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update week 13
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‎doc/pub/week13/html/week13-bs.html‎

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@@ -168,15 +168,6 @@
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2,
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None,
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'what-will-we-need-in-the-case-of-a-quantum-computer'),
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('The most general ansatz', 2, None, 'the-most-general-ansatz'),
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('Quantum SVM', 2, None, 'quantum-svm'),
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('Defining the Quantum Kernel',
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2,
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None,
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'defining-the-quantum-kernel'),
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('Side note', 2, None, 'side-note'),
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('Feature map', 2, None, 'feature-map'),
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('Classical functions', 2, None, 'classical-functions'),
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('Plans for next week', 2, None, 'plans-for-next-week')]}
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end of tocinfo -->
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@@ -271,12 +262,6 @@
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<!-- navigation toc: --> <li><a href="#back-to-the-more-realistic-cases" style="font-size: 80%;">Back to the more realistic cases</a></li>
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<!-- navigation toc: --> <li><a href="#quantum-svms" style="font-size: 80%;">Quantum SVMs</a></li>
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<!-- navigation toc: --> <li><a href="#what-will-we-need-in-the-case-of-a-quantum-computer" style="font-size: 80%;">What will we need in the case of a quantum computer?</a></li>
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<!-- navigation toc: --> <li><a href="#the-most-general-ansatz" style="font-size: 80%;">The most general ansatz</a></li>
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<!-- navigation toc: --> <li><a href="#quantum-svm" style="font-size: 80%;">Quantum SVM</a></li>
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<!-- navigation toc: --> <li><a href="#defining-the-quantum-kernel" style="font-size: 80%;">Defining the Quantum Kernel</a></li>
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<!-- navigation toc: --> <li><a href="#side-note" style="font-size: 80%;">Side note</a></li>
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<!-- navigation toc: --> <li><a href="#feature-map" style="font-size: 80%;">Feature map</a></li>
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<!-- navigation toc: --> <li><a href="#classical-functions" style="font-size: 80%;">Classical functions</a></li>
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<!-- navigation toc: --> <li><a href="#plans-for-next-week" style="font-size: 80%;">Plans for next week</a></li>
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</ul>
@@ -319,7 +304,7 @@ <h2 id="plans-for-the-week-of-april-21-25" class="anchor">Plans for the week of
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<ol>
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<li> Introduction to QML</li>
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<li> Support Vector Machines (SVM, classical machine learning approach)</li>
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<li> Quantum Support Vector Machine Learning (QSVM)</li>
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<li> Quantum Support Vector Machine Learning (QSVM), short intro</li>
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<li> Video of lecture at <a href="https://youtu.be/C36Kg4eaO7A" target="_self"><tt>https://youtu.be/C36Kg4eaO7A</tt></a></li>
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<li> Whiteboard notes at <a href="https://github.com/CompPhysics/QuantumComputingMachineLearning/blob/gh-pages/doc/HandWrittenNotes/2025/NotesApril23.pdf" target="_self"><tt>https://github.com/CompPhysics/QuantumComputingMachineLearning/blob/gh-pages/doc/HandWrittenNotes/2025/NotesApril23.pdf</tt></a></li>
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</ol>
@@ -1883,107 +1868,6 @@ <h2 id="what-will-we-need-in-the-case-of-a-quantum-computer" class="anchor">What
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</div>
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<!-- !split -->
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<h2 id="the-most-general-ansatz" class="anchor">The most general ansatz </h2>
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<p>Following these steps we can define an ansatz for this kind of problem
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which is
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</p>
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$$
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W(\theta) \mathcal{U}_{\Phi}(\vec{x}) \vert 0 \rangle.
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$$
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<p>These kind of ansatzes are called quantum variational circuits.</p>
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<!-- !split -->
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<h2 id="quantum-svm" class="anchor">Quantum SVM </h2>
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<p>In the case of a quantum SVM we will only use the quantum feature maps</p>
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$$
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\mathcal{U}_{\Phi(\vec{x})},
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$$
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<p>to translate the classical data into
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quantum states and build the Kernel of the SVM out of these quantum
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states. After calculating the Kernel matrix on the quantum computer we
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can train the Quantum SVM the same way as the classical SVM.
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</p>
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<!-- !split -->
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<h2 id="defining-the-quantum-kernel" class="anchor">Defining the Quantum Kernel </h2>
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<p>The idea of the quantum kernel is exactly the same as in the classical
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case. We take the inner product
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</p>
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$$
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K(\vec{x}, \vec{z}) = \vert \langle \Phi (\vec{x}) \vert \Phi(\vec{z}) \rangle \vert^2 = \langle 0^n \vert \mathcal{U}_{\Phi(\vec{x})}^{t} \mathcal{U}_{\Phi(\vec{z})} \vert 0^n \rangle,
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$$
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<p>but now with the quantum feature maps</p>
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$$
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\mathcal{U}_{\Phi(\vec{x})}.
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$$
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<p>The idea is that if we choose a quantum feature maps that is not easy
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to simulate with a classical computer we might obtain a quantum
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advantage.
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</p>
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<!-- !split -->
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<h2 id="side-note" class="anchor">Side note </h2>
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<p>There is no proof yet that the QSVM brings a quantum
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advantage, but the argument the authors of
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<a href="https://arxiv.org/pdf/1804.11326.pdf" target="_self"><tt>https://arxiv.org/pdf/1804.11326.pdf</tt></a> make, is that there is
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for sure no advantage if we use feature maps that are easy to simulate
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classically, because then we would not need a quantum computer to
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construct the Kernel.
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</p>
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<!-- !split -->
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<h2 id="feature-map" class="anchor">Feature map </h2>
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<p>For the feature maps we use the ansatz</p>
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$$
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\mathcal{U}_{\Phi(x)} = U_{\Phi(x)} \otimes H^{\otimes n},
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$$
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<p>where</p>
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$$
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U_{\Phi(x)} = \exp \left( i \sum_{S \in n} \phi_S(x) \prod_{i \in S} Z_i \right),
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$$
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<p>which simplifies a lot when we (like in <a href="https://arxiv.org/pdf/1804.11326.pdf" target="_self"><tt>https://arxiv.org/pdf/1804.11326.pdf</tt></a>) only consider
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\(S \leq 2\) interactions, which means we only let two qubits interact
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at a time. For \(S \leq 2\) the product \(\prod_{i \in S}\) only leads
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to interactions \(Z_i Z_j\) and non interacting terms \(Z_i\). And the
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sum \(\sum_{S \in n}\) over all these terms that are possible with \(n\)
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qubits.
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</p>
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<!-- !split -->
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<h2 id="classical-functions" class="anchor">Classical functions </h2>
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<p>Finally we define the classical functions \(\phi_i(\vec{x}) = x_i\) and
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\(\phi_{i,j}(\vec{x}) = (\pi - x_i)( \pi- x_j)\).
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</p>
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<p>If we write this ansatz for 2 qubits and \(S \leq 2\) we see how it
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simplifies:
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</p>
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$$
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U_{\Phi(x)} = \exp \left(i \left(x_1 Z_1 + x_2 Z_2 + (\pi - x_1)( \pi- x_2) Z_1 Z_2 \right) \right).
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$$
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<p>We won't get into details to much here, why we would take this ansatz.
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It is simply an ansatz that is simple enough an leads to good results.
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</p>
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<p>Finally we can define a depth of these circuits. Depth 2 means we repeat
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this ansatz two times. Which means our feature map becomes
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\(U_{\Phi(x)} \otimes H^{\otimes n} \otimes U_{\Phi(x)} \otimes H^{\otimes n}\)
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</p>
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<!-- !split -->
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<h2 id="plans-for-next-week" class="anchor">Plans for next week </h2>
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<ol>

‎doc/pub/week13/html/week13-reveal.html‎

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<ol>
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<p><li> Introduction to QML</li>
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<p><li> Support Vector Machines (SVM, classical machine learning approach)</li>
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<p><li> Quantum Support Vector Machine Learning (QSVM)</li>
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<p><li> Quantum Support Vector Machine Learning (QSVM), short intro</li>
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<p><li> Video of lecture at <a href="https://youtu.be/C36Kg4eaO7A" target="_blank"><tt>https://youtu.be/C36Kg4eaO7A</tt></a></li>
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<p><li> Whiteboard notes at <a href="https://github.com/CompPhysics/QuantumComputingMachineLearning/blob/gh-pages/doc/HandWrittenNotes/2025/NotesApril23.pdf" target="_blank"><tt>https://github.com/CompPhysics/QuantumComputingMachineLearning/blob/gh-pages/doc/HandWrittenNotes/2025/NotesApril23.pdf</tt></a></li>
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</ol>
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</div>
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</section>
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<section>
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<h2 id="the-most-general-ansatz">The most general ansatz </h2>
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<p>Following these steps we can define an ansatz for this kind of problem
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which is
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</p>
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<p>&nbsp;<br>
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$$
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W(\theta) \mathcal{U}_{\Phi}(\vec{x}) \vert 0 \rangle.
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$$
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<p>&nbsp;<br>
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<p>These kind of ansatzes are called quantum variational circuits.</p>
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</section>
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<section>
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<h2 id="quantum-svm">Quantum SVM </h2>
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<p>In the case of a quantum SVM we will only use the quantum feature maps</p>
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<p>&nbsp;<br>
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$$
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\mathcal{U}_{\Phi(\vec{x})},
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$$
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<p>&nbsp;<br>
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<p>to translate the classical data into
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quantum states and build the Kernel of the SVM out of these quantum
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states. After calculating the Kernel matrix on the quantum computer we
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can train the Quantum SVM the same way as the classical SVM.
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</p>
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</section>
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<section>
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<h2 id="defining-the-quantum-kernel">Defining the Quantum Kernel </h2>
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<p>The idea of the quantum kernel is exactly the same as in the classical
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case. We take the inner product
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</p>
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<p>&nbsp;<br>
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$$
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K(\vec{x}, \vec{z}) = \vert \langle \Phi (\vec{x}) \vert \Phi(\vec{z}) \rangle \vert^2 = \langle 0^n \vert \mathcal{U}_{\Phi(\vec{x})}^{t} \mathcal{U}_{\Phi(\vec{z})} \vert 0^n \rangle,
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$$
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<p>&nbsp;<br>
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<p>but now with the quantum feature maps</p>
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<p>&nbsp;<br>
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$$
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\mathcal{U}_{\Phi(\vec{x})}.
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$$
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<p>&nbsp;<br>
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1991-
<p>The idea is that if we choose a quantum feature maps that is not easy
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to simulate with a classical computer we might obtain a quantum
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advantage.
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</p>
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</section>
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<section>
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<h2 id="side-note">Side note </h2>
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<p>There is no proof yet that the QSVM brings a quantum
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advantage, but the argument the authors of
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<a href="https://arxiv.org/pdf/1804.11326.pdf" target="_blank"><tt>https://arxiv.org/pdf/1804.11326.pdf</tt></a> make, is that there is
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for sure no advantage if we use feature maps that are easy to simulate
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classically, because then we would not need a quantum computer to
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construct the Kernel.
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</p>
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</section>
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<section>
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<h2 id="feature-map">Feature map </h2>
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<p>For the feature maps we use the ansatz</p>
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<p>&nbsp;<br>
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$$
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\mathcal{U}_{\Phi(x)} = U_{\Phi(x)} \otimes H^{\otimes n},
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$$
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<p>&nbsp;<br>
2018-
2019-
<p>where</p>
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<p>&nbsp;<br>
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$$
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U_{\Phi(x)} = \exp \left( i \sum_{S \in n} \phi_S(x) \prod_{i \in S} Z_i \right),
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$$
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<p>&nbsp;<br>
2025-
2026-
<p>which simplifies a lot when we (like in <a href="https://arxiv.org/pdf/1804.11326.pdf" target="_blank"><tt>https://arxiv.org/pdf/1804.11326.pdf</tt></a>) only consider
2027-
\(S \leq 2\) interactions, which means we only let two qubits interact
2028-
at a time. For \(S \leq 2\) the product \(\prod_{i \in S}\) only leads
2029-
to interactions \(Z_i Z_j\) and non interacting terms \(Z_i\). And the
2030-
sum \(\sum_{S \in n}\) over all these terms that are possible with \(n\)
2031-
qubits.
2032-
</p>
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</section>
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2035-
<section>
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<h2 id="classical-functions">Classical functions </h2>
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<p>Finally we define the classical functions \(\phi_i(\vec{x}) = x_i\) and
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\(\phi_{i,j}(\vec{x}) = (\pi - x_i)( \pi- x_j)\).
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</p>
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2042-
<p>If we write this ansatz for 2 qubits and \(S \leq 2\) we see how it
2043-
simplifies:
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</p>
2045-
<p>&nbsp;<br>
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$$
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U_{\Phi(x)} = \exp \left(i \left(x_1 Z_1 + x_2 Z_2 + (\pi - x_1)( \pi- x_2) Z_1 Z_2 \right) \right).
2048-
$$
2049-
<p>&nbsp;<br>
2050-
2051-
<p>We won't get into details to much here, why we would take this ansatz.
2052-
It is simply an ansatz that is simple enough an leads to good results.
2053-
</p>
2054-
2055-
<p>Finally we can define a depth of these circuits. Depth 2 means we repeat
2056-
this ansatz two times. Which means our feature map becomes
2057-
\(U_{\Phi(x)} \otimes H^{\otimes n} \otimes U_{\Phi(x)} \otimes H^{\otimes n}\)
2058-
</p>
2059-
</section>
2060-
20611940
<section>
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<h2 id="plans-for-next-week">Plans for next week </h2>
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<ol>

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