# Quantum logic gates

A ''quantum logic gate'' is a device which performs a fixed unitary operation on selected qubits in a fixed period of time. The gates listed below are common enough to have their own names. The matrices describing $n$ qubit gates are written in the computational basis $\\left\{ |x\rangle\\right\}$, where $x$ is a binary string of length $n$. The diagrams provide schematic representation of the gates. ==Hadamard gate== The Hadamard gate is a common single qubit gate $H$ defined as Image:Img44.png The matrix is written in the computational basis $\\left\{|0\rangle, |1\rangle \\right\}$ and the diagram on the right provides a schematic representation of the gate $H$ acting on a qubit in state $|x\rangle$, with $x=0,1$. ==Phase gate== The phase shift gate $\mathbf\left\{\phi \right\}$ defined as $\left| \,0\right\rangle \mapsto \left| \,0\right\rangle$ and $\left| \,1\right\rangle \mapsto e^\left\{i\phi \right\}\left| \,1\right\rangle$, or, in matrix notation, Image:Img59.png ==Controlled NOT gate== The controlled-NOT (C-NOT) gate, also known as the XOR or the measurement gate is one of the most popular two-qubit gate. It flips the second (target) qubit if the first (control) qubit is $\left| \,1\right\rangle$ and does nothing if the control qubit is $\left| \,0\right\rangle$. The gate is represented by the unitary matrix Image:Img76.png where $x,y=0\mbox\left\{ or \right\}1$ and $\oplus$ denotes XOR or addition modulo 2. If we apply the C-NOT to Boolean data in which the target qubit is $|0\rangle$ and the control is either $|0\rangle$ or $|1\rangle$ then the effect is to leave the control unchanged while the target becomes a copy of the control, ''i.e.'' $|x\rangle |0\rangle \mapsto |x\rangle |x\rangle \qquad x=0,1.$ Category:Evolutions and Operations Category:Quantum Computation Category:Handbook of Quantum Information