added some more stuff to general QC

thesis/chapters/quantum_computing.tex

@@ -14,8 +14,8 @@
     are used in these cases.
 \end{definition}
 
-A gate acting on a qbit is a unitary operator $G \in U(2)$. One can show that
-$\forall G \in U(2)$ $G$ can be approximated arbitrarily good as a product of unitary generator matrices
+A gate acting on a qbit is a unitary operator $G \in SU(2)$. One can show that
+$\forall G \in SU(2)$ $G$ can be approximated arbitrarily good as a product of unitary generator matrices
 \cite[Chapter 4.3]{kaye_ea2007}\cite[Chapter 2]{marquezino_ea_2019},
 common choices for the generators are $ X, H, R_{\phi}$ and $Z, H, R_{\phi}$ with 
 \label{ref:singleqbitgates}
@@ -108,10 +108,25 @@ is the $U$ gate acting on qbit $j$.
 \begin{definition}\label{def:CU}
     For two qbits $i,j = 0, 1, ..., n - 1$, $i \neq j$ and a gate $U_i$ acting on $i$
     the controlled version of $U$ is defined by
-    \begin{equation}\label{eq:CU}
-        CU_{i, j} = \ket{0}\bra{0}_j\otimes I_i + \ket{1}\bra{1}_j \otimes U_i 
+    \begin{equation}
+        \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}
 
+    where without loss of generality $j < i$; the other case is analogous.
+
     In particular for $X, Z$:
     \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 
@@ -122,7 +137,7 @@ is the $U$ gate acting on qbit $j$.
 
 \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.
 
 One can show that the gates in \ref{ref:singleqbitgates} together with an entanglement gate, such as $CX$ or $CZ$ are enough