diff --git a/presentation/main.tex b/presentation/main.tex
index 46e88b5..27362bd 100644
--- a/presentation/main.tex
+++ b/presentation/main.tex
@@ -309,8 +309,8 @@
 \begin{frame}{Example: The $5$ Qbit EPR State}
     
     \begin{itemize}
-        \item{Start from $\ket{0}^{\otimes n}$.}
-        \item{Get to the state $\ket{\psi} = \frac{\ket{0}^{\otimes n} + \ket{1}^{\otimes n}}{\sqrt{2}}$.}
+        \item{Start from $\ket{0}^{\otimes 5}$.}
+        \item{Get to the state $\ket{\psi} = \frac{\ket{0}^{\otimes 5} + \ket{1}^{\otimes 5}}{\sqrt{2}}$.}
         \item{Use the circuit \\
             \[
             \Qcircuit @C=1em @R=.7em {
@@ -319,12 +319,12 @@
             & \qw & \qw & \qw & \gate{X} & \qw & \qw & \qw & \qw &\qw \\
             & \qw & \qw & \qw & \qw & \qw & \gate{X} & \qw & \qw &\qw \\
             & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \gate{X} &\qw \\
-            }
+            }.
             \]
         }
         \item{The state has the form
             \begin{equation}
-                \ket{\psi} = \left(\prod\limits_{1 < i < 5} CX_{i,0}\right) H_0 \ket{0}.
+                \ket{\psi} = \left(\prod\limits_{1 < i < 5} CX_{i,0}\right) H_0 \ket{0}^{\otimes 5}.
             \end{equation}}
     \end{itemize}
 \end{frame}
@@ -342,10 +342,10 @@
             & \gate{H} & \qw & \qw & \gate{Z} & \qw & \qw & \qw & \qw & \gate{H} &\qw \\
             & \gate{H} & \qw & \qw & \qw & \qw & \gate{Z} & \qw & \qw & \gate{H} &\qw \\
             & \gate{H} & \qw & \qw & \qw & \qw & \qw & \qw & \gate{Z} & \gate{H} &\qw \\
-            }
+            }.
             \]
         }
-        \item{Switching from starting state $\ket{0}^{\otimes n}$ to $\ket{+}^{\otimes n}$ gives the
+        \item{Switching from starting state $\ket{0}^{\otimes 5}$ to $\ket{+}^{\otimes 5}$ gives the
             graphical representation.}
     \end{itemize}
 
@@ -465,6 +465,8 @@
     \item{
         If just one vertex operator has been cleared the other vertex is isolated
         and one can precompute all resulting states.}
+    \item{One can show that the probability amplitudes when measuring a qbit of a graphical state
+        are either $0$, $1$ or $\frac{1}{2}$.}
 \end{itemize}
 \end{frame}
 }
@@ -500,8 +502,10 @@
         \item{To increase simulation efficiency the core of both simulators has been
             implemented in \lstinline{C}.}
         \item{The dense vector states are stored in \lstinline{numpy} arrays.}
-        \item{The graph is stored in an length $n$ array of linked lists. The vertex operators
-            are stored in a \lstinline{uint8_t} array.}
+        \item{The graph is stored in an length $n$ array of linked lists.}
+        \item{The vertex operators are local Clifford operators. The local Clifford 
+                group as $24$ elements, they are represented by integers
+            stored in a \lstinline{uint8_t} array.}
     \end{itemize}
 \end{frame}
 }
diff --git a/presentation/spin_chain/hamiltonian.py b/presentation/spin_chain/hamiltonian.py
index 8274bee..b6438a6 100644
--- a/presentation/spin_chain/hamiltonian.py
+++ b/presentation/spin_chain/hamiltonian.py
@@ -16,7 +16,7 @@ def Mi(nqbits, i, M):
 
 
 def H_interaction(nqbits):
-    interaction_terms = [Mi(nqbits, i, Z) @ Mi(nqbits, i+1, Z) for i in range(nqbits)]
+    interaction_terms = [Mi(nqbits, i, Z) @ Mi(nqbits, i+1, Z) for i in range(nqbits - 1)]
     return sum(interaction_terms)
 
 def H_field(nqbits, g):
diff --git a/presentation/spin_chain/time_evolution.py b/presentation/spin_chain/time_evolution.py
index fef4539..02c5399 100644
--- a/presentation/spin_chain/time_evolution.py
+++ b/presentation/spin_chain/time_evolution.py
@@ -20,8 +20,8 @@ matplotlib.rcParams.update(
 nqbits = 6
 g = 3
 N_trot = 80
-t_stop = 9
-delta_t = 0.09
+t_stop = 29
+delta_t = 0.1
 qbits = list(range(nqbits))
 
 n_sample = 2200
@@ -49,8 +49,8 @@ for t in np.arange(0, t_stop, delta_t):
 
     errors_sampling.append(bootstrap(result[0], n_sample, n_sample, n_sample // 2, np.average))
 
-    #amplitude = np.sum(np.abs(state._qm_state[measure_coefficient_mask])**2)
-    #amplitudes_qc.append(amplitude)
+    amplitude = np.sum(np.abs(state._qm_state[measure_coefficient_mask])**2)
+    amplitudes_qc.append(amplitude)
 
     # Simulation using matrices
     np_zero_state = np.zeros(2**nqbits)
@@ -68,11 +68,17 @@ print()
 print("done.")
 
 results_qc = np.array(results_qc)
+amplitudes_qc = np.array(amplitudes_qc)
 
-errors_trotter = (np.arange(0, t_stop, delta_t) * g)**3 / N_trot**3
+errors_trotter = (np.arange(0, t_stop, delta_t))**3 / N_trot**3
 errors_sampling = np.array(errors_sampling)
+#errors_sampling = np.abs(results_qc - amplitudes_qc)
+#errors_sampling = (amplitudes_qc * (1 - amplitudes_qc))**2 / n_sample
 
-
+#hm1 = plt.errorbar(np.arange(0, t_stop, delta_t)
+#                , results_qc
+#                , yerr=(errors_sampling + errors_trotter)
+#                , color="red")
 h0 = plt.errorbar(np.arange(0, t_stop, delta_t)
                 , results_qc
                 , yerr=errors_sampling