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- Abstract
- Engineering Correlations Through Spin Currents
- Combinatorial Optimization
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Sci Rep. 2017; 7: 44370.

Published online 2017 March 15. doi: 10.1038/srep44370

PMCID: PMC5353626

Received 2016 September 28; Accepted 2017 February 7.

Copyright © 2017, The Author(s)

This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

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.

The use of Ising computers to solve NP-hard problems has a rich heritage in both theory^{1} and practice. These computers seek to solve a wide range of optimization problems by encoding the solution to the problem as the ground-state of an Ising energy expression. Many diverse systems have been proposed to solve NP-hard optimization problems such as those based on simulated annealing^{2}, DNA^{3}^{,4}, quantum annealing^{5}^{,6}, Cellular Neural Networks^{7}^{,8}^{,9}, CMOS^{10}, trapped ions^{11}, electromechanics^{12}, optics^{13}^{,14}^{,15}^{,16}^{,17}^{,18}^{,19}^{,20}, and magnets^{21}^{,22}^{,23}. A common objective of many of the Ising-based approaches is the identification of hardware configurations that can efficiently solve optimization problems of interest.

In this letter, we demonstrate the possibility of a hardware implementation that does not just mimic the Ising model, but embodies it as a part of its natural physics^{21}^{,22}^{,23}. It uses a network of *N* “soft” nanomagnets operating in a stochastic manner^{24}, each with an energy barrier Δ comparable to *k*_{B}*T* so that they switch between the two Ising states, ±1, on time scales where *τ*_{0}~0.1–1ns. The natural laws of statistical mechanics guide the network through the 2^{N} collective states at GHz rates, with an emphasis on low energy states. We show how an optimization problem of interest is solved by engineering the spin-mediated magnet-magnet interactions to encode the problem solution and to simulate annealing without any change in temperature simply by continuously adjusting their overall strength. As proof-of-concept for the potential applications of this natural Ising computer, we present detailed simulation results for standard NP-complete examples, including a 16-city traveling salesman problem. This involves using experimentally benchmarked modules to simulate a suitably designed network of 225 stochastic nanomagnets and letting the hardware itself rapidly identify solutions within the 2^{225} possibilities. It should be possible to integrate such hardware into standard solid state circuits, which will govern the scalability of the solution.

The Ising Hamiltonian for a collection of spins, *S*_{i}, which can take on one of two values, ±1,

was originally developed to describe ferromagnetism where the *J*_{ij} are positive numbers representing an exchange interaction between neighboring spins *S*_{i} and *S*_{j}, while *h*_{i} represents a local magnetic field for spin *S*_{i}. Classically, different spin configurations *σ*{*S*_{i}} have a probability proportional to , *T* being the temperature, and *k*_{B}, the Boltzmann constant. At low temperatures, the system should be in its ground state *σ*_{G}, the state with the lowest energy *H(σ*). With *h*_{i}=0, and positive *J*_{ij}, it is easy to see that the ground state is the ferromagnetic configuration *σ*_{F} with all spins parallel.

Much of the interest in the Ising Hamiltonian arises from the demonstration of many direct mappings of NP-complete and NP-hard problems to the model^{1}^{,25}^{,26} such that the desired solution is represented by the spin configuration *σ* corresponding to the ground state. However, in general this mapping may require a large number of spins, and may require the parameters *J*_{ij} and *h*_{i} to take on a wide range of values, both positive and negative. Finding the ground state of this artificial spin glass is the essence of Ising computing, and broadly speaking it involves abstractly representing an array of spins, their coupling, and thermal noise through software and hardware that attempts to harness the efficiencies of physical equivalence^{27}. These representations may take the form of abstract models of the spins, the use of random number generators to produce noise, and logical or digital adders for the weighted summing. If enough layers of abstraction can be eliminated, the underlying hardware will inherently solve a given problem as part of its natural, intrinsic operation and this should be reflected in increased speed and efficiency.

Here we describe a natural hardware for an Ising computer based on the representation of an Ising spin *S*_{k} by the magnetization *m* of a stochastic nanomagnet (SNM), which we believe will compare well with other alternative representations. These SNMs are in the “telegraphic” switching regime^{24}^{,28} requiring the existence of a small barrier in the magnetic energy (Δ≈*k*_{B}*T*), that gives a small, but definite preference for a given axis, with two preferred states ±1. In the absence of currents, these SNMs continually switch between +1 and −1 on the order of nanoseconds, and can be physically realized by a reduction of the magnetic grain volume^{29} or by designing weak perpendicular magnetic anisotropy (PMA) magnets^{30}. Figure 1 shows the response of such a monodomain PMA magnet in the presence of an external spin current in the direction of the magnet’s easy axis.

How do we couple the SNMs to implement the Ising Hamiltonian of Equation (1)? The usual forms of coupling involve dipolar or exchange interactions that are too limited in range and weightability. Instead, one possibility is an architecture^{23} that uses charge currents which can be readily converted locally into spin currents through the spin Hall effect (SHE). These charge currents can be arbitrarily long-range and the total number of cross-couplings is only limited by considerations of routing congestion and delay. The couplings may also be confined to nearest-neighbors, simplifying the hardware design complexity while promoting scalability and retaining universality^{26}.

The Ising Hamiltonian of Equation (1) can be implemented by exposing each SNM *m*_{k} to a spin current *I*_{k}

which has a constant bias determined by *h*_{k} together with a term proportional to the magnetization of the *j*^{th} SNM *m*_{j}. The future state of magnet *m*_{k} at time (*t*+Δ*t*) is related to the state of the other magnets at time *t* through the current *I*_{k}. This expression is derived analytically in the following section using the Fokker-Planck equation for the system^{31}.

The spin current *I*_{k} can be generated using well-established phenomena and the prospects for physical realization of such a system are discussed later in this paper. The distinguishing feature of the present proposal arises from the intrinsic stochasticity of SNMs and their biasing through the use of weighted spin currents (Fig. 1(a)). How the SNMs are interconnected to implement Equation (2) can evolve as the field progresses.

Getting a large system to reach its true ground state is non-trivial as it tends to get stuck in local minima^{32}. It is common to guide the system towards the ground state through a process of “annealing”^{2} which is carried out differently in different hardware implementations. For example, systems based on superconducting flux qubits make use of quantum tunneling, which is referred to as quantum annealing^{33}, whereas classical CMOS approaches make use of random number generators^{34} to produce random transitions out of local minima.

For our system of coupled SNMs, random noise is naturally present and can be easily controlled (Fig. 1(a)), causing the system of SNMs coupled according to Equation (2) to explore the configuration space of the problem on a nanosecond timescale. Annealing could be performed through a controlled lowering of the actual temperature, or equivalently through a controlled increase in the magnitude of the current *I*_{k}, even at room temperature. It has been noted that certain annealing schedules can guarantee convergence to the true ground state, but these schedules may be too slow to be used in practice^{35}. This paper only presents a straightforward annealing process and does not seek out optimal annealing schedules. Consequently, as we show in one of our combinatorial optimization examples, we may find only an approximate solution which, however, may be adequate for many practical problems.

Our goal is to interconnect magnets such that their equilibrium state is governed by Boltzmann statistics with thermal noise as an inherent characteristic of the system. To see that this is possible, consider a system of *N* magnets where we want

and

where *m*_{k} represents the z-component of the magnets.

Suppose each magnet is driven by a spin current derived from the others. We start with the Fokker-Planck equation^{31} for the *N*-magnet system:

where and *i*_{k}=*I*_{k}/*I*_{0} with *I*_{0} as the critical switching spin current . At equilibrium, yielding from (3) and (5):

respectively. Comparing equations (6) and (7) while assuming symmetric coupling, , for the system we find

and arrive at (2):

In this section we briefly describe the simulation framework and stochastic LLG model used throughout this paper. We start with the LLG equation^{31} for a monodomain magnet with magnetization *m*_{i} in the presence of a spin current

The magnetic thermal noise enters the equation through the effective field of the magnet, *H*_{i}=*H*_{0}+*H*_{n}, as an uncorrelated external magnetic field in three dimensions with the following mean and variance:

The numerical model is implemented as an equivalent circuit for SPICE-like simulators and reproduces the equilibrium (Boltzmann) distribution from a Fokker-Planck Equation^{31}.

A given system of magnets is simulated using a collection of independent, though current-coupled, stochastic LLG models. Delays associated with the communication from one magnet to the next are neglected assuming that the response time of the nanomagnets is much greater than associated wire-delays. Presently, the attempt time *τ* of experimental nanomagnets is on the order of ~*μ*s to ~ms^{28}^{,29}^{,36}. With additional scaling, the response times of these magnets will continue to improve^{37} and should approach the ~ns times discussed in this paper. With response times ~ns, our simulations show that even routing delays on the order of 100s of ps do not affect the results materially. Using nearest-neighbor Ising approaches or other constraining design decisions it should be possible to limit routing delays to shorter values. However, if the routing delay is comparable to the intrinsic response time of the nanomagnets then it would be important to include their effect in the simulation.

Many options exist, please see the final section, for physical realization of the proposed system of stochastic nanomagnets. For the simulations in this paper we simply use Equation (2) without assuming any specific hardware to implement it, since it is likely that better alternatives will emerge in the near future, given the rapid pace of discovery in the field of spintronics, see for example^{38}^{,39}^{,40}^{,41}.

We will focus on two specific examples to demonstrate the ability of such an engineered spin glass to solve problems of interest^{42}: an instructive example based on the satisfiability problem (SAT), and a representative example based on the traveling salesman problem (TSP). The first known NP-complete problem is the problem of Boolean satisfiability^{43}, namely, deciding if some assignment of boolean variables {*x*_{i}} exists that satisfies a given conjunctive normal form (CNF) expression. Finding the collection of inputs that makes the clauses of the CNF expression true is computationally difficult, but easy to verify.

It is known that any given CNF expression can be mapped to a collection of Ising contraints using the fundamental building blocks of NOT (), AND (), and OR () each subject to the Ising constraints given by^{44}:

Using these building blocks, a network capable of finding the truth table for XOR was prepared (Fig. 2). For simplicity, the solution uses a naive method to construct the network and leverages the use of ancillary spins to represent and respectively (note that four spins could have been used^{45}). The array of spins from Fig. 2(b) are connected as specified by equations (11, 12, 13), driven by a reference current *I*_{0}. As the magnets explore the configuration space, their outputs are digitized and used to compute the overall energy of the system (Fig. 2(c)). The regions of zero energy correspond to solutions of the problem. The digitized outputs are aggregated to determine their probability of occurrence. By looking at the first three bits of the most probable outputs, the solution to the problem can be directly found (Fig. 2(d,e)). While this problem helps convey the essence of the approach, a more demonstrative application is worth considering.

The decision form of the TSP is NP-complete, that is, for a collection of *N* cities, does there exist a closed path for which each city is visited exactly once that has a tour length less than some value *d*? Finding tours that satisfy this problem is computationally challenging and also of great practical interest. There are well-known mappings that translate the TSP to the Ising model^{25}^{,46}. Here we adopt the following:

where *x*_{i,j} is a Boolean variable that is TRUE when city *i* is stop number *j* and FALSE otherwise, and *W*_{(uv)} are directed weights based on the distance between cities *u* and *v*. This Hamiltonian is mapped to a spin system by replacing each *x*_{ij} with 1/2(*m*_{ij}+1) and weights *W*_{(uv)} with *i*_{(uv)} given by Equation (2).

If the interconnections between each city are symmetric, then a Boltzmann machine^{47} with each of the 2^{N×N} states associated with an effective energy *H* is realized, and the probability of the system visiting a particular state is proportional to . In order to find low-energy, optimized states, direct annealing of the glass can be performed. Using the ulysses16 reference dataset^{48}, annealing of a problem specific magnetic array through control of the effective temperature was performed (Fig. 3). Two specific traits of interest arise, namely the energy decays in a sigmoidal relationship with the ln *T*, and the specific heat of the system, , shows a defined peak about a critical temperature. At high temperatures, the system is disordered and corresponds to high energy states (Fig. 3(c)). As the temperature is reduced, the system continues to explore the energy landscape on a nanosecond timescale while gradually converging to a low-energy solution. For the given annealing profile and simulation duration, a low-energy, though not ideal, solution is found to the problem, highlighting the heuristic nature of the optimization^{46}. Note that in principle these simulation results could be obtained directly from actual hardware. For example, Figs 2(d) and 3(d) could be obtained by continuously monitoring the states of the individual SNMs using spin valves.

Physical realization of these engineered spin glasses requires the integration of multiple functional elements as highlighted in Fig. 2(a). The magnetization of each magnet *m*_{i} is first sensed with a read unit. The signal produced by this read unit is then propagated to all of the magnets with couplings dependent on the read magnetization *m*_{i}. Each of these connections is independently weighted with weights *W*_{(ij)} and provided as input, along with an on-site bias *B*_{i} to the write units. The write units in turn influence and control the state of magnet *m*_{j}.

There are a number of design options available for each functional unit as shown in Table 1. Write-control of the magnets can be affected through a number of means including the spin Hall effect (SHE)^{39} or perhaps through voltage control^{40}. The use of the SHE effect provides a convenient mechanism with which to sum several, independently weighted, input currents. Readout of the magnetization can be accomplished using well-established tunnel junctions^{49} which have been demonstrated for stochastic nanomagnets^{29}. Alternatively, readout could perhaps be accomplished using the inverse SHE^{39}. Assuming the use of a SHE material and tunnel junction stack, care must be given to accomodate the simultaneous use of write and read currents. One approach would be to introduce the use of a time-multiplexed scheme that disassociates the write and read operations^{50}. Alternatively, structures that provide write and read isolation may be used^{51}.

The ability to write and read the magnetization is of fundamental importance, however, once read, the likely weak signal must be amplified to satisfy the fanout requirements of the network. This transistor-like gain can be realized using all-spin based approaches^{23}^{,51} or perhaps with the use of a hybrid-CMOS design^{52}^{,53}. These proposed approaches may introduce power dissipation challenges during the read operation, e.g. the short-circuit current produced with the use of amplifying inverters. Power dissipation considerations must be carefully evaluated to assess the viability of scaling the proposed system.

The output from the amplification stage can be selectively weighted so that a wide range of problems based on (1) can be encoded onto the network. The weighting of inputs can be based on an approach using re-programmable floating-gate voltages^{54} that would enable the use of analog weights for the circuit. While floating-gate regulation would enable convenient re-programmability, the design would be complicated with the requirement for peripheral drivers to control the floating-gate array. Others proposals have suggested the use of memristors^{24}^{,53}^{,55} or other programmable elements in a cross-bar like configuration^{50}^{,56}, though with constrained fanout. Note that one weighting scheme that still retains the ability to encode NP-hard problems onto the network is with the use of {−1, 0, 1} weights^{1}. Using this simple approach removes the necessity for tunable weights and instead relegates the problem to one of routing, connectivity, and area.

All of the simulations used in this paper assume a fully connected network of magnets in which each magnet talks to all other magnets. For small networks this is reasonable, however, such an assumption is invalid for large networks as the number of routes grows rapidly. Instead, different topologies^{57} and routing considerations must be made to account for congestion and long-distance communication. By limiting the connections to local-neighbors^{9}^{,10}, the network may still be used to perform NP-hard optimization while also simplifying routing complexity. One design possibility is to leverage the lessons learned from the advances in the design of Field-Programmable Gate Array (FPGA) interconnects^{58}. FPGAs are designed with routing topologies that facilitate both short and long-range interconnections while also providing re-programmability.

The fidelity of the programmed weights and number of high-fanout signals needed for robust solutions may impose challenges on the selected weighting and routing schemes. Additionally, the propagation delay of these high-fanout signals must be balanced with the response time of the magnets in order for the system to be governed by (1). While flexibility in the allowed weights and number of couplings is convenient for encoding problems onto the model^{25}, it is important to note that discrete nearest-neighbor couplings still retain NP-hardness^{1} and may greatly simplify the hardware design, improving scalability at the expense of increased encoding complexity and area.

The main point of this paper is the remarkable high-speed search through Fock space enabled by the intrinsic physics of a network of stochastic nanomagnets interacting via spin-mediated interactions. We hope this work fosters an interest in the physical realization and exploration of stochastic nanomagnets as a viable Ising computer.

Simulations based on the modular framework for spintronics^{38} were used to produce the results in this work. Within the framework, a stochastic Landau-Lifshitz-Gilbert (LLG) model was used to simulate each nanomagnet. The magnetic parameters for the telegraphic PMA magnets used in the simulations are: effective anisotropy of PMA, , saturation magnetization, , damping coefficient, *α*=0.01, and PMA diameter, Φ=45nm, amounting to a barrier height of Δ=1kT. In all simulations, the initial state of the magnetic array was randomly selected. Figure 1 was produced using a modular stochastic LLG simulation element with the input current swept from −2*μ*A to 2*μ*A in increments of 800nA. At each current, the response of the magnet is observed for 10*μ*s. Figure 2 was simulated for 100*μ*s using the coupling depicted in the Figure and a reference current *I*_{0} of 2*μ*A. Figure 3 used an annealing schedule of *T*_{i+1}=0.9*T*_{i} and Lagrange multiplier of . At each temperature the magnets were allowed to randomly walk for 1*μ*s and were measured every 200ps. The SAT and TSP magnetic networks were simulated using coupled stochastic LLG models with the intermagnet-coupling and on-site biases produced via the spin current term of the LLG equation. The magnetization of each magnet was digitized using Schmitt trigger based thresholds. HSPICE was used to solve the simultaneous coupled differential equations of the magnetic network.

**How to cite this article**: Sutton, B. *et al*. Intrinsic optimization using stochastic nanomagnets. *Sci. Rep.*
**7**, 44370; doi: 10.1038/srep44370 (2017).

**Publisher's note:** Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported in part by C-SPIN, one of six centers of STARnet, a Semiconductor Research Corporation program, sponsored by MARCO and DARPA and in part by the National Science Foundation through the NCN-NEEDS program, contract 1227020-EEC.

The authors declare no competing financial interests.

**Author Contributions** B.S. designed the networks, performed the simulations, and wrote the paper. K.C. developed the modules describing the biased superparamagnets, B.B. provided insights on PMA magnets and S.D. developed the quantitative connection between spin-torque coupling and the Ising model. B.B. and S.D. conceived the use of stochastic magnets for solving problems of this type, while K.C. and S.D. helped write the paper. All authors discussed the results and helped refine the manuscript.

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