Learning about quantum CZ gate

Introduction

This time we learn a little about CZ gate.

CZ gate?

CZ gate has controlled gate and target gate. If the controlled gate is 1, qubit on target gate was applied Z gate.

a -----*-----
       |
b -----Z-----

It is realized by applying H gate on CX gate.

a -----*-----
       |
b ---H-X-H---

CZ is reversible.

CZ gate is reversible on controlled gate and target gate.

a -----*-----     a -----Z-----
       |       =         |
b -----Z-----     b -----*-----

so,

a -----*-----
       |
b -----*-----

Usually we can write it like that.

Check the Unitary operator

CZ unitary operator is

CZ = [[1  0  0  0]
      [0  1  0  0]
      [0  0  1  0]
      [0  0  0 -1]]

If we apply state vector [a,b,c,d], we get [a’,b’,c’,d’]

[a'  =  [[1  0  0  0]  @ [a
 b'      [0  1  0  0]     b
 c'      [0  0  1  0]     c
 d']     [0  0  0 -1]]    d]

The cols are input and rows are output

a  b  c  d
[[1  0  0  0]  a' 
 [0  1  0  0]  b'
 [0  0  1  0]  c'
 [0  0  0 -1]] d'

The state vector in other expression

|00> |01> |10> |11>
[[1   0    0    0]  |00>' 
 [0   1    0    0]  |01>'
 [0   0    1    0]  |10>'
 [0   0    0   -1]] |11>'

CZ gate affect only the both qubits are 1.

CZ(01)(a|00>+b|01>+c|10>+d|11>) = a|00>+b|01>+c|10>-d|11>

CZ(10)(a|00>+b|01>+c|10>+d|11>) = a|00>+b|01>+c|10>-d|11>