ControlledNOT

Controlled-NOT, commonly abbreviated CNOT, is a two-qubit quantum gate that performs a NOT on the target qubit conditioned on the control qubit being in the state 1. If the control is |0>, the target is unchanged; if the control is |1>, the target is flipped, i.e., |0> ↔ |1>.

In the computational basis {|00>, |01>, |10>, |11>} the action is: |00> → |00>, |01> → |01>, |10> → |11>, |11> → |10>. The corresponding unitary matrix, written in this basis, is

[[1, 0, 0, 0],

[0, 1, 0, 0],

[0, 0, 0, 1],

[0, 0, 1, 0]].

As a quantum gate, CNOT is unitary and reversible.

CNOT is an entangling gate: when applied to a product state such as (|0> + |1>)/√2 ⊗ |0>, it yields the Bell state (|00> + |11>)/√2. Its behavior on superpositions can generate nonclassical correlations between the qubits, which is a hallmark of quantum interactions.

Role in quantum computing: together with arbitrary single-qubit gates, CNOT forms a universal gate set, capable of constructing any quantum operation. It is central to many quantum algorithms and protocols, including quantum teleportation, superdense coding, and quantum error correction codes.

Implementation: CNOT has been realized across various physical platforms, such as superconducting qubits and trapped ions, through controlled interactions that conditionally flip the target qubit. As a fundamental two-qubit operation, it remains a standard building block in experimental and theoretical quantum computing.