e-Lecture : Introctory Quantum Information

Taksu Cheon
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Quantum Information for Quantum Cats (27)

Workings of Deutsch-Josza Alririthm : One Qubit

As an excersize, we look into the detail workings of the Deutsch-Josza algorithm in the example of one qubit to one qubit oracle. The four unitary matrices U_a, U_b, U_c, and U_d are given by

U_a =
[I
 0]
[0
 I]

U_b =
[ I
 0]
[0
 X]

U_c =
[X
 0]
[0
 I]

U_d =
[X
 0]
[0
 X]
We have
W =
[Z
 0]
[0
 Z]
So, the quantum phase oracle V_f for f=a, b, c, and d are given by


[ 1
0
0
0 ]
V_a =
[ 0
-1
0
0 ]
[ 0
0
1
0 ]
[ 0
‚O
0
-1 ]

[ 1
0
0
0 ]
V_b =
[ 0
-1
0
0 ]
[ 0
0
-1
0 ]
[ 0
‚O
0
1 ]

[ -1
0
0
0 ]
V_c =
[ 0
1
0
0 ]
[ 0
0
1
0 ]
[ 0
‚O
0
-1 ]
[ -1
0
0
0 ]
V_b =
[ 0
1
0
0 ]
[ 0
0
-1
0 ]
[ 0
‚O
0
1 ]

Considering the state

| \psi > = (H | 0 >) | 0 >, namely
[1]
| \psi > =1/\sqrt2
[0]
[1]
[0]

we obtain

< \psi | V_a | \psi > = 1 , < \psi | V_a | \psi > = 0
< \psi | V_c | \psi > = 0 , < \psi | V_d | \psi > = -1

whose squares give P. This furnishes the proof of the assertion of the last section. Readers are encouraged to extend this to the sisteen oracles that appear in the two qubit case.
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