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Title
Intrinsic optimization using stochastic nanomagnets.
Permalink
https://escholarship.org/uc/item/9k25p8kn
Journal
Scientific reports, 7(1)
ISSN
2045-2322
Authors
Sutton, Brian
Camsari, Kerem Yunus
Behin-Aein, Behtash
et al.
Publication Date
2017-03-01
DOI
10.1038/srep44370
Peer reviewed
eScholarship.org Powered by the California Digital Library
University of California
1
Scientific RepoRts | 7:44370 | DOI: 10.1038/srep44370
www.nature.com/scientificreports
Intrinsic optimization using
stochastic nanomagnets
Brian Sutton
1
, Kerem Yunus Camsari
1
, Behtash Behin-Aein
2
& Supriyo Datta
1
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 recongured as needed
to solve specic 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.
e use of Ising computers to solve NP-hard problems has a rich heritage in both theory
1
and practice. ese
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 optimi-
zation problems such as those based on simulated annealing
2
, DNA
3,4
, quantum annealing
5,6
, Cellular Neural
Networks
7–9
, CMOS
10
, trapped ions
11
, electromechanics
12
, optics
13–20
, and magnets
21–23
. A common objective of
many of the Ising-based approaches is the identication of hardware congurations that can eciently 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
21–23
. It uses a network of N “so” nanomagnets operating in
a stochastic manner
24
, each with an energy barrier ∆ comparable to k
B
T so that they switch between the two Ising
states, ± 1, on time scales
τ τ∼∆kTexp( /)
0B
where τ
0
~ 0.1–1 ns. e 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 continu-
ously 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 trave-
ling salesman problem. is 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.
e Ising Hamiltonian for a collection of spins, S
i
, which can take on one of two values, ± 1,
∑∑
=− −HJSS hS
(1)
ij
ij ij
i
ii
,
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 eld for spin S
i
. Classically,
dierent spin congurations σ{S
i
} have a probability proportional to
σ− HkTexp( ()/)
B
, 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
conguration σ
F
with all spins parallel.
1
School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, 47907, USA.
2
GLOBALFOUNDRIES Inc., Santa Clara, CA 95054, USA. Correspondence and requests for materials should be
addressed to B.S. (email: bmsutton@purdue.edu) or S.D. (email: datta@purdue.edu)
Received: 28 September 2016
Accepted: 07 February 2017
Published: 15 March 2017
OPEN
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Scientific RepoRts | 7:44370 | DOI: 10.1038/srep44370
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
conguration 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 articial 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 soware and hard-
ware that attempts to harness the eciencies of physical equivalence
27
. ese 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 reected in increased
speed and eciency.
Engineering Correlations Through Spin Currents
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. ese SNMs are in the “telegraphic” switching regime
24,28
requiring the existence of a small bar-
rier in the magnetic energy (∆ ≈ k
B
T), that gives a small, but denite preference for a given axis, with two pre-
ferred 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
. Figure1 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.
Figure 1. Response of stochastic nanomagnet to spin current. (a) e magnetization of a stochastic
nanomagnet is shown for varying spin currents. e ve number summary of the magnetization
m
z
is shown
throughout the simulation. (b) Obtaining stochastic operation for a magnet can be accomplished with a
reduction of the energy barrier of the magnet E
B
through device geometry or by increasing its temperature. e
response of the magnet to thermal noise under these conditions is modeled using a stochastic Landau-Lifshitz-
Gilbert (LLG) circuit element based on the input spin current I
S
and magnetic eld H. (c) Sample time slices are
shown at various set points along the sigmoid in order to visualize the magnetization dynamics.
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Scientific RepoRts | 7:44370 | DOI: 10.1038/srep44370
How do we couple the SNMs to implement the Ising Hamiltonian of Equation(1)? e usual forms of cou-
pling involve dipolar or exchange interactions that are too limited in range and weightability. Instead, one pos-
sibility is an architecture
23
that uses charge currents which can be readily converted locally into spin currents
through the spin Hall eect (SHE). ese charge currents can be arbitrarily long-range and the total number of
cross-couplings is only limited by considerations of routing congestion and delay. e couplings may also be con-
ned to nearest-neighbors, simplifying the hardware design complexity while promoting scalability and retaining
universality
26
.
e Ising Hamiltonian of Equation(1) can be implemented by exposing each SNM m
k
to a spin current I
k
∑
α
=
+
()
Im
q
hJm{}
2
2
(2)
kj k
j
kj j
which has a constant bias determined by h
k
together with a term proportional to the magnetization of the j
th
SNM
m
j
. e future state of magnet m
k
at time (t + ∆ t) is related to the state of the other magnets at time t through the
current I
k
. is expression is derived analytically in the following section using the Fokker-Planck equation for
the system
31
.
e spin current I
k
can be generated using well-established phenomena and the prospects for physical reali-
zation of such a system are discussed later in this paper. e 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 eld 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 dif-
ferently in dierent hardware implementations. For example, systems based on superconducting ux 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 conguration space of the problem
on a nanosecond timescale. Annealing could be performed through a controlled lowering of the actual tempera-
ture, 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
. is paper only presents a straightforward annealing pro-
cess and does not seek out optimal annealing schedules. Consequently, as we show in one of our combinatorial
optimization examples, we may nd only an approximate solution which, however, may be adequate for many
practical problems.
Steady-State Fokker-Planck Description. Our goal is to interconnect magnets such that their equilib-
rium 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
ρρ…=
−…
mm e(, ,)
(3)
N
Em mkT
1
0
(,,)/
NB1
and
∑∑
…= ++
Em mAmhmJmm(, ,)
()
(4)
N
i
ii ii
ij
ij ij1
2
,
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:
ρ
τ
ρ
ρ∂
∂
=
∂
∂
−
−+
∆
∂
∂
m
mi m
m
(1 )( )
1
2
(5)
k
kk k
kk
2
where
µ∆= HMVkT/2
kks
0
B
and i
k
= I
k
/I
0
with I
0
as the critical switching spin current
α=∆Iq kT(2 /)2
k0B
. At
equilibrium,
ρτ∂∂=/0
yielding from (3) and (5):
ρ∂
∂
=− ∆−
m
im
(ln)
2( )
(6)
k
kk k
∑
ρ∂
∂
=−
++ +
mkT
Am hJJm
(ln) 1
2()
(7)
kB
kk k
j
kj jk j
respectively. Comparing equations(6) and (7) while assuming symmetric coupling,
≡JJ
kj jk
, for the system we
nd
∆=−AkT/
(8)
kkB
and arrive at (2):
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Scientific RepoRts | 7:44370 | DOI: 10.1038/srep44370
µ
=
+∑
i
hJm
HMV
2
k
k
j
kj j
KS
0
Stochastic Landau-Lifshitz-Gilbert (LLG) Model. In this section we briey 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
=
ˆ
IIz()
s 0
αγαγ
α
+=−×−××+ ××
+
×
ˆ
ˆˆˆˆˆ
ˆ
dm
dt
mH mmH
qN
mI m
qN
mI
(1 )()
1
()
()
(9)
i
ii iii
i
iSii
i
iSi
2
e magnetic thermal noise enters the equation through the eective eld of the magnet, H
i
= H
0
+ H
n
, as an
uncorrelated external magnetic eld in three dimensions with the following mean and variance:
α
γ
==
.
HH0,
2kT
MVol
(10)
n
r
n
r
2
s
e numerical model is implemented as an equivalent circuit for SPICE-like simulators and reproduces the equi-
librium (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 ∼ µs 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 aect
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 eect in the simulation.
Many options exist, please see the nal 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 specic hard-
ware to implement it, since it is likely that better alternatives will emerge in the near future, given the rapid pace
of discovery in the eld of spintronics, see for example
38–41
.
Combinatorial Optimization
We will focus on two specic examples to demonstrate the ability of such an engineered spin glass to solve prob-
lems of interest
42
: an instructive example based on the satisability problem (SAT), and a representative example
based on the traveling salesman problem (TSP). e rst known NP-complete problem is the problem of Boolean
satisability
43
, namely, deciding if some assignment of boolean variables {x
i
} exists that satises a given conjunc-
tive normal form (CNF) expression. Finding the collection of inputs that makes the clauses of the CNF expression
true is computationally dicult, but easy to verify.
It is known that any given CNF expression can be mapped to a collection of Ising contraints using the funda-
mental building blocks of NOT (
=m m
12
), AND (
=∧m mm
123
), and OR (
=∨m mm
123
) each subject to the
Ising constraints given by
44
:
=−−Hmm1( )
(11)
NOT12
=−−+ +−−++Hmmmmmmmmm3( 22)(2)
(12)
AND231213123
=−−+ +−−−Hmmmmmmmmm3( 22)(2)
(13)
OR 23 12 13 123
Using these building blocks, a network capable of nding the truth table for XOR
=∨∧∧mmmmm(( )( ))
12323
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
∨mm()
23
and
∧mm()
23
respectively (note that four spins could have been used
45
). e
array of spins from Fig.2(b) are connected as specied by equations(11–13), driven by a reference current I
0
. As the
magnets explore the conguration space, their outputs are digitized and used to compute the overall energy of the sys-
tem (Fig.2(c)). e regions of zero energy correspond to solutions of the problem. e digitized outputs are aggregated
to determine their probability of occurrence. By looking at the rst 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.
e 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. ere are well-known mappings
that translate the TSP to the Ising model
25,46
. Here we adopt the following: