Applied Physics CSIE II: UNIT IV: Quantum Computing
CNOT Gate (Controlled NOT Gate)
Definition, Action, Matrix form, Operation, Bloch Sphere action | Quantum gates
CNOT gate (Controlled‒NOT gate) is a two‒qubit quantum gate that flips the target qubit only when the control qubit is in state |1>.
CNOT GATE (CONTROLLED
NOT GATE)
Definition
CNOT
gate (Controlled‒NOT gate) is a two‒qubit quantum gate that flips the target
qubit only when the control qubit is in state |1>.
It
basically implements a reversible EX‒OR. It can be used to generate
entanglement.
Circuit Symbol
Operation
|c, t > → |c, t 
c>
where
c is the control, t is the target, and is XOR.
If
control = |0>: target stays the same
If
control = |1>: target flips |0> ↔ |1>
It
is the quantum version of a conditional NOT operation and is essential for
creating entanglement.
Truth Table
The
Truth Table for CNOT is shown below
Flipping
happens only when c=1
Matrix form
CNOT
is a 4 x 4 matrix
In
the basis: |00>, |01>, |10>, |11>.
Action on Computational basis
CNOT
| 00>= |00>
CNOT
| 01> = |01>
CNOT
| 10> = |11>
CNOT
| 11> = |10>
CNOT Input and Output on General state
Given input:
|Ψ >= a | 00> + b | 01 >+ c | 10>+d
|11>
Output:
CNOT | Ψ >=a | 00> + b | 01
>+c|11>+d|10>
Bloch Sphere action
1. Control qubit = |0>
If
the control qubit is |0>, nothing happens to the target qubit.
So
when control = 0
|0>c 
| Ψ >t
→ |0 >c | Ψ >t
2. Control qubit = |1>
If
the control qubit is |1>, the gate applies NOT (Pauli‒X) on the target qubit.
So
when control = 1:
|1>c |Ψ >t
→ |1>c | Ψ>t
Since,
CNOT affects the target only conditionally, the Bloch‒sphere effect is not representable
as a single rotation for the whole 2‒qubit system.
Applied Physics CSIE II: UNIT IV: Quantum Computing : Tag: Applied Physics : Definition, Action, Matrix form, Operation, Bloch Sphere action | Quantum gates - CNOT Gate (Controlled NOT Gate)
Applied Physics CSIE II: UNIT IV: Quantum Computing
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