The CNOT Gate
The controlled-NOT gate flips a target qubit when the control is 1; combined with single-qubit gates it enables universal computation and entanglement.
A two-qubit gate
The controlled-NOT (CNOT or CX) gate acts on two qubits. It flips the target qubit if and only if the control qubit is |1>, and does nothing if the control is |0>. It is the most common entangling gate in the circuit model.
CNOT (control, target)
Truth table on basis states
| control | target | target-out |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
On computational basis states CNOT looks like a classical XOR written into the target. The quantum power appears when the control is in superposition.
Creating entanglement
Put the control in superposition first. Apply H to qubit 1 of |00> to get (|00>+|10>)/sqrt(2), then CNOT to get (|00>+|11>)/sqrt(2) — a Bell state. The control and target are now entangled and neither has a definite value alone. CNOT is the standard entangler.
Universality and no-cloning
CNOT together with arbitrary single-qubit gates forms a universal set. It might look like a copier — it maps |x>|0> to |x>|x> for x in {0,1} — but on a superposition it entangles rather than copies, consistent with the no-cloning theorem. Applying CNOT to (|0>+|1>)|0> gives the entangled Bell state, not two independent copies.