The CNOT gate is one of the most important 2-qubit gates and is represented in the standard basis {∣0⟩, ∣1⟩} by the following 4 × 4 matrix:

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

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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 **1**_{2}, **0**_{2} are the 2 × 2 identity and null matrices respectively and *σ*_{1} is the Pauli matrix

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

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We can also rewrite the action of *U*_{CNOT} on two qubits operationally. We take two qubits ∣*x*⟩ and ∣*y*⟩, 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*⟩ ⊗ ∣*y*⟩] = ∣*x*⟩ ⊗ ∣*x* ⊕ *y*⟩,

where *x* ⊕ *y* = (*x* + *y*)*m**o**d* 2.

The CNOT together with the Hadamard gate and all phase gates form an infinite *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