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5.5 QUANTUM GATES
In quantum computation, there are circuits that make and carry out compu-
tation processes. A quantum logic gate is a function performed by a unitary
operator in a set of selected qubits at a certain period of time. In a classical
way, the logic gates establish an infinite set, but in quantum computation
the number of possible unitary transformations is also the same, and as a
consequence there are infinite quantum gates [9]. Few examples of fre-
quently encountered quantum gates are the Hadamard gate H , the not gate
X , the Pauli- Z gate Z , and the p /8 gate.
Is possible to prove [9-12] that in any unitary transformation, a set of
n qubits can be performed by successive application of only two quantum
gates: the XOR operation and the
rotation.
The XOR operator is a particular case of any unitary transformation on
only one qubit. XOR can be described as
ˆ (, )
R
θφ
I
U .
(12)
XOR
=
0
0
+
1 1
Namely, while the first qubit remains with no change, the second is ap-
plied to I or U depending on the state of the first. Specifically, the gate
XOR turns the ordered sequence of states x
xy
y into x , where
identifies the logic O-exclusive operation. On the other hand, the rotation
operation ˆ (, )
can be described as
R
θφ
R ( q , f )
(
[
]
[
]
) + 1
(
[
]
[
]
)
.
(13)
=
0
cos q /2
0
ie i q sin q /2
1
ie i q sin q /2
1
+
cos q /2
0
In this case, it is necessary that
θ
and φ are real; this is in order to get
ˆ (, )
a
transformation through the continuous use of the same quantum
gate with these values perfectly defi ned.
R
θφ
5.6 QUANTUM PARALLELISM
The feature that has turned quantum computation into one of the most
promising areas is the power to support in the shortest time problems that
result, and are intractable for the classical computation, by processing in-
formation in a parallel way. This idea was presented by David Deutsch
[13] in 1985 and is outlined as follows: Suppose we have a quantum com-
puter that is able to evolve any two qubits in accordance with the unitary
operator U whose transformation is represented by
 
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