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‎doc/pub/week1/html/week1-bs.html‎

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

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@@ -206,7 +206,6 @@ <h2 id="overview-of-first-week-basic-notions-of-quantum-mechanics">Overview of f
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<p><li> States in Hilbert space, pure and mixed states</li>
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<p><li> Operators and simple gates</li>
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<p><li> <a href="https://youtu.be/" target="_blank">Video of lecture to be added</a></li>
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<p><li> Test your background knowledge (to be added)</li>
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</ol>
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<p>
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<p><b>Reading recommendation</b>: <a href="https://link.springer.com/book/10.1007/978-3-030-12358-1" target="_blank">Scherer, Mathematics of Quantum Computations, chapter 2</a></p>
@@ -227,7 +226,7 @@ <h2 id="practicalities">Practicalities </h2>
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</ul>
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<p>
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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</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><li> All info at the GitHub address <a href="https://github.com/CompPhysics/QuantumComputingMachineLearning" target="_blank"><tt>https://github.com/CompPhysics/QuantumComputingMachineLearning</tt></a></li>
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</ol>
@@ -240,7 +239,6 @@ <h2 id="possible-textbooks">Possible textbooks </h2>
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<p><li> Wolfgang Scherer, Mathematics of Quantum Computing, see <a href="https://link.springer.com/book/10.1007/978-3-030-12358-1" target="_blank"><tt>https://link.springer.com/book/10.1007/978-3-030-12358-1</tt></a></li>
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<p><li> Robert Hundt, Quantum Computing for Programmers, <a href="https://www.cambridge.org/core/books/quantum-computing-for-programmers/BA1C887BE4AC0D0D5653E71FFBEF61C6" target="_blank"><tt>https://www.cambridge.org/core/books/quantum-computing-for-programmers/BA1C887BE4AC0D0D5653E71FFBEF61C6</tt></a></li>
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<p><li> Robert Loredo, Learn Quantum Computing with Python and IBM Quantum Experience, see <a href="https://github.com/PacktPublishing/Learn-Quantum-Computing-with-Python-and-IBM-Quantum-Experience" target="_blank"><tt>https://github.com/PacktPublishing/Learn-Quantum-Computing-with-Python-and-IBM-Quantum-Experience</tt></a></li>
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<p><li> More texts with links will be added.</li>
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</ol>
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</section>
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@@ -1403,11 +1401,358 @@ <h2 id="more-on-orthogonality">More on orthogonality </h2>
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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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<h2 id="entanglement">Entanglement </h2>
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<p>In order to study entanglement and why it is so important for quantum
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computing, we need to introduce some basic measures and useful
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quantities. These quantities are the spectral decomposition of
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hermitian operators, how these are then used to define measurements
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and how we can define so-called density operators (matrices). These
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are all quantities which will become very useful when we discuss
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entanglement and in particular how to quantify it. In order to define
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these quantities we need first to remind ourselves about some basic linear
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algebra properties of hermitian operators and matrices.
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</p>
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</section>
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<section>
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<h2 id="basic-properties-of-hermitian-operators">Basic properties of hermitian operators </h2>
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<p>The operators we typically encounter in quantum mechanical studies are</p>
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<ol>
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<p><li> Hermitian (self-adjoint) meaning that for example the elements of a Hermitian matrix \( \boldsymbol{U} \) obey \( u_{ij}=u_{ji}^* \).</li>
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<p><li> Unitary \( \boldsymbol{U}\boldsymbol{U}^{\dagger}=\boldsymbol{U}^{\dagger}\boldsymbol{U}=\boldsymbol{I} \), where \( \boldsymbol{I} \) is the unit matrix</li>
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<p><li> The oparator \( \boldsymbol{U} \) and its self-adjoint commute (often labeled as normal operators), that is \( [\boldsymbol{U},\boldsymbol{U}^{\dagger}]=0 \). An operator is <b>normal</b> if and only if it is diagonalizable. A Hermitian operator is normal.</li>
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</ol>
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<p>
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<p>Unitary operators in a Hilbert space preserve the norm and orthogonality. If \( \boldsymbol{U} \) is a unitary operator acting on a state \( \vert \psi_j\rangle \), the action of</p>
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<p>&nbsp;<br>
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$$
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\vert \phi_i\rangle=\boldsymbol{U}\vert \psi_j\rangle,
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$$
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<p>&nbsp;<br>
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<p>preserves both the norm and orthogonality, that is \( \langle \phi_i \vert \phi_j\rangle=\langle \psi_i \vert \psi_j\rangle=\delta_{ij} \), as discussed earlier.</p>
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</section>
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<section>
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<h2 id="the-pauli-matrices-again">The Pauli matrices again </h2>
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<p>As example, consider the Pauli matrix \( \sigma_x \). We have already seen that this matrix is a unitary matrix. Consider then an orthogonal and normalized basis \( \vert 0\rangle^{\dagger} =\begin{bmatrix} 1 &amp; 0\end{bmatrix} \) and \( \vert 1\rangle^{\dagger} =\begin{bmatrix} 0 &amp; 1\end{bmatrix} \) and a state which is a linear superposition of these two basis states</p>
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<p>&nbsp;<br>
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$$
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\vert \psi_a\rangle=\alpha_0\vert 0\rangle +\alpha_1\vert 1\rangle.
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$$
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<p>&nbsp;<br>
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<p>A new state \( \vert \psi_b\rangle \) is given by</p>
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<p>&nbsp;<br>
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$$
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\vert \psi_b\rangle=\sigma_x\vert \psi_a\rangle=\alpha_0\vert 1\rangle +\alpha_1\vert 0\rangle.
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$$
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<p>&nbsp;<br>
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</section>
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<section>
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<h2 id="spectral-decomposition">Spectral Decomposition </h2>
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<p>An important technicality which we will use in the discussion of
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density matrices, entanglement, quantum entropies and other properties
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is the so-called spectral decomposition of an operator.
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</p>
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<p>Let \( \vert \psi\rangle \) be a vector in a Hilbert space of dimension \( n \) and a hermitian operator \( \boldsymbol{A} \) defined in this
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space. Assume \( \vert \psi\rangle \) is an eigenvector of \( \boldsymbol{A} \) with eigenvalue \( \lambda \), that is
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</p>
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<p>&nbsp;<br>
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$$
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\boldsymbol{A}\vert \psi\rangle = \lambda\vert \psi\rangle = \lambda \boldsymbol{I}\vert \psi \rangle,
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$$
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<p>&nbsp;<br>
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<p>where we used \( \boldsymbol{I}\vert \psi \rangle = 1 \vert \psi \rangle \).
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Subtracting the right hand side from the left hand side gives
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</p>
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<p>&nbsp;<br>
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$$
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\left[\boldsymbol{A}-\lambda \boldsymbol{I}\right]\vert \psi \rangle=0,
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$$
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<p>&nbsp;<br>
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<p>which has a nontrivial solution only if the determinant
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\( \mathrm{det}(\boldsymbol{A}-\lambda\boldsymbol{I})=0 \).
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</p>
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</section>
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<section>
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<h2 id="exercises-to-test-yourself">Exercises to test yourself </h2>
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<p>To be added</p>
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<h2 id="onb-again-and-again">ONB again and again </h2>
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<p>We define now an orthonormal basis \( \vert i \rangle =\{\vert 0
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\rangle, \vert 1\rangle, \dots, \vert n-1\rangle \) in the same Hilbert
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space. We will assume that this basis is an eigenbasis of \( \boldsymbol{A} \) with eigenvalues \( \lambda_i \)
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</p>
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<p>We expand a new vector using this eigenbasis of \( \boldsymbol{A} \)</p>
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<p>&nbsp;<br>
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$$
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\vert \psi \rangle = \sum_{i=0}^{n-1}\alpha_i\vert i\rangle,
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$$
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<p>&nbsp;<br>
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<p>with the normalization condition \( \sum_{i=0}^{n-1}\vert \alpha_i\vert^2 \).
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Acting with \( \boldsymbol{A} \) on this new state results in
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</p>
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<p>&nbsp;<br>
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$$
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\boldsymbol{A}\vert \psi \rangle = \sum_{i=0}^{n-1}\alpha_i\boldsymbol{A}\vert i\rangle=\sum_{i=0}^{n-1}\alpha_i\lambda_i\vert i\rangle.
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$$
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<p>&nbsp;<br>
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</section>
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<section>
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<h2 id="projection-operators">Projection operators </h2>
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<p>If we then use that the outer product of any state with itself defines a projection operator we have the projection operators</p>
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<p>&nbsp;<br>
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$$
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\boldsymbol{P}_{\psi} = \vert \psi\rangle\langle \psi\vert,
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$$
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<p>&nbsp;<br>
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<p>and</p>
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<p>&nbsp;<br>
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$$
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\boldsymbol{P}_{j} = \vert j\rangle\langle j\vert,
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$$
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<p>&nbsp;<br>
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<p>we have that </p>
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<p>&nbsp;<br>
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$$
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\boldsymbol{P}_{j}\vert \psi\rangle=\vert j\rangle\langle j\vert\sum_{i=0}^{n-1}\alpha_i\vert i\rangle=\sum_{i=0}^{n-1}\alpha_i\vert j\rangle\langle j\vert i\rangle.
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$$
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<p>&nbsp;<br>
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</section>
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<section>
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<h2 id="further-manipulations">Further manipulations </h2>
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<p>This results in</p>
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<p>&nbsp;<br>
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$$
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\boldsymbol{P}_{j}\vert \psi\rangle=\alpha_j\vert j\rangle,
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$$
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<p>&nbsp;<br>
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<p>since \( \langle j\vert i\rangle \).
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With the last equation we can rewrite
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</p>
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<p>&nbsp;<br>
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$$
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\boldsymbol{A}\vert \psi \rangle = \sum_{i=0}^{n-1}\alpha_i\lambda_i\vert i\rangle=\sum_{i=0}^{n-1}\lambda_i\boldsymbol{P}_i\vert \psi\rangle,
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$$
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<p>&nbsp;<br>
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<p>from which we conclude that</p>
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<p>&nbsp;<br>
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$$
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\boldsymbol{A}=\sum_{i=0}^{n-1}\lambda_i\boldsymbol{P}_i.
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$$
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<p>&nbsp;<br>
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</section>
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<section>
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<h2 id="spectral-decomposition">Spectral decomposition </h2>
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<p>This is the spectral decomposition of a hermitian and normal
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operator. It is true for any state and it is independent of the
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basis. The spectral decomposition can in turn be used to exhaustively
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specify a measurement, as we will see in the next section.
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</p>
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<p>As an example, consider two states \( \vert \psi_a\rangle \) and \( \vert
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\psi_b\rangle \) that are eigenstates of \( \boldsymbol{A} \) with eigenvalues
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\( \lambda_a \) and \( \lambda_b \), respectively. In the diagonalization
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process we have obtained the coefficients \( \alpha_0 \), \( \alpha_1 \),
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\( \beta_0 \) and \( \beta_1 \) using an expansion in terms of the orthogonal
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basis \( \vert 0\rangle \) and \( \vert 1\rangle \).
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</p>
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</section>
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<section>
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<h2 id="explicit-results">Explicit results </h2>
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<p>We have then</p>
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<p>&nbsp;<br>
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$$
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\vert \psi_a\rangle = \alpha_0\vert 0\rangle+\alpha_1\vert 1\rangle,
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$$
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<p>&nbsp;<br>
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<p>and</p>
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<p>&nbsp;<br>
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$$
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\vert \psi_b\rangle = \beta_0\vert 0\rangle+\beta_1\vert 1\rangle,
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$$
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<p>&nbsp;<br>
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<p>with corresponding projection operators</p>
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<p>&nbsp;<br>
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$$
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\boldsymbol{P}_a=\vert \psi_a\rangle \langle \psi_a\vert = \begin{bmatrix} \vert \alpha_0\vert^2 &\alpha_0\alpha_1^* \\
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\alpha_1\alpha_0^* & \vert \alpha_1\vert^* \end{bmatrix},
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$$
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<p>&nbsp;<br>
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<p>and</p>
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<p>&nbsp;<br>
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$$
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\boldsymbol{P}_b=\vert \psi_b\rangle \langle \psi_b\vert = \begin{bmatrix} \vert \beta_0\vert^2 &\beta_0\beta_1^* \\
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\beta_1\beta_0^* & \vert \beta_1\vert^* \end{bmatrix}.
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$$
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<p>&nbsp;<br>
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</section>
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<section>
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<h2 id="the-spectral-decomposition">The spectral decomposition </h2>
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<p>The results from the previous slide gives us
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the following spectral decomposition of \( \boldsymbol{A} \)
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</p>
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<p>&nbsp;<br>
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$$
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\boldsymbol{A}=\lambda_a \vert \psi_a\rangle \langle \psi_a\vert+\lambda_b \vert \psi_b\rangle \langle \psi_b\vert,
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$$
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<p>&nbsp;<br>
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<p>which written out in all its details reads</p>
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<p>&nbsp;<br>
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$$
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\boldsymbol{A}=\lambda_a\begin{bmatrix} \vert \alpha_0\vert^2 &\alpha_0\alpha_1^* \\
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\alpha_1\alpha_0^* & \vert \alpha_1\vert^* \end{bmatrix} +\lambda_b\begin{bmatrix} \vert \beta_0\vert^2 &\beta_0\beta_1^* \\
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\beta_1\beta_0^* & \vert \beta_1\vert^* \end{bmatrix}.
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$$
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<p>&nbsp;<br>
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</section>
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<section>
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<h2 id="first-exercise-set">First exercise set </h2>
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<p>The last two exercises are meant to build the basis for
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the two projects we will work on during the semester. The first
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project deals with implementing the so-called
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<b>Variational Quantum Eigensolver</b> algorithm for finding the eigenvalues and eigenvectors of selected Hamiltonians.
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</p>
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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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<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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$$
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\vert \Phi^+\rangle = \frac{1}{\sqrt{2}}\left[\vert 00\rangle +\vert 11\rangle\right]=\frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ 0 \\ 0 \\ 1\end{bmatrix},
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$$
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<p>&nbsp;<br>
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<p>&nbsp;<br>
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$$
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\vert \Phi^-\rangle = \frac{1}{\sqrt{2}}\left[\vert 00\rangle -\vert 11\rangle\right]=\frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ 0 \\ 0 \\ -1\end{bmatrix},
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$$
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<p>&nbsp;<br>
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<!-- --- end exercise --- -->
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</section>
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<section>
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<h2 id="and-the-next-two">And the next two </h2>
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<p>&nbsp;<br>
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$$
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\vert \Psi^+\rangle = \frac{1}{\sqrt{2}}\left[\vert 10\rangle +\vert 01\rangle\right]=\frac{1}{\sqrt{2}}\begin{bmatrix} 0 \\ 1 \\ 1 \\ 0\end{bmatrix},
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$$
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<p>&nbsp;<br>
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<p>and</p>
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<p>&nbsp;<br>
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$$
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\vert \Psi^-\rangle = \frac{1}{\sqrt{2}}\left[\vert 10\rangle -\vert 01\rangle\right]=\frac{1}{\sqrt{2}}\begin{bmatrix} 0 \\ 1 \\ -1 \\ 0\end{bmatrix}.
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$$
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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-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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<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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<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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<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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<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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