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)