CNOT

The controlled-NOT gate, commonly called CNOT or CX, is a two-qubit quantum gate that performs an X (bit-flip) on the target qubit when the control qubit is in state 1, and leaves the target unchanged when the control is 0. In the standard computational basis ordered as |00>, |01>, |10>, |11>, the action is: |00> → |00>, |01> → |01>, |10> → |11>, |11> → |10>. Thus, it can entangle qubits and is essential for many quantum circuits.

Matrix form and properties: The CNOT gate is unitary and its own inverse (CNOT squared equals the identity). In the two-qubit basis, its 4×4 matrix for control on the first qubit and target on the second is

[[1, 0, 0, 0],

[0, 1, 0, 0],

[0, 0, 0, 1],

[0, 0, 1, 0]].

Note that different conventions for qubit ordering can yield a functionally equivalent matrix with rows and columns permuted accordingly.

Applications: CNOT, together with arbitrary single-qubit gates, forms a universal gate set for quantum computation. It is used to prepare entangled states such as Bell states; for example, applying a Hadamard gate to the control qubit followed by a CNOT yields (|00> + |11>)/√2 from the input |00>. CNOT is also fundamental in quantum error correction, quantum teleportation, and superdense coding, and is widely used in reversible classical logic circuits implemented on quantum hardware.