## Sunday, 8 September 2013

### Controlled-U Gates

Another class of gates essential to quantum computation are \emph{controlled$-U$} gates, which act on a register of at least 2 qubits applying a unitary operation $U$ on some subspace of the system condition on the state of some other part the system. Let $\mathcal{H}=\mathcal{H}_A\otimes \mathcal{H}_B$ be the tensor space of two Hilbert spaces $\mathcal{H}_A$ and $\mathcal{H}_B$, and let $U: \mathcal{H}_B\rightarrow \mathcal{H}_B$ be any unitary operator on $\mathcal{H}_B$. Then a controlled$-U$ gate} $c-U$ gate performs the operation $U$ on $\mathcal{H}_B$ condition on the state of the register $\mathcal{H}_A$, and leaves the state in $\mathcal{H}_B$ unchanged otherwise. The register of $\mathcal{H}_A$ is called the control register, and the register of $\mathcal{H}_B$ is called the target register.

For example, suppose $\mathcal{H}_A=\mathcal{H}_B=\mathcal{H}^2$ are both registers representing single qubits. Consider the controlled$-NOT$ gate},  $c-X$,  acting on this system of two qubits defined in terms of the $X$ Pauli gate as follows:
$c-X : \left\{ \begin{array}{l l} \left|0\right>\left|0\right> &\mapsto \left|0\right>\left|0\right> , \\ \left|0\right>\left|1\right> &\mapsto \left|0\right>\left|1\right>, \\ \left|1\right>\left|0\right> &\mapsto \left|1\right>X\left|0\right> = \left|1\right>\left|1\right>, \\ \left|1\right>\left|1\right> &\mapsto \left|1\right>X\left|1\right> =\left|1\right>\left|0\right> \end{array} \right.$
Here, the $u-X$ gate flips the  basis states of the target qubit if the state of the control qubit is in the state $\left|1\right>$, and leaves the target qubit unchanged if the control qubit is in the state $\left|0\right>$.

Controlled gates act on multiple qubit registers and provide a natural example of a gate $U$ that acts on the tensor space $\mathcal{H}_A\otimes \mathcal{H}_B$  of two different systems that cannot be represented as a tensor product of gates of the form $U=U_A\otimes U_B$, where $U_A$ and $U_B$ are unitary gates acting on the spaces $\mathcal{H}_A$ and $\mathcal{H}_B$, respectively. In other words, $U\neq U_A\otimes U_B$ for any gates $U_A$ and $U_B$. Gates of this type will play a crucial role in many of the quantum algorithms.