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Fig. 1.
The inherent structure of qubit neuron
where
w
i
=[
w
j
1
,w
j
2
,...,w
jn
]
T
is the input weight vector, and
b
i
is the threshold
of the
i
th hidden node.
w
i
·
x
j
denotes the inner product of
w
i
and
x
j
. The linear
function is chosen as the activation function of the output nodes here.
And then, the state of the qubit is represented as
=cos(
π
+ sin(
π
|
ϕ
=cos(
θ
)
|
0
+ sin(
θ
)
|
1
2
u
)
|
0
2
u
)
|
1
(5)
When the neuron is triggered, the qubit state collapses into state
|
1
. The neuron
state
z
is the probability with which the qubit will be found in the state
|
1
.
z
=
f
(
θ
)=sin
2
(
θ
)=sin
2
[
π
2
(
w
i
·
x
j
+
b
i
)](
j
=1
,
2
,...,N
)
(6)
According to Equations (4)-(6), the
i
th hidden neuron output is given by
HID
i
=sin
2
[
π
2
(
w
i
·
x
j
+
b
i
)](
j
=1
,
2
,...,N
)
(7)
Finally, we obtain the network output for the
j
th sample:
N
N
β
i
sin
2
[
π
β
i
HID
i
=
2
(
w
i
·
x
j
+
b
i
)](
j
=1
,
2
,...,N
)
o
j
=
(8)
i
=1
i
=1
where
β
i
=[
β
i
1
,β
i
2
,...,β
im
]
T
is the output weight vector.
3.2 ELM Algorithm for QNN
For the feedforward neural network, gradient descent-based methods like back-
propagation (BP) algorithm [14] and evolutionary algorithms [15] are taken
as the traditional learning rule. However, these learning methods are time-
consuming. Comparatively, the ELM algorithm reaches the solutions straight-
forwardly. And ELM is not to take much long time to train the feedforward
network.
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