update week 14

Author: mhjensen

doc/pub/week14/html/week14-bs.html

@@ -0,0 +1,12 @@
+\mode<presentation>
+\usecolortheme[rgb={0.8, 0.2, 0}]{structure}
+\usefonttheme[onlysmall]{structurebold}
+
+\setbeamertemplate{navigation symbols}{}
+%\setbeamertemplate{footline}[frame number]
+
+\usepackage{tikz}
+\usetikzlibrary{arrows,shapes,backgrounds,decorations,mindmap}
+
+\mode
+<all>

doc/pub/week14/html/week14-reveal.html

@@ -49,20 +49,20 @@ system doconce format html $name --html_style=bootstrap --pygments_html_style=de
 # IPython notebook
 system doconce format ipynb $name $opt
 
-# Ordinary plain LaTeX document
-system doconce format pdflatex $name --minted_latex_style=trac --latex_admon=paragraph $opt
+
+# LaTeX Beamer slides
+beamertheme=red_plain
+system doconce format pdflatex $name --latex_title_layout=beamer --latex_table_format=footnotesize $opt
 system doconce ptex2tex $name envir=minted
 # Add special packages
 doconce subst "% Add user's preamble" "\g<1>\n\\usepackage{simplewick}" $name.tex
-doconce replace 'section{' 'section*{' $name.tex
-pdflatex -shell-escape $name
-pdflatex -shell-escape $name
-mv -f $name.pdf ${name}.pdf
+system doconce slides_beamer $name --beamer_slide_theme=$beamertheme
+system pdflatex -shell-escape ${name}
+system pdflatex -shell-escape ${name}
+cp $name.pdf ${name}.pdf
 cp $name.tex ${name}.tex
 
 
-
-
 # Publish
 dest=../../pub
 if [ ! -d $dest/$name ]; then
@@ -96,3 +96,7 @@ cp ${ipynb_tarfile} $dest/$name/ipynb
 
 
 
+
+
+
+

doc/pub/week14/html/week14-solarized.html

@@ -6,8 +6,334 @@ DATE: April 30
 ===== Plans for the week of April 28-May 2 =====
 
 !bblock 
-  O TBA
-  o "Video of lecture TBA":"https://youtu.be/"
+o Classical Support Vector Machines, reminder from last week
+o Quantum Support Vector Machines
+o And if we get time, we may start with Quantum Neural Networks
+#  o "Video of lecture TBA":"https://youtu.be/"
 #  o "Whiteboard notes":"https://github.com/CompPhysics/QuantumComputingMachineLearning/blob/gh-pages/doc/HandWrittenNotes/2024/NotesApril24.pdf"
 !eblock
 
+!bblock Reading recommendations
+o For classical Support Vector Machines, see lecture notes by Morten Hjorth-Jensen at URL:"https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter5.html"
+o Claudio Conti, Quantum Machine Learning (Springer), sections 1.5-1.12, see URL:"https://link.springer.com/book/10.1007/978-3-031-44226-1". With an IP-number from the university of Oslo, you can download this book for free. The material on quantum support vector machines will be discussed next week.
+!eblock
+
+!split
+===== What is Machine Learning? =====
+
+Machine Learning (ML) is the study of algorithms that improve through data experience.
+
+!bblock Types of Machine Learning:
+o \textbf{Supervised Learning:} Labeled data for classification or regression.
+o \textbf{Unsupervised Learning:} No labels; discover hidden patterns.
+o \textbf{Reinforcement Learning:} Learning through interaction with the environment.
+!eblock
+
+!bblock
+\textbf{ML Workflow:}
+!bt
+\[
+\text{Data} \rightarrow \text{Model Training} \rightarrow \text{Prediction}
+\]
+!et
+!eblock
+
+
+!split
+===== What is Quantum Machine Learning? =====
+
+Quantum Machine Learning (QML) integrates quantum computing with machine learning algorithms to exploit quantum advantages.
+
+!bblock Motivation:
+o High-dimensional Hilbert spaces for better feature representation.
+o Quantum parallelism for faster computation.
+o Quantum entanglement for richer data encoding.
+!eblock 
+
+!split
+===== Quantum Speedups in ML =====
+\textbf{Why Quantum?}
+!bblock
+o \textbf{Quantum Parallelism:} Process multiple states simultaneously.
+o \textbf{Quantum Entanglement:} Correlated states for richer information.
+o \textbf{Quantum Interference:} Constructive and destructive interference to enhance solutions.
+!eblock
+
+!bblock
+Example - Quantum Fourier transforms:
+!bt
+\begin{align*}
+\mathrm{Discrete Fourier Transform} & \sim O(N^2)\\
+\mathrm{Fast Fourier Transform} & \sim O(N^2)\\
+\mathrm{Qauntum Fourier Transform} & \sim O(N^2)\\
+\end{align*}
+!et
+!eblock
+
+Advantage: Speedups in high-dimensional optimization and linear algebra problems.
+
+
+!split
+===== Challenges in Quantum Machine Learning =====
+
+
+!bblock Quantum Hardware Limitations:
+o Noisy Intermediate-Scale Quantum (NISQ) devices.
+o Decoherence and limited qubit coherence times.
+!eblock
+
+!bblock Data Encoding:
+o Efficient embedding of classical data into quantum states.
+!eblock
+
+!bblock Scalability:
+o Difficult to scale circuits to large datasets.
+!eblock
+
+
+!split
+===== Classical Support Vector Machines (SVMs) =====
+
+Support Vector Machines (SVM) are supervised learning algorithms used
+for classification tasks, and regression tasks as well. The main goal
+of SVMs is to find the optimal separating hyperplane (in
+high-dimensional space) that provides a maximum margin between
+classes.
+
+!split
+===== First mathematical formulation of SVMs (formal details below) =====
+
+For a dataset \((\mathbf{x}_i, y_i)\) where \(\mathbf{x}_i \in
+\mathbb{R}^n\) and \(y_i \in \{-1, 1\}\), the decision boundary is
+defined as:
+!bt
+\[
+f(\mathbf{x}) = \mathbf{w} \cdot \mathbf{x} + b = 0.
+\]
+!et
+
+The goal is to optimize
+!bt
+\[
+\min_{\mathbf{w}, b} \frac{1}{2} ||\mathbf{w}||^2,
+\]
+!et
+subject to
+!bt
+\[
+y_i (\mathbf{w} \cdot \mathbf{x}_i + b) \geq 1, \quad \forall i
+\]
+!et
+
+!split
+===== Kernels and more =====
+
+In classical SVMs, kernels help with non-linear data
+separations. Quantum computers can speed up the computation of complex
+kernel evaluations by efficiently simulating an inner product in an
+exponentially large Hilbert space.
+
+
+
+!split
+=====  Quantum Support Vector Machines (QSVMs) =====
+
+!bblock Quantum Kernel Estimation:
+o Maps classical data to a quantum Hilbert space.
+o Quantum kernel measures similarity in high-dimensional space.
+!eblock
+
+!bblock Quantum-enhanced  kernel:
+!bt
+\[
+K(\bm{x}, \bm{x}') = |\langle \psi(\bm{x}) \vert  \psi(\bm{x}')\rangle|^2
+\]
+!et
+!eblock
+
+Here, $\vert \phi(\bm{x})\rangle$ is the quantum state encoding of
+the classical data $\bm{x}$.
+
+Advantage: Potentially exponential speedup over classical SVMs.
+
+
+!split
+===== Quantum Neural Networks (QNNs) =====
+
+Quantum Neural Networks replace classical neurons with parameterized quantum circuits.
+!bblock  Key Concepts:
+o Quantum Gates as Activation Functions.
+o Variational Quantum Circuits (VQCs) for optimization.
+!eblock
+!bblock
+\textbf{Parameterized Quantum Circuit:}
+!bt
+\[
+U(\theta) = \prod_i R_y(\theta_i) \cdot CNOT \cdot R_x(\theta_i)
+\]
+!et
+!eblock
+Advantage:Quantum gradients enable exploration of non-convex landscapes.
+
+!split
+===== Quantum Boltzmann Machines (QBMs) =====
+
+Quantum Boltzmann Machines leverage quantum mechanics to sample from a probability distribution.
+
+!bblock
+o Quantum tunneling aids in escaping local minima.
+o Quantum annealing for optimization problems.
+!eblock
+!bblock Quantum Hamiltonian:
+!bt
+\[
+H = -\sum_i b_i \sigma_i^z - \sum_{ij} w_{ij} \sigma_i^z \sigma_j^z
+\]
+!et
+!eblock
+Advantage: Efficient sampling in complex probability distributions.
+
+
+
+
+!split
+===== Quantum SVMs =====
+
+The idea of a classical SVM is that we have a set of points
+that are in either one group or another and we want to find a line that
+separates these two groups. This line can be linear, but it can also be
+much more complex, which can be achieved by the use of Kernels.
+
+!split
+===== What will we need in the case of a quantum computer? =====
+
+!bblock
+We will have to translate the classical data point \(\vec{x}\)
+into a quantum datapoint \(\vert \Phi{(\vec{x})} \rangle\). This can
+be achieved by a circuit $\mathcal{U}_{\Phi(\vec{x})} \vert 0\rangle$.
+
+
+Here $\Phi()$ could be any classical function applied
+on the classical data $\vec{x}$.
+!eblock
+!bblock 
+We need a parameterized quantum circuit $W(\theta)$ that
+processes the data in a way that in the end we
+can apply a measurement that returns a classical value \(-1\) or
+\(1\) for each classical input \(\vec{x}\) that indentifies the label
+of the classical data.
+!eblock
+
+
+!split
+===== The most general ansatz =====
+
+Following these steps we can define an ansatz for this kind of problem
+which is
+!bt
+\[
+W(\theta) \mathcal{U}_{\Phi}(\vec{x}) \vert 0 \rangle.
+\]
+!et
+
+These kind of ansatzes are called quantum variational circuits.
+
+!split
+===== Quantum SVM =====
+
+In the case of a quantum SVM we will only use the quantum feature maps
+!bt
+\[
+\mathcal{U}_{\Phi(\vec{x})},
+\]
+!et
+to translate the classical data into
+quantum states and build the Kernel of the SVM out of these quantum
+states. After calculating the Kernel matrix on the quantum computer we
+can train the Quantum SVM the same way as the classical SVM.
+
+
+!split
+===== Defining the Quantum Kernel =====
+
+The idea of the quantum kernel is exactly the same as in the classical
+case. We take the inner product
+!bt
+\[
+K(\vec{x}, \vec{z}) = \vert \langle \Phi (\vec{x}) \vert \Phi(\vec{z}) \rangle \vert^2 = \langle 0^n \vert \mathcal{U}_{\Phi(\vec{x})}^{t} \mathcal{U}_{\Phi(\vec{z})} \vert 0^n \rangle,
+\]
+!et
+but now with the quantum feature maps
+!bt
+\[
+\mathcal{U}_{\Phi(\vec{x})}.
+\]
+!et
+
+The idea is that if we choose a quantum feature maps that is not easy
+to simulate with a classical computer we might obtain a quantum
+advantage.
+
+
+!split
+===== Side note =====
+
+There is no proof yet that the QSVM brings a quantum
+advantage, but the argument the authors of
+URL:"https://arxiv.org/pdf/1804.11326.pdf" make, is that there is
+for sure no advantage if we use feature maps that are easy to simulate
+classically, because then we would not need a quantum computer to
+construct the Kernel.
+
+!split
+===== Feature map =====
+
+For the feature maps we use the ansatz
+!bt
+\[
+\mathcal{U}_{\Phi(x)} = U_{\Phi(x)} \otimes H^{\otimes n},
+\]
+!et
+where
+!bt
+\[
+U_{\Phi(x)} = \exp \left( i \sum_{S \in n} \phi_S(x) \prod_{i \in S} Z_i \right),
+\]
+!et
+which simplifies a lot when we (like in URL:"https://arxiv.org/pdf/1804.11326.pdf") only consider
+\(S \leq 2\) interactions, which means we only let two qubits interact
+at a time. For \(S \leq 2\) the product \(\prod_{i \in S}\) only leads
+to interactions \(Z_i Z_j\) and non interacting terms \(Z_i\). And the
+sum \(\sum_{S \in n}\) over all these terms that are possible with \(n\)
+qubits.
+
+!split
+===== Classical functions =====
+
+Finally we define the classical functions \(\phi_i(\vec{x}) = x_i\) and
+\(\phi_{i,j}(\vec{x}) = (\pi - x_i)( \pi- x_j)\).
+
+If we write this ansatz for 2 qubits and \(S \leq 2\) we see how it
+simplifies:
+!bt
+\[
+U_{\Phi(x)} = \exp \left(i \left(x_1 Z_1 + x_2 Z_2 + (\pi - x_1)( \pi- x_2) Z_1 Z_2 \right) \right).
+\]
+!et
+We won't get into details to much here, why we would take this ansatz.
+It is simply an ansatz that is simple enough an leads to good results.
+
+Finally we can define a depth of these circuits. Depth 2 means we repeat
+this ansatz two times. Which means our feature map becomes
+\(U_{\Phi(x)} \otimes H^{\otimes n} \otimes U_{\Phi(x)} \otimes H^{\otimes n}\)
+
+
+!split
+===== Plans for next week =====
+o Introducing quantum neural networks
+
+
+
+
+
+