Update week4.do.txt

Author: mhjensen

doc/src/week4/week4.do.txt

@@ -714,11 +714,25 @@ plt.show()
 !split
 ===== Gates, the whys and hows =====
 
+In quantum computing it is common to rewrite various unitary
+transformations acting on a given state in terms of so-called gates
+(one-qubit, two-qubit or more qubit gates). These unitary
+transformations do actually represent specific interactions (for
+example with extenrally applied probes) of the system with the
+environment.
+
+Each such operation is by convention written in terms of gates and a
+chain of such gates represents a circuit. The latter represents then a
+specific set of operations on an initial state.
+
+The aim of the first set of notes this week is to link these gates
+(and thereby circuits) to their respective unitary
+transformation. These unitary transformations represent selected
+physical processes.
 
+!split
+===== Mathematical background =====
 
-\frametitle{Schr\"odinger picture}
-\begin{small}
-{\scriptsize
 The time-dependent Schr\"odinger equation (or equation of motion) reads
 \[
 \imath \hbar\frac{\partial }{\partial t}|\Psi_S(t)\rangle = \hat{H}\Psi_S(t)\rangle,
@@ -730,14 +744,10 @@ A formal solution is given by
 \]
 The Hamiltonian $\hat{H}$ is hermitian and the exponent represents a unitary 
 operator with an operation carried ut on the wave function at a time $t_0$.
-}
-\end{small}
-}
-\frame
-{
-\frametitle{Interaction picture}
-\begin{small}
-{\scriptsize
+
+!split
+===== Interaction picture  =====
+
 Our Hamiltonian is normally written out as the sum of an unperturbed part $\hat{H}_0$ and an interaction part $\hat{H}_I$, that is
 \[
 \hat{H}=\hat{H}_0+\hat{H}_I.
@@ -749,23 +759,20 @@ We wish now to define a unitary transformation in terms of $\hat{H}_0$ by defini
 \]
 which is again a unitary transformation carried out now at the time $t$ on the 
 wave function in the Schr\"odinger picture. 
-}
-\end{small}
-}
-\frame
-{
-\frametitle{Interaction picture}
-\begin{small}
-{\scriptsize
+
+!split
+===== Interaction picture  =====
+
+
 We can easily find the equation of motion by taking the time derivative
 \[
 \imath \hbar\frac{\partial }{\partial t}|\Psi_I(t)\rangle = -\hat{H}_0\exp{(\imath\hat{H}_0t/\hbar)}\Psi_S(t)\rangle+\exp{(\imath\hat{H}_0t/\hbar)}
 \imath \hbar\frac{\partial }{\partial t}\Psi_S(t)\rangle.
 \]
-}
-\end{small}
-}
-\frame
+
+!split
+===== Interaction picture  =====
+
 {
 \frametitle{Interaction picture}
 \begin{small}
@@ -786,6 +793,13 @@ which gives us
 }
 \end{small}
 }
+
+
+!split
+===== Interaction picture  =====
+
+
+
 \frame
 {
 \frametitle{Interaction picture}
@@ -813,6 +827,13 @@ stating that a unitary transformation does not change expectation values!
 }
 \end{small}
 }
+
+
+!split
+===== Interaction picture  =====
+
+
+
 \frame
 {
 \frametitle{Interaction picture}
@@ -827,6 +848,13 @@ together with the observation that any function of an operator commutes with the
 }
 \end{small}
 }
+
+
+!split
+===== Interaction picture  =====
+
+
+
 \frame
 {
 \frametitle{Interaction picture}
@@ -844,6 +872,12 @@ with the obvious value $\hat{U}(t_0,t_0)=1$.
 }
 \end{small}
 }
+
+!split
+===== Interaction picture  =====
+
+
+
 \frame
 {
 \frametitle{Interaction picture}
@@ -873,6 +907,12 @@ which leads to
 }
 \end{small}
 }
+
+
+!split
+===== Interaction picture  =====
+
+
 \frame
 {
 \frametitle{Interaction picture}
@@ -895,6 +935,13 @@ or
 }
 \frame
 {
+
+
+!split
+===== Interaction picture  =====
+
+
+
 \frametitle{Interaction picture}
 \begin{small}
 {\scriptsize
@@ -906,6 +953,14 @@ It is then easy to convince oneself that the properties defined above are satisf
 }
 \end{small}
 }
+
+
+
+!split
+===== Interaction picture  =====
+
+
+
 \frame
 {
 \frametitle{Interaction picture}
@@ -929,6 +984,15 @@ which can be rewritten as
 }
 \frame
 {
+
+
+
+!split
+===== Interaction picture  =====
+
+
+
+
 \frametitle{Interaction picture}
 \begin{small}
 {\scriptsize
@@ -947,6 +1011,13 @@ The third term can be written as
 }
 \frame
 {
+
+
+!split
+===== Interaction picture  =====
+
+
+
 \frametitle{Interaction picture}
 \begin{small}
 {\scriptsize
@@ -968,6 +1039,14 @@ with $\Theta(t''-t')$ being the standard Heavyside or step function. The step fu
 }
 \frame
 {
+
+
+!split
+===== Interaction picture  =====
+
+
+
+
 \frametitle{Interaction picture}
 \begin{small}
 {\scriptsize
@@ -982,6 +1061,14 @@ where $\Hat{T}$ is the so-called time-ordering operator.
 }
 \frame
 {
+
+
+!split
+===== Interaction picture  =====
+
+
+
+
 \frametitle{Interaction picture}
 \begin{small}
 {\scriptsize
@@ -998,6 +1085,14 @@ to derive various contributions to many-body perturbation theory. See also exerc
 }
 \frame
 {
+
+
+!split
+===== Interaction picture  =====
+
+
+
+
 \frametitle{Heisenberg picture}
 \begin{small}
 {\scriptsize
@@ -1019,6 +1114,13 @@ meaning that $|\Psi_H(t)\rangle$ is time independent. An operator in this pictur
 }
 \end{small}
 }
+
+
+!split
+===== Interaction picture  =====
+
+
+
 \frame
 {
 \frametitle{Heisenberg picture}
@@ -1040,6 +1142,12 @@ operator in the interaction picture as
 }
 \end{small}
 }
+
+!split
+===== Interaction picture  =====
+
+
+
 \frame
 {
 \frametitle{Heisenberg picture}
@@ -1064,6 +1172,13 @@ We can relate this wave function to that a given time $t$ via the time evolution
 }
 \frame
 {
+
+
+!split
+===== Interaction picture  =====
+
+
+
 \frametitle{Adiabatic hypothesis}
 \begin{small}
 {\scriptsize
@@ -1081,6 +1196,12 @@ and its interaction term is meant to simulate the switching of the interaction.
 }
 \frame
 {
+
+!split
+===== Interaction picture  =====
+
+
+
 \frametitle{Adiabatic hypothesis}
 \begin{small}
 {\scriptsize
@@ -1093,14 +1214,11 @@ with
 \hat{U}_{\varepsilon}(t,t_0)=\sum_{n=0}^{\infty}\left(\frac{-\imath}{\hbar}\right)^n\frac{1}{n!}
 \int_{t_0}^t dt_1\dots \int_{t_0}^t dt_N \exp{(-\varepsilon(t_1+\dots+t_n)/\hbar)}\hat{T}\left[\hat{H}_I(t_1)\dots\hat{H}_I(t_n)\right]
 \]
-}
-\end{small}
-}
-\frame
-{
-\frametitle{Adiabatic hypothesis}
-\begin{small}
-{\scriptsize
+
+
+!split
+===== Interaction picture  =====
+
 In the limit $t_0\rightarrow -\infty$, the solution ot Schr\"odinger's equation is
 $|\Phi_0\rangle$, and the eigenenergies are given by 
 \[
@@ -1114,14 +1232,12 @@ with the corresponding interaction picture wave function given by
 \[
 |\Psi_I(t_0)\rangle = \exp{(\imath \hat{H}_0t_0/\hbar)}|\Psi_S(t_0)\rangle=|\Phi_0\rangle.
 \]
-}
-\end{small}
-}
-\frame
-{
-\frametitle{Adiabatic hypothesis}
-\begin{small}
-{\scriptsize
+
+
+
+!split
+===== Interaction picture  =====
+
 The solution becomes time independent in the limit $t_0\rightarrow -\infty$.
 The same conclusion can be reached by looking at 
 \[
@@ -1133,6 +1249,3 @@ We can rewrite the equation for the wave function at a time $t=0$ as
 \[
 |\Psi_I(0) \rangle = \hat{U}_{\varepsilon}(0,-\infty)|\Phi_0\rangle.
 \]
-}
-\end{small}
-}