t b a b a

c
t
c
t ⊕ c
c c
Figure 2.3: Circuit representations of control-NOT and control-U gate
representation is shown in Figure 2.3. The top line represents the control qubit while
the bottom line represents the target qubit. This gate leaves the target qubit unchanged
if the control qubit is set to | 0〉, and flips the target qubit if the control qubit is set to
| 1〉. That corresponds to the state transformations:
| 00〉 →| 00〉; | 01〉 →| 01〉; | 10〉 →| 11〉; | 11〉 →| 10〉.
The gate’s matrix representation is:
⎛
U
⎞
⎜ 1
⎜ 0
UCNOT = ⎜ 0
⎜
⎝
0
1
0
0
0
0
0 ⎟
0
⎟
1
⎟
⎠
0 0 1 0
The function of a CNOT gate can also be summarized as | c, t〉 →| c, c ⊕ t〉, where ⊕
denotes addition modulo two.
• Controlled-U gate
18