# CNOT Gate (Controlled-NOT or CX)

**CNOT** is the most common two-qubit gate: flips the target qubit if the control qubit is $|1\rangle$. Essential for creating entanglement.

Matrix (control on qubit 0, target on qubit 1):

$$\text{CNOT} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix}$$

Action on basis states:

$$|00\rangle \to |00\rangle, \quad |01\rangle \to |01\rangle, \quad |10\rangle \to |11\rangle, \quad |11\rangle \to |10\rangle$$

## Bell States

CNOT creates Bell (maximally entangled) states:

$$\text{CNOT}(H \otimes I)|\psi\rangle = \frac{|00\rangle + |11\rangle}{\sqrt{2}} \quad \text{(if } |\psi\rangle = |00\rangle\text{)}$$

## Properties

- **Self-inverse**: $\text{CNOT}^2 = I$
- **CX notation**: CNOT and CX are the same (CX = controlled X)
- **Control/target interchangeable** (with Hadamards): $\text{CX}_{01} = (I \otimes H) \text{CZ}_{01} (I \otimes H)$

## Uses

- **Entanglement**: primary entangling gate on most quantum computers
- **Parity**: CNOT chains measure parity (XOR of qubits)
- **Quantum algorithms**: Deutsch-Jozsa, Grover, VQE, QAOA
- **Error correction**: enables quantum information distribution via entanglement

## Decomposition

- **From SWAPs**: $\text{SWAP} = \text{CX}_{01} \text{CX}_{10} \text{CX}_{01}$ (three CNOTs)
- **Performance note**: minimize CNOT count in circuit design (bottleneck for depth and fidelity)

## Implementation

- **Superconducting qubits**: parametric interaction (flux pulse) or resonant coupling; gate time ~20–100 ns; fidelity 98–99.5%
- **Trapped ions**: Mølmer-Sørensen or similar entangling laser pulses; gate time ~1–10 μs; fidelity 99.5–99.9%
- **Photonic**: challenging; probabilistic schemes or nonlinear media; lower fidelity