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ˆ
(14)
Uxy
=
xyf
().
x
This operator changes the second y qubit to zero if f is 1 during the first
x qubit; if f is 0 1 no change is seen. Considering y
1
2
(
)
=
0
1
ˆ
Uxy
=
xyf
().
x
(15)
where the function f has been isolated in an x phase-dependent. If
x
1
2
1
2
(
)
(
)
=
0
+
1
y
=
0
1
and
is immediately checked that
1
1
1
1
ˆ :
(
)
(
)
(
)
()
f
(0)
()
f
(1)
(16)
U
0
1
0
+
1
1
0
+ −
1
1
0
1
.
2
2
2
2
In this way, it is only necessary to make an orthogonal projection of the
first qubit in a
{
}
+
,
base, where
1
(
)
±=
01.
±
(17)
2
The problem proposed in Eq. (14) has been resolved with an only one
computation. This is possible because at the moment of using a quantum
gadget, this has no limit to evaluate f (0) or f (1) , but instead, acting on
a superposition of the states 0 and 1 allowing to take out global in-
formation of function, namely, information that depends on combination
between f (0) and f (1) . This is the quantum parallelism.
Similarly, parallel processing can be used to identify some properties
of more complex functions. Thus, if we want to calculate all the possible
combinations on a set of n bits, the value of whichever function f , a
classical computer will need to perform 2 n evaluations of the function.
However, using a quantum computer, we only need to perform on an n
given set: only one transformation given by the unitary operator
U r , in
which case
ˆ
r Ux
:
0
xf
( ) .
x
(18)
The operation
denotes the binary addition or module 2, which is defined by
0
⊕=
0
0, 0
⊕=
1
1, 1
⊕=
0
1
and 11 0.
⊕=
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