CNOT

CNOT ("controlled-NOT") is the most common two-qubit gate: it flips a target qubit, but only when a control qubit is \( |1\rangle \). It's the standard way to create entanglement between two qubits.

Qubits	2 (one control, one target)
Parameters	None
Inverse	Itself (CNOT² = I)
Also called	CX, controlled-X

What it does

If the control qubit is \( |0\rangle \), CNOT does nothing. If the control is \( |1\rangle \), CNOT applies an X gate to the target. Applied to a control in superposition, this correlates the two qubits - the essence of entanglement.

Matrix

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

Action on basis states

(control first, target second)

• \( |00\rangle \rightarrow |00\rangle \)
• \( |01\rangle \rightarrow |01\rangle \)
• \( |10\rangle \rightarrow |11\rangle \)
• \( |11\rangle \rightarrow |10\rangle \)

Example

On the circuit canvas, CNOT is drawn as a solid dot on the control wire connected by a vertical line to a circled + on the target wire - see Multi-qubit gate symbols. H on q[0] followed by CNOT from q[0] to q[1] is the standard Bell state circuit:

Other notations

Naming conventions across ecosystems; exact syntax can vary by library version, and this does not claim to be Qompile's generated output unless otherwise noted.

Language	Typical form
OpenQASM 2.0	cx q[0], q[1];
Qiskit	circuit.cx[q[0], q[1]]
Cirq	cirq.CNOT(q[0], q[1])
Q#	CNOT(q[0], q[1]);

Related

• Toffoli - the three-qubit generalization (two controls)
• Control - add a standalone control point to build custom controlled gates
• Bell states