 000 times faster than conventional digital approaches. With their speed and low energy dissipation (10^{-17}J/spike), this set of proof-of-concept experiments establishes Josephson junction neurons as a viable approach for improvements in neuronal computation as well as applications in neuromorphic computing.

Holmes D.S., Kadin A.M., Johnson M.W.

Once focused solely on computation speed, superconducting computing is now proving useful in hybrid systems where its unique capabilities complement conventional computing technologies. Energy efficiency has become a strong motivation for developing large-scale superconducting systems.

Segall K., Guo S., Crotty P., Schult D., Miller M.

Aiming to understand group behaviors and dynamics of neural networks, we have previously proposed the Josephson junction neuron (JJ neuron) as a fast analog model that mimics a biological neuron using superconducting Josephson junctions. In this study, we further analyze the dynamics of the JJ neuron numerically by coupling one JJ neuron to another. In this coupled system we observe a phase-flip bifurcation, where the neurons synchronize out-of-phase at weak coupling and in-phase at strong coupling. We verify this by simulation of the circuit equations and construct a bifurcation diagram for varying coupling strength using the phase response curve and spike phase difference map. The phase-flip bifurcation could be observed experimentally using standard digital superconducting circuitry.

Merolla P.A., Arthur J.V., Alvarez-Icaza R., Cassidy A.S., Sawada J., Akopyan F., Jackson B.L., Imam N., Guo C., Nakamura Y., Brezzo B., Vo I., Esser S.K., Appuswamy R., Taba B., et. al.

Modeling computer chips on real brains Computers are nowhere near as versatile as our own brains. Merolla et al. applied our present knowledge of the structure and function of the brain to design a new computer chip that uses the same wiring rules and architecture. The flexible, scalable chip operated efficiently in real time, while using very little power. Science, this issue p. 668 A large-scale computer chip mimics many features of a real brain. Inspired by the brain’s structure, we have developed an efficient, scalable, and flexible non–von Neumann architecture that leverages contemporary silicon technology. To demonstrate, we built a 5.4-billion-transistor chip with 4096 neurosynaptic cores interconnected via an intrachip network that integrates 1 million programmable spiking neurons and 256 million configurable synapses. Chips can be tiled in two dimensions via an interchip communication interface, seamlessly scaling the architecture to a cortexlike sheet of arbitrary size. The architecture is well suited to many applications that use complex neural networks in real time, for example, multiobject detection and classification. With 400-pixel-by-240-pixel video input at 30 frames per second, the chip consumes 63 milliwatts.

Onomi T., Nakajima K.

We have proposed a superconducting Hopfield-type neural network for solving the N-Queens problem which is one of combinatorial optimization problems. The sigmoid-shape function of a neuron output is represented by the output of coupled SQUIDs gate consisting of a single-junction and a double-junction SQUIDs. One of the important factors for an improvement of the network performance is an improvement of a threshold characteristic of a neuron circuit. In this paper, we report an improved design of coupled SQUID gates for a superconducting neural network. A step-like function with a steep threshold at a rising edge is desirable for a neuron circuit to solve a combinatorial optimization problem. A neuron circuit is composed of two coupled SQUIDs gates with a cascade connection in order to obtain such characteristics. The designed neuron circuit is fabricated by a 2.5 kA/cm2 Nb/AlOx/Nb process. The operation of a fabricated neuron circuit is experimentally demonstrated. Moreover, we discuss about the performance of the neural network using the improved neuron circuits and delayed negative self-connections.

Chiarello F., Carelli P., Castellano M.G., Torrioli G.

We propose a scheme for the realization of artificial neural networks based on superconducting quantum interference devices (SQUIDs). In order to demonstrate the operation of this scheme we designed and successfully tested a small network that implements an XOR gate and is trained by means of examples. The proposed scheme can be particularly convenient as support for superconducting applications such as detectors for astrophysics, high energy experiments, medicine imaging and so on.

Yamanashi Y., Umeda K., Yoshikawa N.

A superconductive perceptron, an artificial neural network, has been investigated using single flux quantum (SFQ) stochastic logic. A superconductive pseudo sigmoid function generator that corresponds to an artificial neuron device for the perceptron has been proposed and implemented using an SFQ current comparator and a frequency-to-current converter, which generates current that is proportional to the average input SFQ frequency. A frequency-to-current converter has been implemented using a dc-SQUID voltage driver coupled with a Josephson transmission line. We implemented and tested the pseudo sigmoid function generator using the SRL 2.5 kA/cm2 Nb process. The measured input-output characteristic agreed with the ideal sigmoid function with an average error of 0.063%.

Liu Z., Chen S., Qu P., Tang G., You H.

Schneider M.L., Jué E.M., Pufall M.R., Segall K., Anderson C.W.

Abstract Neuromorphic computing takes biological inspiration to the device level aiming to improve computational efficiency and capabilities. One of the major issues that arises is the training of neuromorphic hardware systems. Typically training algorithms require global information and are thus inefficient to implement directly in hardware. In this paper we describe a set of reinforcement learning based, local weight update rules and their implementation in superconducting hardware. Using SPICE circuit simulations, we implement a small-scale neural network with a learning time of order one nanosecond per update. This network can be trained to learn new functions simply by changing the target output for a given set of inputs, without the need for any external adjustments to the network. Further, this architecture does not require programing explicit weight values in the network, alleviating a critical challenge with analog hardware implementations of neural networks.

Karamuftuoglu M.A., Ucpinar B.Z., Razmkhah S., Fayyazi A., Kamal M., Pedram M.

Abstract A novel high-fan-in differential superconductor neuron structure designed for ultra-high-performance Spiking Neural Network (SNN) accelerators is presented. Utilizing a high-fan-in neuron structure allows us to design SNN accelerators with more synaptic connections, enhancing the overall network capabilities. The proposed neuron design is based on superconductor electronics fabric, incorporating multiple superconducting loops, each with two Josephson Junctions. This arrangement enables each input data branch to have positive and negative inductive coupling, supporting excitatory and inhibitory synaptic data. Compatibility with synaptic devices and thresholding operation is achieved using a single flux quantum (SFQ) pulse-based logic style. The neuron design, along with ternary synaptic connections, forms the foundation for a superconductor-based SNN inference. To demonstrate the capabilities of our design, we train the SNN using snnTorch, augmenting the PyTorch framework. After pruning, the demonstrated SNN inference achieves an impressive 96.1% accuracy on MNIST images. Notably, the network exhibits a remarkable throughput of 8.92 GHz while consuming only 1.5 nJ per inference, including the energy consumption associated with cooling to 4K. These results underscore the potential of superconductor electronics in developing high-performance and ultra-energy-efficient neural network accelerator architectures.

Schegolev Andrey E., Bastrakova Marina V., Sergeev Michael A., Maksimovskaya Anastasia A., Klenov Nikolay V., Soloviev Igor

The extensive development of the field of spiking neural networks has led to many areas of research that have a direct impact on people’s lives. As the most bio-similar of all neural networks, spiking neural networks not only allow for the solution of recognition and clustering problems (including dynamics), but they also contribute to the growing understanding of the human nervous system. Our analysis has shown that hardware implementation is of great importance, since the specifics of the physical processes in the network cells affect their ability to simulate the neural activity of living neural tissue, the efficiency of certain stages of information processing, storage and transmission. This survey reviews existing hardware neuromorphic implementations of bio-inspired spiking networks in the ”semiconductor”, ”superconductor”, and ”optical” domains. Special attention is given to the potentials for effective ”hybrids” of different approaches.

Primavera B.A., Khan S., Adler S.R., Shainline J.M.

Karamuftuoglu M.A., Ucpinar B.Z., Razmkhah S., Kamal M., Pedram M.

Ucpinar B.Z., Karamuftuoglu M.A., Razmkhah S., Pedram M.

Karimov T., Ostrovskii V., Rybin V., Druzhina O., Kolev G., Butusov D.

Josephson junctions (JJs) are superconductor-based devices used to build highly sensitive magnetic flux sensors called superconducting quantum interference devices (SQUIDs). These sensors may vary in design, being the radio frequency (RF) SQUID, direct current (DC) SQUID, and hybrid, such as D-SQUID. In addition, recently many of JJ’s applications were found in spiking models of neurons exhibiting nearly biological behavior. In this study, we propose and investigate a new circuit model of a sensory neuron based on DC SQUID as part of the circuit. The dependence of the dynamics of the designed model on the external magnetic flux is demonstrated. The design of the circuit and derivation of the corresponding differential equations that describe the dynamics of the system are given. Numerical simulation is used for experimental evaluation. The experimental results confirm the applicability and good performance of the proposed magnetic-flux-sensitive neuron concept: the considered device can encode the magnetic flux in the form of neuronal dynamics with the linear section. Furthermore, some complex behavior was discovered in the model, namely the intermittent chaotic spiking and plateau bursting. The proposed design can be efficiently applied to developing the interfaces between circuitry and spiking neural networks. However, it should be noted that the proposed neuron design shares the main limitation of all the superconductor-based technologies, i.e., the need for a cryogenic and shielding system.

Kovalyuk V.V., Venediktov I.O., Sedykh K.O., Svyatodukh S.S., Hydyrova S., Moiseev K.M., Florya I.N., Prokhodtsov A.I., Galanova V.S., Kobtsev D.M., Kuzin A.Y., Golikov A.D., Goltsman G.N.

We consider superconducting single-photon detectors, which are the key element of quantum optical technologies due to their unique characteristics not available in other technologies today. Since the first demonstration in Russia in 2001, such detectors have evolved significantly, and their waveguide-based versions are ready for scaling both in the fields of classical technologies (attenuated light) and of quantum optical applications (non-classical light). The paper studies the operating principle of such detectors and their main characteristics, analyzes superconducting materials and dielectric waveguide platforms, highlights the design principles, considers various levels of integration of on-chip waveguide superconductor detectors, and presents important new areas of application towards the implementation of photonic and ion quantum processors, as well as energy-efficient neuromorphic computing.

Edwards A.J., Krylov G., Friedman J.S., Friedman E.G.

Abstract Conventional semiconductor-based integrated circuits are gradually approaching fundamental scaling limits. Many prospective solutions have recently emerged to supplement or replace both the technology on which basic devices are built and the architecture of data processing.
Neuromorphic circuits are a promising approach to computing where techniques used by the brain to achieve high efficiency are exploited. 
Many existing neuromorphic circuits rely on unconventional and useful properties of novel technologies to better mimic the operation of the brain.
One such technology is single flux quantum (SFQ) logic -- a cryogenic superconductive technology in which the data are represented by quanta of magnetic flux (fluxons) produced and processed by Josephson junctions embedded within inductive loops.
The movement of a fluxon within a circuit produces a quantized voltage pulse (SFQ pulse), resembling a neuronal spiking event. These circuits routinely operate at clock frequencies of tens to hundreds of gigahertz, making SFQ a natural technology for processing high frequency pulse trains.

This work harnesses thermal stochasticity in superconducting synapses to emulate stochasticity in biological synapses in which the synapse probabilistically propagates or blocks incoming spikes. The authors also present neuronal, fan-in, and fan-out circuitry inspired by the literature that seamlessly cascade with the synapses for deep neural network construction. Synapse weights and neuron biases are set with bias current, and the authors propose multiple mechanisms for training the network and storing weights. The network primitives are successfully demonstrated in simulation in the context of a rate-coded multi-layer XOR neural network which achieves a wide classification margin. 
The proposed methodology is based solely on existing SFQ technology and does not employ unconventional superconductive devices or semiconductor transistors, making this proposed system an effective approach for scalable cryogenic neuromorphic computing.

Ionin A.S., Karelina L.N., Shuravin N.S., Sidel'nikov M.S., Razorenov F.A., Egorov S.V., Bol'ginov V.V.

The transfer function of a shunted two-junction interferometer, which was previously proposed as a basic element of superconducting neural networks based on radial basis functions, has been measured for the first time. The sample has been implemented in the form of a multilayer thin-film structure over a thick superconducting screen with the inductive supply of an input signal and the readout of an output signal. It has been found that the transfer function is the sum of the linear and periodic bell-shaped components. The linear component is likely due to the direct transfer of the input magnetic flux to the measuring circuit. The shape of the nonlinear component, which is the output signal of a Gauss neuron, can be approximately described by a Gaussian distribution function or, more precisely, by a parametric dependence derived theoretically in previous works. It has been shown that the transfer function of the Gauss neuron can depend on the choice of the working point of the measuring circuit, which promotes the development of integrated neural networks based on implemented elements.

Krylov G., Jabbari T., Friedman E.G.

The increasing complexity of modern rapid single flux quantum (RSFQ) circuits has made the issue of multiple fanout of growing importance. Most RSFQ gates can only drive a single output. Splitter gates can however distribute an SFQ pulse to multiple fanout. To drive N SFQ gates, N $$-$$ 1 splitters with a fanout of two are required. Large splitter trees are often used in high speed VLSI complexity SFQ systems. These splitters require significant area and increase the path delay. In this chapter, three area and power efficient splitter topologies for large scale RSFQ circuits are introduced. These SFQ splitters are an active splitter tree with fewer JJs, a passive splitter, and a multi-output active splitter. A methodology is presented for determining when to use passive or active splitters. Tradeoffs among the number of JJs, bias current of each stage, and delay are reported along with a margin analysis. The proposed splitters greatly reduce the required bias currents and delay of large scale RSFQ circuits by enabling multiple fanout. The methodologies and techniques are applicable to automated layout and clock tree synthesis for large scale SFQ integrated circuits.

Ionin A.S., Karelina L.N., Shuravin N.S., Sidel’nikov M.S., Razorenov F.A., Egorov S.V., Bol’ginov V.V.

The transfer function of a shunted two-junction interferometer, which was previously proposed as a basic element of superconducting neural networks based on radial basis functions, has been measured for the first time. The sample has been implemented in the form of a multilayer thin-film structure over a thick superconducting screen with the inductive supply of an input signal and the readout of an output signal. It has been found that the transfer function is the sum of the linear and periodic bell-shaped components. The linear component is likely due to the direct transfer of the input magnetic flux to the measuring circuit. The shape of the nonlinear component, which is the output signal of a Gauss neuron, can be approximately described by a Gaussian distribution function or, more precisely, by a parametric dependence derived theoretically in previous works. It has been shown that the transfer function of the Gauss neuron can depend on the choice of the working point of the measuring circuit, which promotes the development of integrated neural networks based on implemented elements.

Volk J., Tzimpragos G., Wynn A., Golden E., Sherwood T.

Crotty P., Segall K., Schult D.