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 $\backslash \{\; |x\backslash rangle\backslash \}$, 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 $\backslash \{|0\backslash rangle,\; |1\backslash rangle\; \backslash \}$ and the diagram on the right provides a schematic representation of the gate $H$ acting on a qubit in state $|x\backslash rangle$, with $x=0,1$.
==Phase gate==
The phase shift gate
$\backslash mathbf\{\backslash phi\; \}$ defined as $\backslash left|\; \backslash ,0\backslash right\backslash rangle\; \backslash mapsto\; \backslash left|\; \backslash ,0\backslash right\backslash rangle$ and $\backslash left|\; \backslash ,1\backslash right\backslash rangle\; \backslash mapsto\; e^\{i\backslash phi\; \}\backslash left|\; \backslash ,1\backslash right\backslash rangle$, 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 $\backslash left|\; \backslash ,1\backslash right\backslash rangle$ and does nothing if the control qubit is $\backslash left|\; \backslash ,0\backslash right\backslash rangle$. The gate is represented by the unitary matrix
Image:Img76.png
where $x,y=0\backslash mbox\{\; or\; \}1$ and $\backslash oplus$ denotes XOR or addition modulo 2. If
we apply the C-NOT to Boolean data in which the target qubit
is $|0\backslash rangle$
and the control is either $|0\backslash rangle$ or $|1\backslash rangle$ then the effect is to
leave the control unchanged while the target becomes a copy of the control, ''i.e.''
$|x\backslash rangle\; |0\backslash rangle\; \backslash mapsto\; |x\backslash rangle\; |x\backslash rangle\; \backslash qquad\; x=0,1.$
Category:Evolutions and Operations
Category:Quantum Computation
Category:Handbook of Quantum Information

## Last modified:

Monday, October 26, 2015 - 17:56