<s>
Quantum	B-Device
neural	I-Device
networks	I-Device
are	O
computational	O
neural	B-Architecture
network	I-Architecture
models	I-Architecture
which	O
are	O
based	O
on	O
the	O
principles	O
of	O
quantum	O
mechanics	O
.	O
</s>
<s>
However	O
,	O
typical	O
research	O
in	O
quantum	B-Device
neural	I-Device
networks	I-Device
involves	O
combining	O
classical	O
artificial	B-Architecture
neural	I-Architecture
network	I-Architecture
models	O
(	O
which	O
are	O
widely	O
used	O
in	O
machine	O
learning	O
for	O
the	O
important	O
task	O
of	O
pattern	O
recognition	O
)	O
with	O
the	O
advantages	O
of	O
quantum	O
information	O
in	O
order	O
to	O
develop	O
more	O
efficient	O
algorithms	O
.	O
</s>
<s>
One	O
important	O
motivation	O
for	O
these	O
investigations	O
is	O
the	O
difficulty	O
to	O
train	O
classical	O
neural	B-Architecture
networks	I-Architecture
,	O
especially	O
in	O
big	B-Application
data	I-Application
applications	I-Application
.	O
</s>
<s>
The	O
hope	O
is	O
that	O
features	O
of	O
quantum	B-Architecture
computing	I-Architecture
such	O
as	O
quantum	B-Architecture
parallelism	I-Architecture
or	O
the	O
effects	O
of	O
interference	O
and	O
entanglement	O
can	O
be	O
used	O
as	O
resources	O
.	O
</s>
<s>
Since	O
the	O
technological	O
implementation	O
of	O
a	O
quantum	B-Architecture
computer	I-Architecture
is	O
still	O
in	O
a	O
premature	O
stage	O
,	O
such	O
quantum	B-Device
neural	I-Device
network	I-Device
models	O
are	O
mostly	O
theoretical	O
proposals	O
that	O
await	O
their	O
full	O
implementation	O
in	O
physical	O
experiments	O
.	O
</s>
<s>
Most	O
Quantum	B-Device
neural	I-Device
networks	I-Device
are	O
developed	O
as	O
feed-forward	B-Algorithm
networks	I-Algorithm
.	O
</s>
<s>
This	O
structure	O
is	O
trained	O
on	O
which	O
path	O
to	O
take	O
similar	O
to	O
classical	O
artificial	B-Architecture
neural	I-Architecture
networks	I-Architecture
.	O
</s>
<s>
Quantum	B-Device
neural	I-Device
networks	I-Device
refer	O
to	O
three	O
different	O
categories	O
:	O
Quantum	B-Architecture
computer	I-Architecture
with	O
classical	O
data	O
,	O
classical	O
computer	O
with	O
quantum	O
data	O
,	O
and	O
quantum	B-Architecture
computer	I-Architecture
with	O
quantum	O
data	O
.	O
</s>
<s>
Quantum	B-Device
neural	I-Device
network	I-Device
research	O
is	O
still	O
in	O
its	O
infancy	O
,	O
and	O
a	O
conglomeration	O
of	O
proposals	O
and	O
ideas	O
of	O
varying	O
scope	O
and	O
mathematical	O
rigor	O
have	O
been	O
put	O
forward	O
.	O
</s>
<s>
Most	O
of	O
them	O
are	O
based	O
on	O
the	O
idea	O
of	O
replacing	O
classical	O
binary	O
or	O
McCulloch-Pitts	B-Algorithm
neurons	I-Algorithm
with	O
a	O
qubit	O
(	O
which	O
can	O
be	O
called	O
a	O
“	O
quron	O
”	O
)	O
,	O
resulting	O
in	O
neural	O
units	O
that	O
can	O
be	O
in	O
a	O
superposition	O
of	O
the	O
state	O
‘	O
firing’	O
and	O
‘	O
resting’	O
.	O
</s>
<s>
A	O
lot	O
of	O
proposals	O
attempt	O
to	O
find	O
a	O
quantum	O
equivalent	O
for	O
the	O
perceptron	B-Algorithm
unit	O
from	O
which	O
neural	B-Architecture
nets	I-Architecture
are	O
constructed	O
.	O
</s>
<s>
Ideas	O
to	O
imitate	O
the	O
perceptron	B-Algorithm
activation	O
function	O
with	O
a	O
quantum	O
mechanical	O
formalism	O
reach	O
from	O
special	O
measurements	O
to	O
postulating	O
non-linear	O
quantum	O
operators	O
(	O
a	O
mathematical	O
framework	O
that	O
is	O
disputed	O
)	O
.	O
</s>
<s>
A	O
direct	O
implementation	O
of	O
the	O
activation	O
function	O
using	O
the	O
circuit-based	B-Application
model	I-Application
of	I-Application
quantum	I-Application
computation	I-Application
has	O
recently	O
been	O
proposed	O
by	O
Schuld	O
,	O
Sinayskiy	O
and	O
Petruccione	O
based	O
on	O
the	O
quantum	B-Algorithm
phase	I-Algorithm
estimation	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
At	O
a	O
larger	O
scale	O
,	O
researchers	O
have	O
attempted	O
to	O
generalize	O
neural	B-Architecture
networks	I-Architecture
to	O
the	O
quantum	O
setting	O
.	O
</s>
<s>
One	O
way	O
of	O
constructing	O
a	O
quantum	O
neuron	O
is	O
to	O
first	O
generalise	O
classical	O
neurons	O
and	O
then	O
generalising	O
them	O
further	O
to	O
make	O
unitary	B-Algorithm
gates	O
.	O
</s>
<s>
Interactions	O
between	O
neurons	O
can	O
be	O
controlled	O
quantumly	O
,	O
with	O
unitary	B-Algorithm
gates	O
,	O
or	O
classically	O
,	O
via	O
measurement	O
of	O
the	O
network	O
states	O
.	O
</s>
<s>
This	O
high-level	O
theoretical	O
technique	O
can	O
be	O
applied	O
broadly	O
,	O
by	O
taking	O
different	O
types	O
of	O
networks	O
and	O
different	O
implementations	O
of	O
quantum	O
neurons	O
,	O
such	O
as	O
photonically	O
implemented	O
neurons	O
and	O
quantum	O
reservoir	O
processor	O
(	O
quantum	O
version	O
of	O
reservoir	B-Algorithm
computing	I-Algorithm
)	O
.	O
</s>
<s>
Most	O
learning	O
algorithms	O
follow	O
the	O
classical	O
model	O
of	O
training	O
an	O
artificial	B-Architecture
neural	I-Architecture
network	I-Architecture
to	O
learn	O
the	O
input-output	O
function	O
of	O
a	O
given	O
training	O
set	O
and	O
use	O
classical	O
feedback	O
loops	O
to	O
update	O
parameters	O
of	O
the	O
quantum	O
system	O
until	O
they	O
converge	O
to	O
an	O
optimal	O
configuration	O
.	O
</s>
<s>
Learning	O
as	O
a	O
parameter	O
optimisation	O
problem	O
has	O
also	O
been	O
approached	O
by	O
adiabatic	O
models	O
of	O
quantum	B-Architecture
computing	I-Architecture
.	O
</s>
<s>
Quantum	B-Device
neural	I-Device
networks	I-Device
can	O
be	O
applied	O
to	O
algorithmic	O
design	O
:	O
given	O
qubits	O
with	O
tunable	O
mutual	O
interactions	O
,	O
one	O
can	O
attempt	O
to	O
learn	O
interactions	O
following	O
the	O
classical	O
backpropagation	B-Algorithm
rule	O
from	O
a	O
training	O
set	O
of	O
desired	O
input-output	O
relations	O
,	O
taken	O
to	O
be	O
the	O
desired	O
output	O
algorithm	O
's	O
behavior	O
.	O
</s>
<s>
The	O
authors	O
do	O
not	O
attempt	O
to	O
translate	O
the	O
structure	O
of	O
artificial	B-Architecture
neural	I-Architecture
network	I-Architecture
models	O
into	O
quantum	O
theory	O
,	O
but	O
propose	O
an	O
algorithm	O
for	O
a	O
circuit-based	B-Application
quantum	I-Application
computer	I-Application
that	O
simulates	O
associative	O
memory	O
.	O
</s>
<s>
The	O
memory	O
states	O
(	O
in	O
Hopfield	B-Algorithm
neural	I-Algorithm
networks	I-Algorithm
saved	O
in	O
the	O
weights	O
of	O
the	O
neural	O
connections	O
)	O
are	O
written	O
into	O
a	O
superposition	O
,	O
and	O
a	O
Grover-like	B-Algorithm
quantum	I-Algorithm
search	I-Algorithm
algorithm	I-Algorithm
retrieves	O
the	O
memory	O
state	O
closest	O
to	O
a	O
given	O
input	O
.	O
</s>
<s>
An	O
advantage	O
lies	O
in	O
the	O
exponential	O
storage	O
capacity	O
of	O
memory	O
states	O
,	O
however	O
the	O
question	O
remains	O
whether	O
the	O
model	O
has	O
significance	O
regarding	O
the	O
initial	O
purpose	O
of	O
Hopfield	B-Algorithm
models	I-Algorithm
as	O
a	O
demonstration	O
of	O
how	O
simplified	O
artificial	B-Architecture
neural	I-Architecture
networks	I-Architecture
can	O
simulate	O
features	O
of	O
the	O
brain	O
.	O
</s>
<s>
A	O
substantial	O
amount	O
of	O
interest	O
has	O
been	O
given	O
to	O
a	O
“	O
quantum-inspired	O
”	O
model	O
that	O
uses	O
ideas	O
from	O
quantum	O
theory	O
to	O
implement	O
a	O
neural	B-Architecture
network	I-Architecture
based	O
on	O
fuzzy	O
logic	O
.	O
</s>
<s>
Quantum	B-Device
Neural	I-Device
Networks	I-Device
can	O
be	O
theoretically	O
trained	O
similarly	O
to	O
training	O
classical/artificial	O
neural	O
networks	O
.	O
</s>
<s>
A	O
key	O
difference	O
lies	O
in	O
communication	O
between	O
the	O
layers	O
of	O
a	O
neural	B-Architecture
networks	I-Architecture
.	O
</s>
<s>
For	O
classical	O
neural	B-Architecture
networks	I-Architecture
,	O
at	O
the	O
end	O
of	O
a	O
given	O
operation	O
,	O
the	O
current	O
perceptron	B-Algorithm
copies	O
its	O
output	O
to	O
the	O
next	O
layer	O
of	O
perceptron(s )	O
in	O
the	O
network	O
.	O
</s>
<s>
However	O
,	O
in	O
a	O
quantum	B-Device
neural	I-Device
network	I-Device
,	O
where	O
each	O
perceptron	B-Algorithm
is	O
a	O
qubit	O
,	O
this	O
would	O
violate	O
the	O
no-cloning	O
theorem	O
.	O
</s>
<s>
A	O
proposed	O
generalized	O
solution	O
to	O
this	O
is	O
to	O
replace	O
the	O
classical	O
fan-out	B-General_Concept
method	O
with	O
an	O
arbitrary	O
unitary	B-Algorithm
that	O
spreads	O
out	O
,	O
but	O
does	O
not	O
copy	O
,	O
the	O
output	O
of	O
one	O
qubit	O
to	O
the	O
next	O
layer	O
of	O
qubits	O
.	O
</s>
<s>
Using	O
this	O
fan-out	B-General_Concept
Unitary	B-Algorithm
(	O
)	O
with	O
a	O
dummy	O
state	O
qubit	O
in	O
a	O
known	O
state	O
(	O
Ex	O
.	O
</s>
<s>
This	O
process	O
adheres	O
to	O
the	O
quantum	O
operation	O
requirement	O
of	O
reversibility	B-Application
.	O
</s>
<s>
Using	O
this	O
quantum	O
feed-forward	B-Algorithm
network	I-Algorithm
,	O
deep	O
neural	B-Architecture
networks	I-Architecture
can	O
be	O
executed	O
and	O
trained	O
efficiently	O
.	O
</s>
<s>
A	O
deep	O
neural	B-Architecture
network	I-Architecture
is	O
essentially	O
a	O
network	O
with	O
many	O
hidden-layers	O
,	O
as	O
seen	O
in	O
the	O
sample	O
model	O
neural	B-Architecture
network	I-Architecture
above	O
.	O
</s>
<s>
Since	O
the	O
Quantum	B-Device
neural	I-Device
network	I-Device
being	O
discussed	O
utilizes	O
fan-out	B-General_Concept
Unitary	B-Algorithm
operators	I-Algorithm
,	O
and	O
each	O
operator	O
only	O
acts	O
on	O
its	O
respective	O
input	O
,	O
only	O
two	O
layers	O
are	O
used	O
at	O
any	O
given	O
time	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
no	O
Unitary	B-Algorithm
operator	I-Algorithm
is	O
acting	O
on	O
the	O
entire	O
network	O
at	O
any	O
given	O
time	O
,	O
meaning	O
the	O
number	O
of	O
qubits	O
required	O
for	O
a	O
given	O
step	O
depends	O
on	O
the	O
number	O
of	O
inputs	O
in	O
a	O
given	O
layer	O
.	O
</s>
<s>
Since	O
Quantum	B-Architecture
Computers	I-Architecture
are	O
notorious	O
for	O
their	O
ability	O
to	O
run	O
multiple	O
iterations	O
in	O
a	O
short	O
period	O
of	O
time	O
,	O
the	O
efficiency	O
of	O
a	O
quantum	B-Device
neural	I-Device
network	I-Device
is	O
solely	O
dependent	O
on	O
the	O
number	O
of	O
qubits	O
in	O
any	O
given	O
layer	O
,	O
and	O
not	O
on	O
the	O
depth	O
of	O
the	O
network	O
.	O
</s>
<s>
To	O
determine	O
the	O
effectiveness	O
of	O
a	O
neural	B-Architecture
network	I-Architecture
,	O
a	O
cost	O
function	O
is	O
used	O
,	O
which	O
essentially	O
measures	O
the	O
proximity	O
of	O
the	O
network	O
’s	O
output	O
to	O
the	O
expected	O
or	O
desired	O
output	O
.	O
</s>
<s>
In	O
a	O
Classical	O
Neural	B-Architecture
Network	I-Architecture
,	O
the	O
weights	O
(	O
)	O
and	O
biases	O
(	O
)	O
at	O
each	O
step	O
determine	O
the	O
outcome	O
of	O
the	O
cost	O
function	O
.	O
</s>
<s>
When	O
training	O
a	O
Classical	O
Neural	B-Architecture
network	I-Architecture
,	O
the	O
weights	O
and	O
biases	O
are	O
adjusted	O
after	O
each	O
iteration	O
,	O
and	O
given	O
equation	O
1	O
below	O
,	O
where	O
is	O
the	O
desired	O
output	O
and	O
is	O
the	O
actual	O
output	O
,	O
the	O
cost	O
function	O
is	O
optimized	O
when	O
=	O
0	O
.	O
</s>
<s>
For	O
a	O
quantum	B-Device
neural	I-Device
network	I-Device
,	O
the	O
cost	O
function	O
is	O
determined	O
by	O
measuring	O
the	O
fidelity	O
of	O
the	O
outcome	O
state	O
(	O
)	O
with	O
the	O
desired	O
outcome	O
state	O
(	O
)	O
,	O
seen	O
in	O
Equation	O
2	O
below	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
Unitary	B-Algorithm
operators	I-Algorithm
are	O
adjusted	O
after	O
each	O
iteration	O
,	O
and	O
the	O
cost	O
function	O
is	O
optimized	O
when	O
C	O
=	O
1	O
.	O
</s>
