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u
an
tu
m
tech
n
o
lo
g
ies
[
1
9
-
22]
.
W
eb
p
l
atf
o
r
m
is
ch
o
s
en
f
o
r
th
e
r
ea
liz
atio
n
o
f
th
e
q
u
an
tu
m
alg
o
r
ith
m
s
to
allo
w
ea
s
e
o
f
u
s
e
an
d
ac
ce
s
s
.
T
h
e
q
u
a
n
tu
m
tech
n
o
l
o
g
ies
th
at
a
r
e
u
s
ed
f
o
r
th
e
web
-
a
p
p
r
ea
lizatio
n
a
r
e
R
ig
etti
Fo
r
est
f
o
r
Gr
o
v
er
’
s
q
u
an
t
u
m
s
ea
r
c
h
al
g
o
r
ith
m
an
d
E
T
H
Z
u
r
ic
h
Pr
o
jectQ
f
o
r
Sh
o
r
’
s
q
u
an
tu
m
f
ac
to
r
in
g
alg
o
r
ith
m
.
T
h
e
two
q
u
an
tu
m
co
m
p
u
tin
g
tech
n
o
lo
g
ies
ar
e
ch
o
s
en
d
u
e
to
th
eir
s
u
p
p
o
r
t
an
d
av
ailab
ilit
y
f
o
r
g
e
n
er
al
u
s
er
s
:
d
o
cu
m
e
n
tatio
n
an
d
e
x
am
p
les.
B
o
th
Fo
r
est
an
d
Pro
jectQ
u
s
e
Py
th
o
n
as
th
eir
p
r
o
g
r
am
m
i
n
g
lan
g
u
ag
e,
h
en
ce
Flas
k
m
icr
o
web
f
r
am
ew
o
r
k
co
u
ld
b
e
u
s
ed
f
o
r
th
e
web
-
a
p
p
d
e
v
elo
p
m
e
n
t
s
in
ce
it
is
also
wr
itten
in
Py
th
o
n
.
T
h
e
p
er
f
o
r
m
a
n
ce
o
f
th
e
web
-
a
p
p
r
ea
lizatio
n
in
te
r
m
s
o
f
th
e
e
x
ec
u
tio
n
tim
e
is
co
m
p
a
r
ed
with
th
e
QC
P’s
r
esu
lt
u
n
d
er
t
h
e
s
am
e
s
im
u
latio
n
s
ce
n
ar
io
an
d
p
ar
a
m
eter
s
.
2.
RE
S
E
ARCH
M
E
T
H
O
D
T
h
e
r
esear
ch
m
eth
o
d
s
u
s
ed
ar
e
liter
atu
r
e
r
ev
iews,
d
esig
n
an
d
im
p
lem
en
tatio
n
,
an
d
te
s
tin
g
an
d
ev
alu
atio
n
.
T
h
e
liter
atu
r
e
r
ev
iews
in
clu
d
e
q
u
a
n
tu
m
b
i
t,
u
n
iv
er
s
al
q
u
an
tu
m
g
ate,
Sh
o
r
’
s
q
u
an
tu
m
f
ac
to
r
i
n
g
alg
o
r
ith
m
[
2
3
]
,
Gr
o
v
e
r
’
s
q
u
a
n
tu
m
s
ea
r
ch
alg
o
r
ith
m
,
Qu
a
n
tu
m
C
o
m
p
u
tin
g
Play
g
r
o
u
n
d
,
E
T
H
Z
u
r
ich
Pro
jectQ
f
r
am
ewo
r
k
,
an
d
R
ig
etti
Fo
r
est
SDK.
T
h
e
d
esig
n
o
f
th
e
im
p
l
em
en
tatio
n
o
f
t
h
e
q
u
a
n
tu
m
alg
o
r
ith
m
is
d
escr
ib
ed
u
s
in
g
a
f
lo
wch
a
r
t.
I
n
t
h
is
r
esear
ch
,
th
e
q
u
an
tu
m
er
r
o
r
co
r
r
ec
ti
o
n
an
d
an
cilla
q
u
b
its
ar
e
n
o
t ta
k
en
in
to
ac
c
o
u
n
t.
T
h
e
web
-
ap
p
r
ea
lizatio
n
o
f
th
e
Sh
o
r
’
s
q
u
a
n
tu
m
f
ac
t
o
r
in
g
al
g
o
r
ith
m
is
d
o
n
e
u
s
in
g
t
h
e
E
T
H
Z
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r
ich
Pro
jectQ
f
r
am
ewo
r
k
an
d
Flas
k
.
Me
an
wh
ile
f
o
r
G
r
o
v
e
r
’
s
q
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an
tu
m
s
ea
r
ch
alg
o
r
ith
m
is
d
o
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e
u
s
in
g
R
ig
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r
est
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th
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y
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il
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r
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r
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d
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wo
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m
p
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ter
s
with
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if
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er
en
t
h
a
r
d
war
e
s
p
ec
if
icatio
n
s
ar
e
u
s
ed
f
o
r
th
e
im
p
lem
e
n
tatio
n
p
ar
t
.
T
h
e
test
in
g
is
d
o
n
e
u
s
in
g
th
e
wh
ite
-
b
o
x
test
in
g
m
e
th
o
d
o
lo
g
y
an
d
th
e
p
er
f
o
r
m
an
ce
o
f
th
e
we
b
-
ap
p
i
s
m
ea
s
u
r
ed
b
y
th
e
e
x
ec
u
tio
n
t
im
e
an
d
co
m
p
ar
ed
with
th
e
Q
u
an
tu
m
C
o
m
p
u
tin
g
Play
g
r
o
u
n
d
u
n
d
er
th
e
s
am
e
s
im
u
latio
n
s
ce
n
ar
io
a
n
d
p
ar
am
e
ter
s
as p
r
esen
ted
in
[
2
4
,
2
5
]
.
2
.
1
.
Sh
o
r’
s
im
plem
ent
a
t
io
n desi
g
n
Fig
u
r
e
1
s
h
o
ws
Sh
o
r
’
s
q
u
a
n
t
u
m
f
ac
to
r
in
g
alg
o
r
ith
m
f
l
o
wch
ar
t.
First,
th
e
u
s
er
is
ask
e
d
to
in
p
u
t
a
p
o
s
itiv
e
in
teg
er
v
alu
e
i.e
.
N.
T
h
e
in
p
u
t
is
ev
alu
ated
an
d
r
e
tu
r
n
ed
with
tr
u
e
if
N
m
o
d
u
lo
b
y
2
eq
u
als
0
,
an
d
r
etu
r
n
s
f
alse
o
th
er
wis
e.
T
h
e
n
ex
t
p
r
o
ce
s
s
is
th
e
in
s
p
ec
tio
n
p
r
o
ce
s
s
if
N
is
p
r
im
e
o
r
n
o
t.
Sam
e
as
th
e
p
r
ev
io
u
s
p
r
o
ce
s
s
,
th
is
p
r
o
ce
s
s
will
r
etu
r
n
tr
u
e
o
r
f
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ep
e
n
d
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g
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n
th
e
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al
u
e
o
f
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e
n
ter
ed
.
I
f
th
e
v
alu
e
o
f
N
ca
n
b
e
u
s
ed
u
p
m
o
d
u
lated
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y
a
n
u
m
b
er
s
m
aller
th
an
its
elf
,
th
is
p
r
o
ce
s
s
will
r
etu
r
n
a
f
alse
v
alu
e.
Fu
r
th
er
m
o
r
e,
th
e
v
alu
e
o
f
th
e
v
a
r
iab
le
q
is
d
eter
m
in
ed
b
y
f
in
d
i
n
g
a
r
a
n
k
o
f
2
th
at
is
g
r
ea
ter
o
r
eq
u
al
to
th
e
v
alu
e
o
f
N
2
.
T
h
is
p
r
o
ce
s
s
will
ad
d
o
n
e
t
o
th
e
p
o
wer
v
ar
iab
le
u
n
til
a
d
is
p
lace
m
en
t
o
f
two
is
g
r
ea
te
r
th
an
t
h
e
t
ar
g
et.
T
h
is
p
r
o
ce
s
s
f
lo
wch
ar
t
ca
n
b
e
s
ee
n
in
Fig
u
r
e
2
n
o
tated
b
y
o
f
f
-
p
ag
e
r
ef
er
en
ce
1
.
Af
ter
th
at,
th
e
v
alu
e
o
f
th
e
v
ar
iab
le
x
is
d
eter
m
in
ed
r
an
d
o
m
ly
.
T
h
e
s
u
b
-
q
u
a
n
tu
m
r
o
u
tin
e
o
f
Sh
o
r
'
s
q
u
an
tu
m
f
ac
to
r
in
g
alg
o
r
ith
m
will
b
e
r
u
n
if
x
is
co
p
r
im
e
with
N
an
d
N
is
n
o
t p
r
im
e
n
o
r
ev
en
.
On
-
p
ag
e
r
ef
e
r
en
ce
C
d
escr
ib
e
s
th
e
in
itializatio
n
p
r
o
ce
s
s
o
f
v
ar
iab
les
wh
ich
will
later
b
e
u
s
ed
in
th
e
q
u
an
tu
m
co
m
p
u
tatio
n
p
r
o
ce
s
s
wh
ich
i
s
n
o
tated
b
y
th
e
o
n
-
p
ag
e
r
ef
e
r
en
ce
D
in
Fig
u
r
e
2
.
T
h
e
n
v
alu
e
in
th
is
p
r
o
ce
s
s
s
to
r
es
th
e
b
it
s
ize
n
e
ed
ed
to
r
e
p
r
esen
t
in
te
g
er
s
u
p
to
t
h
e
v
ar
iab
le
q
m
in
u
s
o
n
e.
Nex
t,
t
h
e
q
u
an
tu
m
r
eg
is
ter
is
in
itialized
to
s
ize
n
,
th
e
b
it
s
ize
p
r
ev
i
o
u
s
ly
ca
lcu
lated
.
T
h
is
q
u
a
n
t
u
m
r
e
g
is
ter
will
th
en
b
e
p
ass
ed
th
r
o
u
g
h
th
e
Ha
d
am
ar
d
g
ate
to
en
ter
th
e
s
u
p
er
p
o
s
itio
n
s
tate
o
f
all
n
u
m
b
er
s
f
r
o
m
ze
r
o
to
q
m
in
u
s
o
n
e.
A
q
u
b
it
will
also
b
e
in
itialized
as
a
ct
r
l_
q
u
b
it
v
ar
iab
le.
T
h
e
m
ea
s
u
r
em
en
t
v
a
r
iab
les
ar
e
cr
ea
ted
a
s
n
-
s
ized
ar
r
ay
s
to
s
to
r
e
th
e
v
alu
e
o
f
m
o
d
u
lar
e
x
p
o
n
en
tiatio
n
at
a
later
s
tep
.
T
h
e
n
e
x
t
p
r
o
ce
s
s
is
m
o
d
u
l
ar
ex
p
o
n
e
n
tiatio
n
.
T
h
is
s
tep
is
r
ep
ea
ted
f
o
r
n
n
u
m
b
er
o
f
tim
es.
T
h
e
cu
r
r
e
n
t_
x
v
alu
e
is
f
illed
with
th
e
v
alu
e
f
r
o
m
t
h
e
ca
l
cu
latio
n
o
f
p
o
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(
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n
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1
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k
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wh
er
e
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r
o
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etitio
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m
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la
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o
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s
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ith
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ich
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h
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its
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u
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o
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th
e
im
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lem
en
tatio
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alg
o
r
ith
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in
Pro
jectQ
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r
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Fig
u
r
e
10
.
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o
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s
alg
o
r
ith
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i
m
u
latio
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o
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f
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.
2
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ro
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ith
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.
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h
ese
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s
ar
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in
itializatio
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,
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lific
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d
m
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u
r
e.
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g
u
r
e
1
1
s
h
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im
p
le
m
en
tatio
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co
d
e
f
o
r
in
itializatio
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in
R
ig
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Fo
r
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h
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itializa
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m
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k
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its
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n
th
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iece
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e,
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ate
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it.
Fig
u
r
e
1
2
s
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am
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e
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m
p
lific
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R
ig
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Fo
r
est.
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√
.
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e
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lied
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Fig
u
r
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11
.
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itio
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itializatio
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f
o
r
all
q
u
b
its
Fig
u
r
e
12
.
Am
p
litu
d
e
am
p
lific
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co
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e
T
h
e
Or
ac
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f
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n
ctio
n
is
n
o
t
p
a
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th
e
Gr
o
v
er
’
s
q
u
an
tu
m
s
ea
r
ch
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o
r
ith
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.
Fig
u
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e
1
3
s
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iec
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co
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at
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ig
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ally
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th
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Or
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th
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2
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im
en
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io
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al
ar
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y
.
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f
u
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atr
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ar
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Gr
o
v
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ter
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im
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ted
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ter
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ac
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.
Fig
u
r
e
1
4
s
h
o
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o
f
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ak
e
a
Dif
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u
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th
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io
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ar
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ay
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Fig
u
r
e
1
5
is
th
e
im
p
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en
tatio
n
o
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ag
o
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ith
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h
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T
h
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r
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en
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e
q
u
b
its
to
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llap
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e
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o
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e
o
f
its
eig
en
s
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h
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u
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en
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r
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e
s
to
r
ed
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e
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o
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wit
h
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ize
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al
to
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e
n
u
m
b
er
o
f
th
e
q
u
b
its
u
s
ed
.
Fig
u
r
e
1
6
s
h
o
ws th
e
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im
u
latio
n
r
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lt f
o
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Fig
u
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1
7
wh
er
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ataset
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s
1
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6
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2
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4
,
an
d
3
a
n
d
th
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et
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alu
e
is
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.
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u
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14
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15
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Fig
u
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16
.
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Fig
u
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17
.
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ith
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2
Evaluation Warning : The document was created with Spire.PDF for Python.