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For a quantum circuit Born machine (QCBM) on
n
n
n
qubits using circuit
U
(
θ
)
U(\theta)
U
(
θ
)
on
∣
0
⟩
n
|0\rangle^n
∣0
⟩
n
, what gives the probability of observing bitstring
x
x
x
?
p
θ
(
x
)
=
1
/
2
n
p_\theta(x) = 1/2^n
p
θ
(
x
)
=
1/
2
n
for every
x
x
x
, since a parameterized circuit always outputs the uniform distribution
p
θ
(
x
)
=
⟨
x
∣
U
(
θ
)
∣
0
⟩
n
p_\theta(x) = \langle x| U(\theta) |0\rangle^n
p
θ
(
x
)
=
⟨
x
∣
U
(
θ
)
∣0
⟩
n
, the raw amplitude without squaring
p
θ
(
x
)
=
p_\theta(x) =
p
θ
(
x
)
=
a separately trained classical likelihood model fit to the circuit's samples
p
θ
(
x
)
=
∣
⟨
x
∣
U
(
θ
)
∣
0
⟩
n
∣
2
p_\theta(x) = |\langle x| U(\theta) |0\rangle^n|^2
p
θ
(
x
)
=
∣
⟨
x
∣
U
(
θ
)
∣0
⟩
n
∣
2
, directly from the Born rule -- the circuit itself is the distribution
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