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added some more stuff to general QC

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Daniel Knüttel 2020-02-19 11:19:06 +01:00
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thesis/chapters/quantum_computing.tex
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@ -14,8 +14,8 @@
are used in these cases. are used in these cases.
\end{definition} \end{definition}
A gate acting on a qbit is a unitary operator $G \in U(2)$. One can show that A gate acting on a qbit is a unitary operator $G \in SU(2)$. One can show that
$\forall G \in U(2)$ $G$ can be approximated arbitrarily good as a product of unitary generator matrices $\forall G \in SU(2)$ $G$ can be approximated arbitrarily good as a product of unitary generator matrices
\cite[Chapter 4.3]{kaye_ea2007}\cite[Chapter 2]{marquezino_ea_2019}, \cite[Chapter 4.3]{kaye_ea2007}\cite[Chapter 2]{marquezino_ea_2019},
common choices for the generators are $ X, H, R_{\phi}$ and $Z, H, R_{\phi}$ with common choices for the generators are $ X, H, R_{\phi}$ and $Z, H, R_{\phi}$ with
\label{ref:singleqbitgates} \label{ref:singleqbitgates}
@ -108,10 +108,25 @@ is the $U$ gate acting on qbit $j$.
\begin{definition}\label{def:CU} \begin{definition}\label{def:CU}
For two qbits $i,j = 0, 1, ..., n - 1$, $i \neq j$ and a gate $U_i$ acting on $i$ For two qbits $i,j = 0, 1, ..., n - 1$, $i \neq j$ and a gate $U_i$ acting on $i$
the controlled version of $U$ is defined by the controlled version of $U$ is defined by
\begin{equation}\label{eq:CU} \begin{equation}
CU_{i, j} = \ket{0}\bra{0}_j\otimes I_i + \ket{1}\bra{1}_j \otimes U_i \begin{aligned}\label{eq:CU}
CU_{i, j} &:= &\ket{0}\bra{0}_j\otimes I_i
+ \ket{1}\bra{1}_j \otimes U_i \\
&:= &\left(\bigotimes\limits_{l < j} I\right)
\otimes \ket{0}\bra{0}
\otimes \left(\bigotimes\limits_{j < l < i} I\right)
\otimes I
\otimes \left(\bigotimes\limits_{j > l} I \right)\\
& &+ \left(\bigotimes\limits_{l < j} I\right)
\otimes \ket{1}\bra{1}
\otimes \left(\bigotimes\limits_{j < l < i} I\right)
\otimes U
\otimes \left(\bigotimes\limits_{j > l} I \right)\\
\end{aligned}
\end{equation} \end{equation}
where without loss of generality $j < i$; the other case is analogous.
In particular for $X, Z$: In particular for $X, Z$:
\begin{equation}\label{eq:CX_pr} \begin{equation}\label{eq:CX_pr}
CX(i, j) = \ket{0}\bra{0}_j\otimes I_i + \ket{1}\bra{1}_j \otimes X_i CX(i, j) = \ket{0}\bra{0}_j\otimes I_i + \ket{1}\bra{1}_j \otimes X_i
@ -122,7 +137,7 @@ is the $U$ gate acting on qbit $j$.
\end{definition} \end{definition}
In the definition \ref{def:CU} $i$ is called the act-qbit and $j$ the control-qbit. In words In Definition \ref{def:CU} $i$ is called the act-qbit and $j$ the control-qbit. In words
$CU$ applies the gate $U$ to the act-qbit if the control-qbit is in its $\ket{1}$ state. $CU$ applies the gate $U$ to the act-qbit if the control-qbit is in its $\ket{1}$ state.
One can show that the gates in \ref{ref:singleqbitgates} together with an entanglement gate, such as $CX$ or $CZ$ are enough One can show that the gates in \ref{ref:singleqbitgates} together with an entanglement gate, such as $CX$ or $CZ$ are enough