CNOT Quantum Gate

quantum-computing

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

$$|c⟩|t⟩\mapsto |c⟩|c\oplus t⟩,$$

where $\oplus$ is XOR, i.e. addition modulo $2$. Equivalently, on the computational basis:

$$\begin{array}{rl}|00⟩& \mapsto |00⟩,\\ |01⟩& \mapsto |01⟩,\\ |10⟩& \mapsto |11⟩,\\ |11⟩& \mapsto |10⟩.\end{array}$$

The control qubit is never changed. When the control is $|0⟩$, the target is left alone; when the control is $|1⟩$, the target is flipped by $X$.

Its matrix in the computational basis $\{|00⟩,|01⟩,|10⟩,|11⟩\}$ is

$$\mathrm{CNOT}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right)=I\otimes I-|0⟩⟨0|\otimes I+|1⟩⟨1|\otimes X.$$

CNOT is its own unitary inverse: applying it twice returns to the input. It is also a controlled-”$X$” 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

$$|\psi ⟩={\alpha }_{00}|00⟩+{\alpha }_{01}|01⟩+{\alpha }_{10}|10⟩+{\alpha }_{11}|11⟩,$$

CNOT swaps the amplitudes of $|10⟩$ and $|11⟩$ while leaving the amplitudes of $|00⟩$ and $|01⟩$ alone:

$$|\psi ⟩\mapsto {\alpha }_{00}|00⟩+{\alpha }_{01}|01⟩+{\alpha }_{11}|10⟩+{\alpha }_{10}|11⟩.$$

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 $|00⟩$ and apply the Hadamard gate to the first qubit:

$$|0⟩|0⟩\stackrel{H\otimes I}{\to }\frac{1}{\sqrt{2}}(|0⟩+|1⟩)|0⟩=\frac{1}{\sqrt{2}}(|00⟩+|10⟩).$$

Apply CNOT. The basis state $|10⟩$ has control $1$, so its target flips to $|11⟩$:

$$\frac{1}{\sqrt{2}}(|00⟩+|10⟩)\stackrel{\mathrm{CNOT}}{\to }\frac{1}{\sqrt{2}}(|00⟩+|11⟩).$$

The result is the Bell state $|{\Phi }^{+}⟩$, 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.