Update week1.do.txt

Author: mhjensen

doc/src/week1/week1.do.txt

@@ -13,7 +13,6 @@ DATE: January 22, 2025
   o States in Hilbert space, pure and mixed states
   o Operators and simple gates
   o "Video of lecture to be added":"https://youtu.be/"
-  o Test your background knowledge (to be added)
 _Reading recommendation_: "Scherer, Mathematics of Quantum Computations, chapter 2":"https://link.springer.com/book/10.1007/978-3-030-12358-1"
 !eblock
 
@@ -27,7 +26,7 @@ o We plan to work on two projects which will define the content of the course, t
  * Second project: Applications and implementations of quantum machine learning algorithms
  * Second project: studies of entanglement and physical realization of quantum gates and circuits
 o Two projects which count $50\%$ each for the final grade
-o Deadline first project March 21
+o Deadline first project March 21, see URL:"https://github.com/CompPhysics/QuantumComputingMachineLearning/tree/gh-pages/doc/Projects/2025/Project1"
 o Deadline second project June 1
 o All info at the GitHub address URL:"https://github.com/CompPhysics/QuantumComputingMachineLearning"
 
@@ -1063,6 +1062,115 @@ If we stay with this notation we have
 Unitary transformations are rotations in state space which preserve the
 length (the square root of the inner product) of the state vector.
 
+
+!split
+===== First exercise set =====
+
+The exercises we present each week are meant to build the basis for
+the two projects we will work on during the semester.  The first
+project deals with implementing the so-called
+_Variational Quantum Eigensolver_ algorithm for finding the eigenvalues and eigenvectors of selected Hamiltonians.
+Feel free to use the above codes in order to get started.
+
+
+
+!split
+===== Exercise: One-qubit basis and  Pauli matrices  =====
+
+Write a function which sets up a one-qubit basis and apply the various Pauli matrices to these basis states.
+
+!split
+===== Exercise: Hadamard and Phase gates  =====
+
+Apply the Hadamard and Phase gates to the same one-qubit basis states and study their actions on these states.
+
 !split
-===== Exercises to test yourself =====
-To be added
+===== Exercise: Rewrite simple one-qubit Hamiltonian in terms of Pauli matrices  =====
+
+We define a  symmetric matrix  $H\in {\mathbb{R}}^{2\times 2}$
+!bt
+\[
+H = \begin{bmatrix} H_{11} & H_{12} \\ H_{21} & H_{22}
+\end{bmatrix},
+\]
+!et
+We  let $H = H_0 + H_I$, where
+!bt
+\[
+H_0= \begin{bmatrix} E_1 & 0 \\ 0 & E_2\end{bmatrix},
+\]
+!et
+is a diagonal matrix. Similarly,
+!bt
+\[
+H_I= \begin{bmatrix} V_{11} & V_{12} \\ V_{21} & V_{22}\end{bmatrix},
+\]
+!et
+where $V_{ij}$ represent various interaction matrix elements.
+We can view $H_0$ as the non-interacting solution
+!bt
+\begin{equation}
+       H_0\vert 0 \rangle =E_1\vert 0 \rangle,
+\end{equation}
+!et
+and
+!bt
+\begin{equation}
+       H_0\vert 1\rangle =E_2\vert 1\rangle,
+\end{equation}
+!et
+where we have defined the orthogonal computational one-qubit basis states $\vert 0\rangle$ and $\vert 1\rangle$.
+
+
+!bsubex
+Show that you can rewrite the above Hamiltonian in terms of the Pauli $x$ and $z$ matrices
+!bsol
+We rewrite $H$ (and $H_0$ and $H_I$)  via Pauli matrices
+!bt
+\[
+H_0 = \mathcal{E} I + \Omega \sigma_z, \quad \mathcal{E} = \frac{E_1
+  + E_2}{2}, \; \Omega = \frac{E_1-E_2}{2},
+\]
+!et
+and
+!bt
+\[
+H_I = c \bm{I} +\omega_z\sigma_z + \omega_x\sigma_x,
+\]
+!et
+with $c = (V_{11}+V_{22})/2$, $\omega_z = (V_{11}-V_{22})/2$ and $\omega_x = V_{12}=V_{21}$.
+Study the behavior of these eigenstates as functions of the interaction strength $\lambda$.
+!esol
+!esubex
+
+!split
+===== Exercise: Develop code for two and more qubit basis sets =====
+
+Using the one-qubit basis write a code which sets up a two-qubit basis
+and encodes this basis.
+
+!split
+===== Exercise: Two-qubit Hamiltonian =====
+
+Use the Hamiltonian for the two-qubit example to find the eigenpairs
+as functions of the interaction strength $\lambda$ and study the final
+eigenvectors as functions of the admixture of the original basis
+states.
+
+
+
+
+
+!split
+===== The next lecture, February 5 =====
+
+In our next lecture, we will discuss
+o Reminder and review of  density matrices and measurements
+o Schmidt decomposition and entanglement
+o Discussion of entropies, classical information entropy (Shannon entropy) and von Neumann entropy
+Chapters 3 and 4 of Scherer's text contains useful discussions of several of these topics.
+
+
+
+
+