Update week4.do.txt

Author: mhjensen

doc/src/week4/week4.do.txt

@@ -539,8 +539,42 @@ We can rewrite the equation for the wave function at a time $t=0$ as
 !split
 ===== Part 2: Specific realizations and famous gates =====
 
-This part is not yet ready and will be part of the whiteboard notes.
-I will try to get it ready by the end of Tuesday, Feb 11
+Nuclear magnetic resonance (NMR) quantum computing is one of the several
+proposed approaches for constructing a quantum computer. It uses the
+spin states of nuclei within molecules as qubits. The quantum states
+are probed through the nuclear magnetic resonances, allowing the
+system to be implemented as a variation of nuclear magnetic resonance
+spectroscopy. NMR differs from other implementations of quantum
+computers in that it uses an ensemble of systems, in this case
+molecules, rather than a single pure state.
+
+You can read more about this at URL:"https://cba.mit.edu/docs/papers/98.06.sciqc.pdf"
+
+!split
+===== Spin Hamiltonian =====
+
+In order to understand in terms of a given Hamiltonian how the
+different gates arise, we consider now the Hamiltonian of a nuclear
+spin in a magnetic field. Since the spin provides provides a magnetic
+dipole moment, a nucleus with a spin will interact with the magnetic field. The Haniltonian of a nucleus with spin interacting with a magnetic field $\bm{B}$ is
+!bt
+\[
+H = -\bm{\mu}\bm{B},
+\]
+!et
+with $\bm{\mu}=\gamma\bm{S}$, $\gamma$ being the so-called gyromagnetic ratio and $\bm{S}$ the spin.
+
+!split
+===== Field along the $z$-axis =====
+
+It is common to let the spin interact with a constant magnetic field
+along the $z$-axis. This gives an effecitve Hamiltonian
+!bt
+\[
+H_z = -\frac{\hbar\omega_L}{2}\sigma_z,,
+\]
+!et
+where $\omega_L$ is the so-called Larmor precession frequency. This quantity includes also the constant magnetif field along the $z$-axis. For all practical purposes it suffices for us to have an expression of the Hamiltonian in terms of the Pauli-Z matrix.
 
 
 !split
@@ -1487,5 +1521,68 @@ o We will extend the above one-qubit Hamiltonian to a two-qubit problem and anal
 o Before we introduce the Variational Quantum Eigensolver (VQE), we need to discuss entropy as a measure of entanglement
 o If we get time, we start our discussion of the VQE algorithm
 
+
+!split
+===== Exercises this week: Hamiltonians and project 1 =====
+
+As an initial test, we consider a simple $2\times 2$ real
+Hamiltonian consisting of a diagonal part $H_0$ and off-diagonal part
+$H_I$, playing the roles of a non-interactive one-body and interactive
+two-body part respectively. Defined through their matrix elements, we
+express them in the Pauli basis $\vert 0\rangle$ and $\vert 1 \rangle$
+
+!bt
+\begin{align*}
+    \begin{split} 
+        H &= H_0 + H_I \\
+        H_0 = \begin{bmatrix}
+            E_1 & 0 \\
+            0 & E_2
+        \end{bmatrix}&, \hspace{20px}
+        H_I = \lambda \begin{bmatrix}
+            V_{11} & V_{12} \\
+            V_{21} & V_{22}
+        \end{bmatrix}
+    \end{split}
+\end{align*}
+!et
+Where $\lambda \in [0,1]$ is a coupling constant parameterizing the strength of the interaction. 
+
+!split
+===== Rewriting in terms of Pauli matrices =====
+
+Define
+!bt
+\[
+    E_+ = \frac{E_1 + E_2}{2},\hspace{20px} E_- = \frac{E_1 - E_2}{2}
+\]
+!et
+show  that by combining the identity and $Z$ Pauli matrix, this can be expressed as
+
+!bt
+\[
+    H_0 = E_+ I + E_- Z
+\]
+!et
+
+!split
+===== The interaction part =====
+
+For $H_1$ we use the same trick to fill the diagonal, defining $V_+ = (V_{11} + V_{22})/2, V_- = (V_{11} - V_{22})/2$. From the hermiticity requirements of $H$, we note that $V_{12} = V_{21} \equiv V_o$. Use this to simplify the problem to a simple $X$ term. 
+
+!bt
+\[
+    H_I = V_+ I + V_- Z + V_o X
+\]
+!et
+
 !split
-===== Exercises this week =====
+===== Measurement basis =====
+
+For the above system show that the Pauli $X$ matrix can be rewritten in terms of the Hadamard matrices and the Pauli $Z$ matrix, that is
+!bt
+\[
+X=HZH.
+\]
+!et
+