Adiabatic superconducting artificial neural network: Basic cells
Igor I. Soloviev, Andrey E. Schegolev, Nikolay V. Klenov, Sergey V. Bakurskiy, Mikhail Yu. Kupriyanov, Maxim V.
Tereshonok, Anton V. Shadrin, Vasily S. Stolyarov, and Alexander A. Golubov
Citation: Journal of Applied Physics 124, 152113 (2018); doi: 10.1063/1.5042147
View online: https://doi.org/10.1063/1.5042147
View Table of Contents: http://aip.scitation.org/toc/jap/124/15
Published by the American Institute of Physics
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Adiabatic superconducting artificial neural network: Basic cells
Igor I. Soloviev,
1,2,3,a)
Andrey E. Schegolev,
1,2,4,5
Nikolay V. Klenov,
1,2,4,5,6
Sergey V. Bakurskiy,
1,2,3
Mikhail Yu. Kupriyanov,
1
Maxim V. Tereshonok,
2,5
Anton V. Shadrin,
3
Vasily S. Stolyarov,
3,6,7,8
and Alexander A. Golubov
3,9
1
Lomonosov Moscow State University Skobeltsyn Institute of Nuclear Physics, 119991 Moscow, Russia
2
MIREA—Russian Technological University, 119454 Moscow, Russia
3
Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Russia
4
Physics Department, Lomonosov Moscow State University, 119991 Moscow, Russia
5
Moscow Technical University of Communications and Informatics (MTUCI), 111024 Moscow, Russia
6
Dukhov Research Institute of Automatics (VNIIA), 127055 Moscow, Russia
7
Institute of Solid State Physics RAS, 142432 Chernogolovka, Russia
8
Solid State Physics Department, KFU, 420008 Kazan, Russia
9
Faculty of Science and Technology and MESA+ Institute of Nanotechnology, 7500 AE Enschede,
The Netherlands
(Received 30 May 2018; accepted 25 July 2018; published online 26 September 2018)
We consider adiabatic superconducting cells operating as an artificial neuron and synapse of a multi-
layer 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 non-
linear 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 adi-
abatic superconducting artificial neural network, which promises to be a compact and the most
energy-efficient implementation of MLP. Published by AIP Publishing.
https://doi.org/10.1063/1.5042147
I. INTRODUCTION
Artificial neural network (ANN) is the key technology in
the fast developing area of artificial intelligence. It has been
already broadly introduced in our everyday life. Further pro-
gress requires an increase in complexity and depth of ANNs.
However, modern implementations of the neural networks
are commonly based on conventional computer hardware
which is not well suited for neuromorphic operation. This
leads to excessive power consumption and hardware over-
head. Ideal basic elements of ANNs should combine the mul-
tiple properties like one-shot calculation of their functions,
operation with energy near the thermal noise floor, and nano-
scale dimensions.
The most energy ef ficient computing today can be per-
formed using the superconductor digital technology.
1
The first ever practical logic gates capable of operating down
to and below the Landauer thermal limit
2
were realized
recently
3
on the basis of adiabatic superconductor logic.
Besides the several attempts to the implementation of the
superconducting ANNs proposed since the 1990s,
4–12
the
idea to adopt the adiabatic logic cells to neuromorphic cir-
cuits was presented only recently.
13,14
In this paper, we con-
sider operation principles of adiabatic superconducting basic
cells which comply with the above-mentioned properties for
ANN implementation. We focus on a particular multilayer
perceptron (MLP) because of a wide range of its applicability
and well-developed learning algorithms for such a network.
II. BASIC CELLS
The basic element of superconducting circuits is the
Josephson junction. Its characteristic energy typically lies
below aJ level while switching frequency is several hundred
GHz. Contrary to semiconductor transistor, the Josephson
junction is not fabricated in a substrate but between two
superconductor layers deposited on a substrate utilized as a
mechanical support. This provides opportunity for supercon-
ducting circuits to benefit from 3D topology which can be
especially suitable for deep ANNs. The minimal feature size
of superconducting circuits is progressively decreased down
to nanoscales in recent years.
15
Another attractive feature of the Josephson junction is its
inherently strong nonlinearity. Indeed, the current flowing
through the junction, I, is commonly related to the supercon-
ducting phase difference between the superconducting banks,
w
,as
I ¼ I
c
sin
w
, (1)
where I
c
is the junction critical current. We show below that
this current-phase relation (CPR) having both linear and non-
linear parts is well suited for implementation of supercon-
ducting artifi cial neuron with one-shot calculation of sigmoid
or hyperbolic tangent activation functions
σ (x) ¼
1
1 þ e
x
, (2a)
a)
isol@phys.msu.ru
JOURNAL OF APPLIED PHYSICS 124, 152113 (2018)
0021-8979/2018/124(15)/152113/5/$30.00 124, 152113-1 Published by AIP Publishing.
or
τ (x) ¼ tanh (x), (2b)
utilized in MLP and superconducting synapse enabling
signal transfer with both positive and negative coefficients.
Unlike most of their predecessors,
4–9,11,12
both cells are oper-
ating in a pure superconducting mode featured by minimal
power consumption.
A. Artificial neuron
One of the simplest superconducting cells is parametric
quantron proposed in 1982 for adiabatic operation.
16
It is the
superconducting loop consisted of a Josephson junction and
a superconducting inductance. According to the Josephson
junction CPR (1), the relation between the input magnetic
flux and the Josephson junction phase in its circuit has a
simple expression:
w
þ l sin
w
¼
f
in
, (3)
where we use normalization of current to critical current of
the Josephson junction, I
c
, and input magnetic flux Φ
in
to the
magnetic flux quantum Φ
0
,
f
in
¼ 2πΦ
in
=Φ
0
, inductance, L,
is normalized to characteristic inductance, l ¼ L=L
c
,
L
c
¼ Φ
0
=2πI
c
, accordingly.
It is seen from (1) and (3) that the current circulating in
the loop has a tilted sine dependence on input magnetic flux.
The way to transform this dependence close to the desired
one [(2a) or (2b)] is the addition of a linear term compensat-
ing the sine slope on the initial section (where sin
w
w
)in
the vicinity of zero input flux,
f
in
0.
This can be done by attaching another superconducting
loop with a part of its inductance, l
out
, being common with
the initial circuit [see Fig. 1(a)]. The synthesized cell was
named a “sigma-cell”
13
because its transformation of mag-
netic flux can be very close to sigmoid function. Here, we
are interested in a transfer function,
f
out
(
f
in
), where output
magnetic flux,
f
out
, is proportional to output current,
f
out
¼ l
out
i
out
.
The system of equations describing the proposed cell is
as follows:
w
þ l sin
w
¼
f
in
=2 þ l
out
i
out
, (4a)
w
þ l sin
w
¼
f
in
þ l
a
i
a
, (4b)
where l
a
is the attached inductance. The corresponding
system implicitly defining the transfer function through
dependencies of
f
out
,
f
in
on
w
has the following form:
f
out
¼ l
out
f
in
2l
a
sin
w
2(l
a
þ l
out
)
, (5a)
f
in
¼ 2
l
a
þ l
out
l
a
þ 2l
out
w
þ l þ
l
a
l
out
l
a
þ l
out
sin
w
: (5b)
Vanishing of the derivative d
f
out
=d
f
in
at
f
in
¼ 0 corre-
sponds to the condition:
l
a
¼ 1 þ l: (6)
One can fit (5) to sigmoid function (2a) taking (6) into
account with the two fitting parameters: l, l
out
.
The result of fitting is shown in Fig. 1(b). The found
optimal values, l ¼ 0:125, l
out
¼ 0:3, provide conformity of
the sigma-cell transfer function with sigmoid one with stan-
dard deviation at the level of 10
3
. Sigmoid function (2a)
was scaled as σ (1:173x) in our fitting process. The transfer
function
f
out
(
f
in
) (5) was normalized by 2πl
out
=(l
a
þ 2l
out
)
to fit a unit height and shifted by a half period. The latter can
be obtained by application of a constant bias flux to the
circuit,
f
b
¼2π(l
a
þ l
out
)=(l
a
þ 2l
out
).
While sigmoid activation function is commonly used for
input data defined in the positive domain, for data defined on
the whole numeric axis around zero, it is convenient to use
hyperbolic tangent. Application of additional bias flux provid-
ing π phase shift into the loop containing Josephson junction
moves the center of the nonlinear part of the cell transfer func-
tion to zero. This allows one to obtain the desired shape of
activation function (2b).Theπ phase shift can also be imple-
mented using the π–Josephson junction
17–20
with π shift of its
CPR (1), I ¼I
c
sin (
w
), instead of the standard one.
One needs to correspondingly change the sign of the
terms containing sine function in (5) to perform the fitting
FIG. 1. (a) Scheme of an artificial neuron cell. (b) The cell transfer function
(line) fitted to sigmoid and hyperbolic tangent functions (dots). Scaling of
the functions (2) is shown in the figure. The transfer function
f
out
(
f
in
)is
normalized by 2πl
out
=(l
a
þ 2l
out
) and shifted by 2π(l
a
þ l
out
)=(l
a
þ 2l
out
)
on the flux axis to fit (2a), and normalized to πl
out
=(l
a
þ 2l
out
) with no addi-
tional shift on flux axis to fit (2b). The optimal values of parameters are
l ¼ 0:125, l
out
¼ 0:3, l
a
¼ 1:125. Consistency of curves in both cases is at
the level of 10
3
. Hyperbolic tangent activation function is fitted with π shift
in the Josephson junction CPR (1).
152113-2 Soloviev et al. J. Appl. Phys. 124, 152113 (2018)
procedure. The fitting result is presented in Fig. 1(b).
Hyperbolic tangent function was scaled as tanh (0:586x)
while the transfer function
f
out
(
f
in
) was normalized by a
factor of two lower value than the previous time,
πl
out
=(l
a
þ 2l
out
). With the same values of parameters l, l
out
,
and zero bias flux, we obtained the same conformity of the
curves.
B. Artificial synapse
Synapse modulates the “weight” of a signal arriving at
the neuron. In our case, the signal corresponds to magnetic
flux and, therefore, synapse can be implemented simply as a
transformer of magnetic flux with desired coupling factor.
Summation of signals can be provided by connecting the
transformers to a single superconducting input loop of the
neuron. However, this solution suits for ANN with a certain
and unchangeable configuration.
In most cases, a configurable ANN would be preferable.
The selected configuration of inter-neuron connections
should be maintained during its entire use if the feature space
dimensions do not vary. However, the weight values should
be configurable if we want to train the ANN on the fly. The
best way to meet this requirement is utilization of some non-
volatile memory elements. In superconducting circuits, such
an element can be implement ed by using the ferromagnetic
(F) materials. In particular, introduction of F-layers into the
Josephson junction weak link area allows us to modulate its
critical current.
1,21,22
This phenomenon was already proposed
for utilization in artificial synapse of superconducting spiking
ANN.
12
In our case of MLP, we can also make use of it.
The synapse scheme presented in Fig. 2(a) is nearly a
mirrored scheme of the proposed neuron [Fig. 1(a)]. The
only differences are the addition of the second Josephson
junction and the possibility to independently modulate criti-
cal currents of the magnetic junctions (marked by boxes),
e.g., by application of tuning magnetic field.
For MLP, it is required to provide both positive and
negative weights of signal. Our synapse is designed accord-
ing to this requirement. The input current, i
in
, induced in
inductance l
in
by input magnetic flux,
f
in
, is split toward
the two Josephson junctions. Magnitude of currents i
1
, i
2
in each branch correspond to critical currents of the junc-
tions, i
c1
, i
c2
, so that the sign of output circulating current,
i
cir
¼ (i
1
i
2
)=2 (and the direction of output magnetic fl ux,
f
out
), is determined by their ratio. Maximum in equality of
i
c1
, i
c2
provides maximum output signal, while equal critical
currents correspond to zero transfer coefficient.
It is convenient to present the system of equations for
the synapse cell in terms of Josephson junctions phase sum,
w
þ
¼ (
w
1
þ
w
2
)=2, and phase difference,
w
¼ (
w
1
w
2
)=2:
w
þ
þ
l
2
þ l
in
i
in
þ
f
in
¼ 0, (7a)
w
þ li
cir
¼ 0: (7b)
Furthermore, introducing the sum Σi
c
¼ i
c1
þ i
c2
and differ-
ence Δi
c
¼ i
c1
i
c2
of the critical currents and taking (1) into
account one can represent (7) in the following form:
w
þ
þ
l
2
þ l
in
ðΣi
c
sin
w
þ
cos
w
þ Δi
c
sin
w
cos
w
þ
Þ
þ
f
in
¼ 0;
(8a)
w
þ
l
2
(Σi
c
sin
w
cos
w
þ
þ Δi
c
sin
w
þ
cos
w
) ¼ 0: (8b)
The dependence of the phase difference on the phase sum,
w
(
w
þ
), can be obtained
23,24
from (8b) with corresponding
function
f (
w
,
w
þ
) ¼
w
þ
l
2
(Σi
c
sin
w
cos
w
þ
þ Δi
c
sin
w
þ
cos
w
),
(9)
as follows:
w
¼
ð
πsgnΔi
c
0
H[ f (x;
w
þ
)sgn Δ i
c
]dx; (10)
where H(x) is the Heaviside step function. Equations (7a),
(8a), and (10) implicitly define the cell transfer function
FIG. 2. (a) Scheme of an artificial synapse cell. Magnetic Josephson junc-
tions are marked by boxes. (b) Synapse cell transfer function for the values
of parameters: l
in
¼ 2, l ¼ 4, Σi
c
¼ 1, and Δi
c
as shown in the figure.
Vertical dotted line shows the boundary of highly linear range where stan-
dard deviation from the linear function is at the level of 10
3
. This range
corresponds to maximum output magnetic flux of the optimized neuron cell.
152113-3 Soloviev et al. J. Appl. Phys. 124, 152113 (2018)
f
out
(
f
in
) through dependencies
f
out
¼ 2li
cir
¼2
w
(
w
þ
)
and
f
in
[
w
(
w
þ
),
w
þ
]on
w
þ
. Here, we are interested in the
range of the phase sum,
w
þ
[ [0, π=2), where the transfer
function might be linear.
Figure 2(b) shows synapse cell transfer function for dif-
ferent values of critical currents difference in the range
Δi
c
[ [ 0:9, 0:9]. The critical current sum is Σi
c
¼ 1. With
the fixed critical currents, the shape of the transfer function is
determined by inductances l
in
, l.
In accordance with (7a), an increase in input inductance
l
in
increases the amplitude of nonlinearity of the dependence
of input current on input flux i
in
(
f
in
) making it more tilted.
This is in complete analogy with parametric quantron
scheme (3). The slope of the linear part of the transfer func-
tion is correspondingly decreased. However, this gives a
stretching of this linear part, which is of use for us, and con-
traction of the nonlinear part.
Increase in inductance l provides the same effect [see
(7a)]. At the same time, it increases the nonlinearity of the
dependence of output flux on phase sum [see (8b)] which
vice versa increases the slope of the linear part though
making it less linear. The goal of optimization of the transfer
function
f
out
(
f
in
) is the maximum modulation of its slope
alongside with the high linearity among the possibly wider
range of input flux.
In our case, the values of inductances were chosen to be
l
in
¼ 2, l ¼ 4. With these parameters magnetic flux can be
transferred through the synapse with coefficients in the range
(0:5, 0:5) depending on the critical currents difference.
For maximum output magnetic flux of optimized neuron,
2πl
out
=(l
a
þ l
out
) 1:1, maximum standard deviation of the
synapse transfer function from the linear function is at the
level of 10
3
. In the whole shown range [0, π], it is of an
order of magnitude worse.
III. DISCUSSION
Both considered cells operate in a pure superconducting
regime. Evolution of their states is fully physically reversible.
Therefore, they can be operated adiabatically with energy per
operation down to the Landauer limit.
2
For standard working
temperature of superconducting circuits, T ¼ 4:2 K, this limit
corresponds to the energy, k
B
T ln 2 4 10
23
J (where k
B
is the Boltzmann constant). Estimations show that the bit
energy can be as low as 10
21
J for adiabatic superconductor
logic at clock frequency of 10 GHz.
25
This is million times
less than characteristic energy consumed by a semiconductor
transistor. In one hand, taking into account the fact that
modern implementation of neuron based on complementary-
metal-oxide semiconductor (CMOS) technology requires a
few dozens of transistors, the possible gap between power
consumption of semiconductor and superconductor ANN is
increased by an order. On the other hand, penalty for super-
conducting circuits cooling is typically several hundred W/W
that cancels out the two to three orders of supremacy.
Nevertheless, the proposed adiabatic superconducting ANN
can be up to 10
4
–10
5
times more energy efficient than its
semiconductor counterparts.
One should note some peculiarities of the proposed
concept. First of all, there is no power supply in these circuits
and so the signal vanishes. Therefore, there is a need for a
flux amplifier which can be implemented on a base of some
standard adiabatic cell like adiabatic quantum flux parame-
tron (AQFP).
1,26
However, such aspects as the linearity of
amplification, the distance of signal propagation without
amplification, and related issues of achievable fan-in and
fan-out should be additionally considered.
Another feature is the periodicity of sigma-cell based
neuron transfer function. Corresponding issues can be miti-
gated by a signal normalization.
Along with the use of standard superconducting inte-
grated circuits fabrication process, the proposed cells require
utilization of magnetic Josephson junctions which are rela-
tively new to superconducting technology. Nevertheless,
modern developments of cryogenic magnetic memory
1,27
and superconducting logic circuits with controlled functional-
ity
28,29
promise their fast introduction.
In particular case of the proposed synapse, one could
benefit from implementation of the magnetic Josephson junc-
tion controlled by direction of magnetic field, like the
Josephson magnetic rotary valve
30
with heterogeneous area
of weak link. Such a valve is featured by high critical current
for a certain direction of its F-layer magnetization and low
critical current for the direction rotated by 90
. Two such
junctions in close proximity to each other with mutual rota-
tion on 90
relative to their axes directed along the boundary
of inhomogeneity allow one to obtain high critical current for
one junction and low critical current for another one with the
same direction of magnetizations of their F-layers. In this
case, rotation of their magnetizations leads to a correspond-
ing decrease and increase of Josephson junction’s critical
currents which means modulation of synapse weight, accord-
ing to Fig. 2. Utilization of the rotary valve reduces the
number of control lines required to program the magnetic
Josephson junctions by half. However, their total number,
which is twice the number of synapses, remains huge for
practical ANNs. Therefore, the effective synapse control
is another urgent task on the way to multilayer adiabatic
superconducting ANN.
IV. CONCLUSION
In this paper, we considered operation principles of
adiabatic superconducting basic cells for implementation of
multilayer perceptron. These are artificial neuron and synapse
which are nonlinear and close-to-linear superconducting trans-
formers of magnetic flux, respectively. Both cells are capable
of operation in the adiabatic regime featured by ultra-low
power consumption at the level of 4 to 5 orders of magnitude
less than that of their modern semiconductor counterparts
(including cooling power penalty). The proposed neuron cell
contains just a single Josephson junction. The neuron provides
one-shot calculation of either sigmoid or hyperbolic tangent
activation function. The certain type of this function is deter-
mined by the type of utilized Josephson junction and can also
be switched on the fly by application of magnetic flux. The
synapse is implemented with two magnetic Josephson
152113-4 Soloviev et al. J. Appl. Phys. 124, 152113 (2018)