CNOT Gate

The standard two-qubit entangling gate. Flips the target qubit if and only if the control qubit is |1>.

The CNOT (Controlled-NOT) gate is the most important two-qubit gate in quantum computing. It’s essential for creating entanglement and is part of every universal gate set.

Definition

The CNOT operates on two qubits: a control and a target.

$$\text{CNOT}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right)$$

Action

If control is $|0⟩$: target unchanged If control is $|1⟩$: target flipped (X gate applied)

Shorthand: $|c,t⟩\to |c,t\oplus c⟩$ where $\oplus$ is XOR.

Circuit Symbol

Control: ──●──
           │
Target:  ──⊕──

The dot (●) marks the control; the circle-plus (⊕) marks the target.

Creating Entanglement

CNOT is the standard way to create entangled states:

|0⟩ ──H────●────  →  |Φ+⟩ = (|00⟩ + |11⟩)/√2
           │
|0⟩ ───────⊕────

1. Start: $|00⟩$
2. After H: $\frac{1}{\sqrt{2}}(|0⟩+|1⟩)|0⟩=\frac{1}{\sqrt{2}}(|00⟩+|10⟩)$
3. After CNOT: $\frac{1}{\sqrt{2}}(|00⟩+|11⟩)$

This is a Bell state, maximally entangled!

Properties

Self-Inverse

$${\text{CNOT}}^2=I$$

Relation to Other Gates

$$\text{CNOT}=|0⟩⟨0|\otimes I+|1⟩⟨1|\otimes X$$

Can be written as controlled-X: $\text{CX}$

Basis Changes

With Hadamards, control and target can be swapped: $$(H\otimes H)\cdot \text{CNOT}\cdot (H\otimes H)=\text{CNOT reversed}$$

Physical Implementation

CNOT is a native gate on some platforms:

• Superconducting: Cross-resonance or iSWAP-based
• Trapped ions: Mølmer-Sørensen or XX gates
• Photonic: Requires non-linearities or measurement

CNOT fidelity is a key benchmark for quantum hardware.

Universal Computation

CNOT + single-qubit gates = universal quantum computing. Any quantum operation can be decomposed into CNOTs and single-qubit rotations.

See also: Entanglement, Quantum Gate, Bell State, Universal Gate Set, Toffoli Gate