CNOTgate

CNOTgate, short for controlled-NOT gate, is a two-qubit quantum gate used to entangle qubits and implement conditional operations. It uses one qubit as the control and a second as the target. The gate flips the state of the target qubit if and only if the control qubit is in the state |1⟩; if the control is in |0⟩, the target remains unchanged.

In the computational basis ordered as |00⟩, |01⟩, |10⟩, |11⟩, the CNOTgate is represented by the unitary matrix

[ [1, 0, 0, 0],

[0, 1, 0, 0],

[0, 0, 0, 1],

[0, 0, 1, 0] ].

Its action on basis states is: |00⟩ → |00⟩, |01⟩ → |01⟩, |10⟩ → |11⟩, |11⟩ → |10⟩. For general superpositions, the gate applies the NOT operation to the target component conditioned on the control.

In circuit diagrams, the CNOTgate is depicted with a solid dot on the control wire and a circled plus sign (⊕) on the target wire. The gate is symmetric under the two-qubit basis and can be applied in any arrangement of qubits as part of larger quantum circuits.

CNOTgate is a unitary and Hermitian operation that satisfies CNOTgate squared equals the identity. It is a standard universal two-qubit gate together with single-qubit operations, enabling universal quantum computation. It plays a key role in creating entanglement, executing quantum error correction routines, and implementing common quantum algorithms.