Explain the CNOT (Controlled-NOT) gate.

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The CNOT (Controlled-NOT) gate is one of the most important two-qubit quantum gates, often used to create entanglement in quantum circuits. It is the quantum version of a conditional operation.

🔑 Definition

• CNOT has two qubits:

  1. Control qubit → decides whether the operation is applied.

  2. Target qubit → is flipped if the control qubit is |1⟩.

🧩 Action on States

• If control = |0⟩ → do nothing (target unchanged).

• If control = |1⟩ → apply X gate (NOT) to target.

Truth Table:

Control (C)	Target (T)	Output (C, T)
|0⟩	|0⟩	|0,0⟩
|0⟩	|1⟩	|0,1⟩
|1⟩	|0⟩	|1,1⟩
|1⟩	|1⟩	|1,0⟩

⚡ Matrix Representation

In 4×4 form:

$$CNOT = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix}$$

This matrix flips the second qubit only when the first is |1⟩.

📌 Applications

1. Entanglement – Applying CNOT after a Hadamard gate creates a Bell state:

   $$H\|0⟩ \otimes \|0⟩ \xrightarrow{CNOT} \frac{1}{\sqrt{2}}(\|00⟩ + \|11⟩)$$

2. Quantum Error Correction – Detecting and correcting qubit errors.

3. Quantum Teleportation – Essential step in transferring qubit states.

4. Universal Gate Sets – Along with single-qubit gates (X, H, etc.), CNOT is required to build any quantum circuit.

👉 In summary:
The CNOT gate is a conditional quantum gate that flips the target qubit only when the control qubit is |1⟩. It is fundamental for entanglement and is a building block for universal quantum computation.

Read More :

Explain the Hadamard (H) gate.

What is the Pauli-Z gate?

What is the difference between classical NOT gate and quantum X gate?

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