circuit example: Deutsch's algorithm

content.tex

@@ -520,9 +520,26 @@ A quantum gate is just a unitary operator on a select set of qubits. Because the
 \begin{definition}
   A $n$-qubit quantum circuit is formally described by a sequence of gates and input mappings $C = (U_l, I_l)_{l=1}^m$ with $n = \abs{\bigcup_{l=1}^m I_l}$. Using \cref{def:gate_nbit_extension}, the circuit description equals a unitary matrix according to \cref{def:unitary_operator}:
   \begin{equation*}
-    C = G_n(U_m,I_m) \cdots G_n(U_1,I_1)
+    U_C = G_n(U_m,I_m) \cdots G_n(U_1,I_1)
   \end{equation*}
   The circuit $C$ is said to be maximally parallelized if all neighboring gate extensions $G_n(U_l,I_l)$ and $G_n(U_{l+1}, I_{l+1})$ with $I_l \cap I_{l+1} = \emptyset$ are reduced to $G_n(U_l \otimes U_{l+1}, I_l I_{l+1})$.
 \end{definition}
 
-\contentsketch{Circuit example Deutsch's algorithm}
+\begin{definition}
+  \label{def:circuit_size_depth}
+  Given a quantum circuit $C=(U_l,I_l)_{l=1}^m$ then
+  \begin{itemize}
+    \item the size of $C$ is the total number of gates $m$
+    \item the depth of $C$ is the number of unitary operator $G_n$ in the maximally parallelized form of $C$
+  \end{itemize}
+\end{definition}
+
+In \cref{sec:deutschs_algorithm} Deutsch's Algorithm already was illustrated using an informal notion of quantum circuits. Let $U_D$ be the unitary operator and $C_D$ the corresponding circuit description of Deutsch's algorithm before measuring the first qubit. The circuit $C_D$ has a size of 4 and a depth of 3. 
+\begin{equation*}
+\begin{aligned}
+  C_D &= \underbrace{(H, 1)}_{(U_1,I_1)},\underbrace{(H,2)}_{(U_2,I_2)},\underbrace{(O_f, (1,2))}_{(U_3,I_3)},\underbrace{(H,1)}_{(U_4,I_4)} \\
+  U_D &= G_2(H,1) G_2(O_f,(1,2)) \underbrace{G_2(H, 2)G_2(H, 1)}_{G_2(H \otimes H, (1,2))}
+\end{aligned}
+\end{equation*}
+
+