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‎doc/pub/week1/html/week1-bs.html‎

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None,
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'overview-of-first-week-basic-notions-of-quantum-mechanics'),
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('Practicalities', 2, None, 'practicalities'),
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('More on projects and final grade',
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2,
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None,
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'more-on-projects-and-final-grade'),
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('Possible textbooks', 2, None, 'possible-textbooks'),
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('Online material of possible interest',
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2,
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2,
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None,
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'exercise-2-entangled-state'),
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('Exercise 3: Commutator identies',
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('Exercise 3: Commutator identities',
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2,
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None,
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'exercise-3-commutator-identities'),
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('Exercise 4: Shared eigenvectors',
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2,
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None,
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'exercise-3-commutator-identies'),
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('Shared eigenvectors', 2, None, 'shared-eigenvectors'),
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('Exercise 4: One-qubit basis and Pauli matrices',
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'exercise-4-shared-eigenvectors'),
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('Exercise 5: One-qubit basis and Pauli matrices',
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2,
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None,
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'exercise-4-one-qubit-basis-and-pauli-matrices'),
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('Exercise 5: Hadamard and Phase gates',
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'exercise-5-one-qubit-basis-and-pauli-matrices'),
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('Exercise 6: Hadamard and Phase gates',
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2,
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None,
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'exercise-5-hadamard-and-phase-gates')]}
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'exercise-6-hadamard-and-phase-gates')]}
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end of tocinfo -->
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<body>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="#overview-of-first-week-basic-notions-of-quantum-mechanics" style="font-size: 80%;">Overview of first week, Basic Notions of Quantum Mechanics</a></li>
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<!-- navigation toc: --> <li><a href="#practicalities" style="font-size: 80%;">Practicalities</a></li>
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<!-- navigation toc: --> <li><a href="#more-on-projects-and-final-grade" style="font-size: 80%;">More on projects and final grade</a></li>
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<!-- navigation toc: --> <li><a href="#possible-textbooks" style="font-size: 80%;">Possible textbooks</a></li>
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<!-- navigation toc: --> <li><a href="#online-material-of-possible-interest" style="font-size: 80%;">Online material of possible interest</a></li>
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<!-- navigation toc: --> <li><a href="#notations-and-definitions" style="font-size: 80%;">Notations and definitions</a></li>
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<!-- navigation toc: --> <li><a href="#exercise-1-bell-states" style="font-size: 80%;">Exercise 1: Bell states</a></li>
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<!-- navigation toc: --> <li><a href="#and-the-next-two" style="font-size: 80%;">And the next two</a></li>
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<!-- navigation toc: --> <li><a href="#exercise-2-entangled-state" style="font-size: 80%;">Exercise 2: Entangled state</a></li>
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<!-- navigation toc: --> <li><a href="#exercise-3-commutator-identies" style="font-size: 80%;">Exercise 3: Commutator identies</a></li>
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<!-- navigation toc: --> <li><a href="#shared-eigenvectors" style="font-size: 80%;">Shared eigenvectors</a></li>
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<!-- navigation toc: --> <li><a href="#exercise-4-one-qubit-basis-and-pauli-matrices" style="font-size: 80%;">Exercise 4: One-qubit basis and Pauli matrices</a></li>
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<!-- navigation toc: --> <li><a href="#exercise-5-hadamard-and-phase-gates" style="font-size: 80%;">Exercise 5: Hadamard and Phase gates</a></li>
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<!-- navigation toc: --> <li><a href="#exercise-3-commutator-identities" style="font-size: 80%;">Exercise 3: Commutator identities</a></li>
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<!-- navigation toc: --> <li><a href="#exercise-4-shared-eigenvectors" style="font-size: 80%;">Exercise 4: Shared eigenvectors</a></li>
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<!-- navigation toc: --> <li><a href="#exercise-5-one-qubit-basis-and-pauli-matrices" style="font-size: 80%;">Exercise 5: One-qubit basis and Pauli matrices</a></li>
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<!-- navigation toc: --> <li><a href="#exercise-6-hadamard-and-phase-gates" style="font-size: 80%;">Exercise 6: Hadamard and Phase gates</a></li>
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</ul>
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</li>
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<h2 id="practicalities" class="anchor">Practicalities </h2>
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<ol>
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<li> Weekly lectures with weekly exercise sessions and assignments. The assignments are meant as background for the twp projects</li>
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<li> Weekly lectures with weekly exercise sessions and assignments. The assignments are meant as background for the two projects</li>
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<li> We plan to work on two projects which will define the content of the course, the format can be agreed upon by the participants but the following topics are those we wish to focus on:</li>
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<ul>
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<li> First project: Quantum computing and simulation of quantum mechanical systems</li>
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<li> Second project: Continuation of the first topic with more realistic systems and using adaptive VQE</li>
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<li> Second project: Applications and implementations of quantum machine learning algorithms</li>
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<li> Second project: studies of entanglement and physical realization of quantum gates and circuits</li>
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</ul>
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</ol>
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<!-- !split -->
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<h2 id="more-on-projects-and-final-grade" class="anchor">More on projects and final grade </h2>
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<ol>
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<li> Two projects which count \( 50\% \) each for the final grade</li>
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<li> Deadline first project March 21, see <a href="https://github.com/CompPhysics/QuantumComputingMachineLearning/tree/gh-pages/doc/Projects/2025/Project1" target="_self"><tt>https://github.com/CompPhysics/QuantumComputingMachineLearning/tree/gh-pages/doc/Projects/2025/Project1</tt></a></li>
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<li> Deadline second project June 1</li>
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length (the square root of the inner product) of the state vector.
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</p>
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<p>gates discussed below are examples of operations we can perform on
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specific states.
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</p>
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<p>We consider the state</p>
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$$
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\vert \psi\rangle = \alpha \vert 0 \rangle +\beta \vert 1 \rangle
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$$
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<!-- !split -->
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<h2 id="entanglement" class="anchor">Entanglement </h2>
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</p>
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<!-- !split -->
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<!-- --- begin exercise --- -->
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<h2 id="exercise-1-bell-states" class="anchor">Exercise 1: Bell states </h2>
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<h2 id="exercise-1-bell-states" class="anchor">Exercise 1: Bell states </h2>
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<p>Show that the so-called Bell states listed here (and to be encountered many times in this course) form an orthogonal basis</p>
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$$
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$$
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<!-- --- end exercise --- -->
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<!-- !split -->
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<h2 id="and-the-next-two" class="anchor">And the next two </h2>
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$$
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<!-- !split -->
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<!-- --- begin exercise --- -->
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<h2 id="exercise-2-entangled-state" class="anchor">Exercise 2: Entangled state </h2>
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<h2 id="exercise-2-entangled-state" class="anchor">Exercise 2: Entangled state </h2>
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<p>Show that the state \( \alpha \vert 00\rangle+\beta\vert 11\rangle \) cannot be written as the product of the tensor product of two states and is thus entangle. The constants \( \alpha \) and \( \beta \) are both nonzero.</p>
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<p>Write a function which sets up a one-qubit basis and apply the various Pauli matrices to these basis states.</p>
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<!-- --- end exercise --- -->
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<!-- !split -->
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<!-- --- begin exercise --- -->
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<h2 id="exercise-3-commutator-identies" class="anchor">Exercise 3: Commutator identies </h2>
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<h2 id="exercise-3-commutator-identities" class="anchor">Exercise 3: Commutator identities </h2>
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<p>Prove the following commutator relations for different operators (marked with a hat)</p>
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<ol>
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<li> \( [\hat{A}+\hat{B},\hat{C}]= [\hat{A},\hat{C}]+[\hat{B},\hat{C}] \);</li>
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<li> \( [\hat{A},\hat{B}\hat{C}]= [\hat{A},\hat{B}]\hat{C}+\hat{B}[\hat{A},\hat{C}] \); and</li>
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<li> \( [\hat{A},[\hat{B}\hat{C}]]= [\hat{B},[\hat{C},\hat{A}]]+[\hat{C},[\hat{A},\hat{B}]]=0 \) (the so-called Jacobi identity).</li>
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</ol>
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<!-- --- end exercise --- -->
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<!-- !split -->
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<h2 id="shared-eigenvectors" class="anchor">Shared eigenvectors </h2>
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<h2 id="exercise-4-shared-eigenvectors" class="anchor">Exercise 4: Shared eigenvectors </h2>
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<p>Prove that if two operators \( \hat{A} \) and \( \hat{B} \) commute they will share a basis of eigenstates</p>
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<!-- !split -->
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<!-- --- begin exercise --- -->
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<h2 id="exercise-4-one-qubit-basis-and-pauli-matrices" class="anchor">Exercise 4: One-qubit basis and Pauli matrices </h2>
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<h2 id="exercise-5-one-qubit-basis-and-pauli-matrices" class="anchor">Exercise 5: One-qubit basis and Pauli matrices </h2>
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<p>Write a function which sets up a one-qubit basis and apply the various Pauli matrices to these basis states.</p>
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<!-- --- end exercise --- -->
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<!-- !split -->
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<!-- --- begin exercise --- -->
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<h2 id="exercise-5-hadamard-and-phase-gates" class="anchor">Exercise 5: Hadamard and Phase gates </h2>
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<h2 id="exercise-6-hadamard-and-phase-gates" class="anchor">Exercise 6: Hadamard and Phase gates </h2>
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<p>Apply the Hadamard and Phase gates to the same one-qubit basis states and study their actions on these states.</p>
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<!-- --- end exercise --- -->
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<!-- ------------------- end of main content --------------- -->
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</div> <!-- end container -->
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<!-- include javascript, jQuery *first* -->

‎doc/pub/week1/html/week1-reveal.html‎

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<h2 id="practicalities">Practicalities </h2>
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<ol>
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<p><li> Weekly lectures with weekly exercise sessions and assignments. The assignments are meant as background for the twp projects</li>
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<p><li> Weekly lectures with weekly exercise sessions and assignments. The assignments are meant as background for the two projects</li>
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<p><li> We plan to work on two projects which will define the content of the course, the format can be agreed upon by the participants but the following topics are those we wish to focus on:</li>
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<ul>
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<p><li> First project: Quantum computing and simulation of quantum mechanical systems</li>
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<p><li> Second project: studies of entanglement and physical realization of quantum gates and circuits</li>
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</ul>
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<p>
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</ol>
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</section>
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<section>
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<h2 id="more-on-projects-and-final-grade">More on projects and final grade </h2>
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<ol>
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<p><li> Two projects which count \( 50\% \) each for the final grade</li>
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<p><li> Deadline first project March 21, see <a href="https://github.com/CompPhysics/QuantumComputingMachineLearning/tree/gh-pages/doc/Projects/2025/Project1" target="_blank"><tt>https://github.com/CompPhysics/QuantumComputingMachineLearning/tree/gh-pages/doc/Projects/2025/Project1</tt></a></li>
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<p><li> Deadline second project June 1</li>
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<p>Unitary transformations are rotations in state space which preserve the
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length (the square root of the inner product) of the state vector.
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</p>
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<p>gates discussed below are examples of operations we can perform on
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specific states.
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</p>
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<p>We consider the state</p>
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<p>&nbsp;<br>
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$$
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\vert \psi\rangle = \alpha \vert 0 \rangle +\beta \vert 1 \rangle
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$$
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<p>&nbsp;<br>
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<section>
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<section>
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<!-- --- begin exercise --- -->
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<h2 id="exercise-1-bell-states">Exercise 1: Bell states </h2>
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<h2 id="exercise-1-bell-states">Exercise 1: Bell states </h2>
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<p>Show that the so-called Bell states listed here (and to be encountered many times in this course) form an orthogonal basis</p>
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<p>&nbsp;<br>
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\vert \Phi^-\rangle = \frac{1}{\sqrt{2}}\left[\vert 00\rangle -\vert 11\rangle\right]=\frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ 0 \\ 0 \\ -1\end{bmatrix},
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$$
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<p>&nbsp;<br>
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<!-- --- end exercise --- -->
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<section>
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<!-- --- begin exercise --- -->
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<h2 id="exercise-2-entangled-state">Exercise 2: Entangled state </h2>
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<h2 id="exercise-2-entangled-state">Exercise 2: Entangled state </h2>
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<p>Show that the state \( \alpha \vert 00\rangle+\beta\vert 11\rangle \) cannot be written as the product of the tensor product of two states and is thus entangle. The constants \( \alpha \) and \( \beta \) are both nonzero.</p>
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<p>Write a function which sets up a one-qubit basis and apply the various Pauli matrices to these basis states.</p>
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<!-- --- end exercise --- -->
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</section>
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<section>
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<!-- --- begin exercise --- -->
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<h2 id="exercise-3-commutator-identies">Exercise 3: Commutator identies </h2>
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<h2 id="exercise-3-commutator-identities">Exercise 3: Commutator identities </h2>
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<p>Prove the following commutator relations for different operators (marked with a hat)</p>
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<ol>
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<p><li> \( [\hat{A}+\hat{B},\hat{C}]= [\hat{A},\hat{C}]+[\hat{B},\hat{C}] \);</li>
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<p><li> \( [\hat{A},\hat{B}\hat{C}]= [\hat{A},\hat{B}]\hat{C}+\hat{B}[\hat{A},\hat{C}] \); and</li>
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<p><li> \( [\hat{A},[\hat{B}\hat{C}]]= [\hat{B},[\hat{C},\hat{A}]]+[\hat{C},[\hat{A},\hat{B}]]=0 \) (the so-called Jacobi identity).</li>
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</ol>
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<p>
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<!-- --- end exercise --- -->
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</section>
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<section>
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<h2 id="shared-eigenvectors">Shared eigenvectors </h2>
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<h2 id="exercise-4-shared-eigenvectors">Exercise 4: Shared eigenvectors </h2>
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<p>Prove that if two operators \( \hat{A} \) and \( \hat{B} \) commute they will share a basis of eigenstates</p>
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</section>
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<section>
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<!-- --- begin exercise --- -->
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<h2 id="exercise-4-one-qubit-basis-and-pauli-matrices">Exercise 4: One-qubit basis and Pauli matrices </h2>
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<h2 id="exercise-5-one-qubit-basis-and-pauli-matrices">Exercise 5: One-qubit basis and Pauli matrices </h2>
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<p>Write a function which sets up a one-qubit basis and apply the various Pauli matrices to these basis states.</p>
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<!-- --- end exercise --- -->
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</section>
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<section>
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<!-- --- begin exercise --- -->
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<h2 id="exercise-5-hadamard-and-phase-gates">Exercise 5: Hadamard and Phase gates </h2>
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<h2 id="exercise-6-hadamard-and-phase-gates">Exercise 6: Hadamard and Phase gates </h2>
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<p>Apply the Hadamard and Phase gates to the same one-qubit basis states and study their actions on these states.</p>
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<!-- --- end exercise --- -->
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</section>
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