e-Lecture : Introctory Quantum Information

Taksu Cheon
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Quantum Information for Quantum Cats (20)

Quantum Circuit : Matrix Representation

We list matrix representations of main elements of quantum circuit.One qubit basis states in vector representation are
| 0 > =
[1]
[0]
| 1> =
[0]
[1]

One qubit identy I, .not. operator X, relative phase operation Z are

I =
[1
 0]
[0
 1]
X =
[0
 1]
[1
 0]
Z =
[1
 0 ]
[0
 -1]

These three matrices are symmetric with respect to the exchange of raw and column, and reduce to the identity when operated twice. The condition is a result of unitarity of quantum operations which is shared by all operations considered here. For the conaiseurs, the unitarity itself is a bit stringent: product of a matrix and its Hermitioan conjugate has to be one.

The two qubit basis states are
[1]
| 0 0 > =
[0]
[0]
[0]
[0]
| 0 1 > =
[1]
[0]
[0]
[0]
| 1 0 > =
[0]
[1]
[0]
[0]
| 1 1> =
[0]
[0]
[1]

on which Control-Not operation is expressed as
[ 1
0
0
0 ]
Cn =
[ 0
1
0
0 ]
[ 0
0
0
1 ]
[ 0
0
1
0 ]
or
Cn =
[ I
 O ]
[O
 X ]

with obvious two-by-two element notation in the right expression.

For three qubits states, with analogous 8 vector extension, the Control-Control-Not operation is expressed by the following eight-by-eight matrix Ccn. We also define phae operation W here:
[ I
O
O
O ]
Ccn =
[ O
I
O
O ]
[ O
O
I
O ]
[ O
O
O
X ]
[ Z
O
O
O ]
W =
[ O
Z
O
O ]
[ O
O
Z
O ]
[ O
O
O
Z ]

Further generalization should be straightforward although cumbersom.
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