Definition
CNOT Quantum Gate
The CNOT (Controlled-NOT) quantum gate is a two-qubit controlled gate whose target operation is the NOT gate. It acts on a control qubit and a target qubit as
where
is XOR, i.e. addition modulo . Equivalently, on the computational basis: The control qubit is never changed. When the control is
, the target is left alone; when the control is , the target is flipped by . Its matrix in the computational basis
is CNOT is its own unitary inverse: applying it twice returns to the input. It is also a controlled-”
” gate, with the first qubit as control and the second as target.
Effect on a General State
For a two-qubit state with arbitrary amplitudes
CNOT swaps the amplitudes of
The control-side amplitudes are preserved; only the target-side amplitudes are conditionally exchanged.
Entanglement
CNOT can generate entanglement when the control is in a superposition.
Bell state preparation
Start from
and apply the Hadamard gate to the first qubit: Apply CNOT. The basis state
has control , so its target flips to : The result is the Bell state
, an entangled state across the two qubits. CNOT is the canonical two-qubit entangling gate: together with single-qubit gates, it is universal for quantum computation.