# Controlled NOT gate

## Quantum logic gate / From Wikipedia, the free encyclopedia

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In computer science, the **controlled NOT gate** (also **C-NOT** or **CNOT**), **controlled- X gate**,

**controlled-bit-flip gate**,

**Feynman gate**or

**controlled Pauli-X**is a quantum logic gate that is an essential component in the construction of a gate-based quantum computer. It can be used to entangle and disentangle Bell states. Any quantum circuit can be simulated to an arbitrary degree of accuracy using a combination of CNOT gates and single qubit rotations.

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^{[2]}The gate is sometimes named after Richard Feynman who developed an early notation for quantum gate diagrams in 1986.

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The CNOT can be expressed in the Pauli basis as:

- ${\mbox{CNOT}}=e^{i{\frac {\pi }{4}}(I_{1}-Z_{1})(I_{2}-X_{2})}=e^{-i{\frac {\pi }{4}}(I_{1}-Z_{1})(I_{2}-X_{2})}.$

Being both unitary and Hermitian, CNOT has the property $e^{i\theta U}=(\cos \theta )I+(i\sin \theta )U$ and $U=e^{i{\frac {\pi }{2}}(I-U)}=e^{-i{\frac {\pi }{2}}(I-U)}$, and is involutory.

The CNOT gate can be further decomposed as products of rotation operator gates and exactly one two qubit interaction gate, for example

- ${\mbox{CNOT}}=e^{-i{\frac {\pi }{4}}}R_{y_{1}}(-\pi /2)R_{x_{1}}(-\pi /2)R_{x_{2}}(-\pi /2)R_{xx}(\pi /2)R_{y_{1}}(\pi /2).$

In general, any single qubit unitary gate can be expressed as $U=e^{iH}$, where *H* is a Hermitian matrix, and then the controlled *U* is $CU=e^{i{\frac {1}{2}}(I_{1}-Z_{1})H_{2}}$.

The CNOT gate is also used in classical reversible computing.