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 $\backslash \{\; |x\backslash rangle\backslash \}$, 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]]