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:

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

CNOT creates Bell (maximally entangled) states:

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

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

• 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

• 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)

• 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