CNOT

The CNOT gate is one of the most important

\; 2

-qubit gates and is represented in the standard basis

\; \{|0\rangle, |1\rangle\}

by the following

\; 4 \times 4

matrix: :

\operatorname{U_{CNOT}} =  \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\  0 & 0 & 1 & 0 \end{bmatrix}

. The operator

\; U_{CNOT}

is Hermitian and unitary, and can be rewritten as a block matrix in the form: :

\operatorname{U_{CNOT}} =  \begin{bmatrix} \mathbf{1}_2 & \mathbf{0}_2 \\   \mathbf{0}_2 & \sigma_1  \end{bmatrix}

, where

\; \mathbf{1}_2 , \mathbf{0}_2

are the

\; 2 \times 2

identity and null matrices respectively and

\; \sigma_1

is the Pauli matrix :

\operatorname{\sigma_1} = \operatorname{\sigma_x} = \begin{bmatrix} 0 & 1 \\   1 & 0  \end{bmatrix}

. We can also rewrite the action of

\; U_{CNOT}

on two qubits operationally. We take two qubits

\; |x\rangle

and

\; |y\rangle

, where the former is the so-called ''control qubit'' and the latter is the ''target qubit''; then the action of

\; U_{CNOT}

on the system of the two qubits is: :

\; U_{CNOT}[|x\rangle \otimes |y\rangle] = |x\rangle \otimes |x \oplus y\rangle ,

where

\; x \oplus y = (x+y)mod\,2

. The CNOT together with the [[Hadamard gate]] and all [[Quantum gates|phase gates]] form an infinite ''[[Quantum gates|universal set of gates]]'', i.e. if the CNOT gate as well as the Hadamard and all phase gates are available then any

\; n

-qubit unitary operation can be simulated exactly with

\; O(4^n n)

such gates. [[Category:Evolutions and Operations]] [[Category:Models of Quantum Computation]]

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