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Author: mhjensen

doc/pub/QCML/html/QCML-bs.html

@@ -106,15 +106,15 @@
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-Quantum Computing and Quantum Machine Learning
+Quantum Machine Learning
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-{\bf Master of Science Thesis Project${}^{}$} \\ [0mm]
+{\bf Master of Science thesis project${}^{}$} \\ [0mm]
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@@ -125,7 +125,7 @@
 
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-Dec 1, 2022
+May 4, 2025
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@@ -136,24 +136,102 @@ \subsection{Quantum Computing and Machine Learning}
 
 \textbf{Quantum Computing and Machine Learning} are two of the most promising
 approaches for studying complex physical systems where several length
-and energy scales are involved.  Traditional many-particle methods,
-either quantum mechanical or classical ones, face huge dimensionality
-problems when applied to studies of systems with many interacting
-particles. To be able to define properly effective potentials for
-realistic molecular dynamics simulations of billions or more
-particles, requires both precise quantum mechanical studies as well as
-algorithms that allow for parametrizations and simplifications of
-quantum mechanical results. Quantum Computing offers now an
-interesting avenue, together with traditional algorithms, for studying
-complex quantum mechanical systems. Machine Learning on the other hand
-allows us to parametrize these results in terms of classical
-interactions. These interactions are in turn suitable for large scale
-molecular dynamics simulations of complicated systems spanning from
-subatomic physics to materials science and life science.
-
-\subsection{Possible  Projects}
-
-\paragraph{Boltzmann machines, from classical ones to quantum Boltzmann machines (Classical and Quantum Machine Learning).}
+and energy scales are involved.  
+
+Quantum computing is an emerging area of computer science that
+leverages the principles of quantum mechanics to perform computations
+beyond the capabilities of classical computers. Unlike classical
+computers, which use bits to represent data as 0s or 1s, quantum
+computers use quantum bits, or qubits. Qubits can exist in multiple
+states simultaneously (superposition) and can be entangled with one
+another, allowing quantum computers to process vast amounts of
+information in parallel.
+
+These unique properties enable quantum computers to tackle problems
+that are currently intractable for classical systems, such as complex
+simulations in chemistry and physics, optimization problems, and
+large-scale data analysis.
+
+Quantum machine learning (QML) is an interdisciplinary field that
+combines quantum computing with machine learning techniques. The goal
+is to enhance the performance of machine learning algorithms by
+utilizing quantum computing’s capabilities.
+
+In QML, quantum algorithms are developed to process and analyze data
+more efficiently than classical algorithms. This includes tasks like
+classification, regression, clustering, and dimensionality
+reduction. By exploiting quantum phenomena, QML has the potential to
+accelerate machine learning processes and handle larger datasets more
+effectively.
+
+Quantum computing and QML hold promise for various applications, including:
+
+\begin{enumerate}
+\item Drug Discovery: Simulating molecular structures to expedite the development of new medications.
+
+\item Financial Modeling: Optimizing portfolios and detecting fraudulent activities through complex data analysis.
+
+\item Artificial Intelligence: Enhancing machine learning algorithms for faster and more accurate predictions. 
+\end{enumerate}
+
+\noindent
+As quantum hardware continues to advance, the integration of quantum
+computing into practical applications is becoming increasingly
+feasible, opening up for  a new era of computational possibilities.
+
+This thesis project deals with the study and implementation of quantum
+machine learning methods applied to classical machine learning data
+for supervised learning.
+The methods we will focus on are
+\begin{enumerate}
+\item Support vector machines and quantum support vector machines
+
+\item Neural networks and quantum neural networks and possibly (if time allows)
+
+\item Classical and quantum Boltzmann machines
+\end{enumerate}
+
+\noindent
+The data sets will span both regression and classification problems,
+with an emphasis on simulating time series of relevance for financial
+problems. The thesis will be done in close collaboration with Norges
+Bank Invenstment Management, Simula Research laboratory and the
+University of Oslo.
+
+\subsection{Support vector machines}
+
+A central model in classical
+supervised learning is the support vector machine (SVM), which is a
+max-margin classifier.  SVMs are widely used for binary classification
+and have extensions to regression problems as well.
+They build on statistical learning
+theory and are known for finding decision boundaries with maximal
+margin.  In particular, SVMs can perform non-linear classification by
+employing the kernel trick, which implicitly maps data into a
+high-dimensional feature space via a kernel function.
+
+A Quantum Support Vector Machine (QSVM) replaces the classical kernel
+or feature map with a quantum procedure.  In QSVM, classical data
+points $\bm{x}$ are encoded into quantum states
+$|\phi(\bm{x})\rangle$ via a quantum feature map (a parameterized
+quantum circuit).  The inner product (overlap) between two such states
+serves as a quantum kernel, measuring data similarity in a
+high-dimensional Hilbert space.
+
+\subsection{Quantum Neural Networks and Variational Circuits}
+
+The Variational Quantum Algorithm (VQA) is a 
+Variational Quantum Circuit (VQC), that is  a quantum circuit with tunable
+parameters and which is trained using a classical optimizer.  In practice, a
+VQC (also called a Parameterized Quantum Circuit (PQC)) is used as a
+Quantum Neural Network (QNN): data are encoded into quantum states, a
+parameterized circuit is applied, and measurements yield outputs.
+For example, it has been shown recently that certain QNNs can exhibit higher
+effective dimension (and thus capacity to generalize) than comparable
+classical networks , suggesting a potential quantum advantage.
+
+\subsection{Boltzmann machines}
+
 Boltzmann Machines (BMs) offer a powerful framework for modeling
 probability distributions.  These types of neural networks use an
 undirected graph-structure to encode relevant information.  More
@@ -174,135 +252,43 @@ \subsection{Possible  Projects}
 domains such as the analysis of quantum many-body systems, statistics,
 biochemistry, social networks, signal processing and finance
 
-BMs are complicated to train in practice because the loss
-function's derivative requires the evaluation of a normalization
-factor, the partition function, that is generally difficult to
-compute.  Usually, it is approximated using Markov Chain Monte Carlo
-methods which may require long runtimes until convergence
-
 Quantum Boltzmann Machines (QBMs) are a natural adaption of BMs to the
 quantum computing framework. Instead of an energy function with nodes
 being represented by binary spin values, QBMs define the underlying
-network using a Hermitian operator, normally a parameterized
-Hamiltonian, see references [1,2] below.
-
-Here we will focus on classification problems such as the famous MNIST
-data set which contains handwritten numbers from $0$ to $9$.  This can
-serve as a starting point.  More data sets can be included at a later
-stage.  The next project parallels this but replaces Boltzmann
-machines with Autoencoders.
+network using a Hermitian operator, normally a parameterized Hamiltonian, see reference [1] below.
 
-\textbf{Literature:}
+\paragraph{Specific tasks and milestones.}
+The aim of this thesis is to study the implementation and development of codes for
+several quantum machine learning methods, including quantum support vector machines, quantum neural networks and possibly  Boltzmann machines, and possibly other classical machine learning algorithms, on a quantum computer. The thesis consists of three basic steps:
 
 \begin{enumerate}
-\item Amin et al., \textbf{Quantum Boltzmann Machines}, Physical Review X \textbf{8}, 021050 (2018).
+\item Develop a classical machine code for studies of classification and regression problems.
 
-\item Zoufal et al., \textbf{Variational Quantum Boltzmann Machines}, ArXiv:2006.06004.
+\item Compare the results from the classical Boltzmann machine with other deep learning methods.
 
-\item Maria Schuld and Francesco Petruccione, \textbf{Supervised Learning with Quantum Computers}, Springer, 2018.
+\item Develop an implementation of a quantum Boltzmann machine code to be run on existing quantum computers and classical computers. Compare the performance of the quantum Boltzmann machines with exisiting classical deep learning methods.
 \end{enumerate}
 
 \noindent
-\paragraph{From Classical Autoenconders to Quantum Autoenconders and classification problems (Classical and Quantum Machine Learning).}
-Classical autoencoders are neural networks that can learn efficient
-low dimensional representations of data in higher dimensional
-space. The task of an autoencoder is, given an input $x$, is to map
-$x$ to a lower dimensional point $y$ such that $x$ can likely be
-recovered from $y$. The structure of the underlying autoencoder
-network can be chosen to represent the data on a smaller dimension,
-effectively compressing the input.
-
-Inspired by this idea, we would like to, following references [1,2]
-below, to introduce Quantum Autoencoders to compress a particular
-dataset like the famous MNIST data set which contains handwritten
-numbers from $0$ to $9$.
-
-\textbf{Literature:}
-
+The milestones are:
 \begin{enumerate}
-\item Carlos Bravo-Prieto, \textbf{Quantum autoencoders with enhanced data encoding}, see \href{{https://arxiv.org/pdf/2010.06599.pdf}}{\nolinkurl{https://arxiv.org/pdf/2010.06599.pdf}} 
+\item Spring 2025: Develop a code for classical Boltzmann machines to be applied to both classification and regression problems. In particular, the latter type of problem can be tailored to solving classical spin problems like the Ising model or quantum mechanical problems. 
 
-\item Jonathan Romero et al, \textbf{Quantum autoencoders for efficient compression of quantum data}, see \href{{https://arxiv.org/pdf/1612.02806.pdf}}{\nolinkurl{https://arxiv.org/pdf/1612.02806.pdf}}
+\item Fall 2025: Develop a code for variational Quantum Boltzmann machines following reference [2] here.  Make comparisons with classical deep learning algorithms on selected classification and regression problems.
 
-\item Maria Schuld and Francesco Petruccione, \textbf{Supervised Learning with Quantum Computers}, Springer, 2018. See \href{{https://www.springer.com/gp/book/9783319964232}}{\nolinkurl{https://www.springer.com/gp/book/9783319964232}}
+\item Spring 2026: The final part is to use the variational Quantum Boltzmann machines to study quantum mechanical systems. Finalize thesis. 
 \end{enumerate}
 
 \noindent
-\paragraph{Bayesian phase difference estimation (Quantum-mechanical many-body Physics).}
-Quantum computers can perform full configuration interaction (full-CI)
-calculations by utilising the quantum phase estimation (QPE)
-algorithms including Bayesian phase estimation (BPE) and iterative
-quantum phase estimation (IQPE). In these quantum algorithms, the time
-evolution of wave functions for atoms and molecules is simulated
-conditionally with an ancillary qubit as the control, which make
-implementation to real quantum devices difficult. Also, most of the
-problems in many-body physics discuss energy differences between two
-states rather than total energies themselves, and thus
-direct calculations of energy gaps in for example atoms and molecules are promising for future
-applications of quantum computers to real quantum mechanical many-body problems. In the
-race of finding efficient quantum algorithms to solve quantum
-chemistry problems, we would like to study  a Bayesian phase difference estimation
-(BPDE) algorithm, which is a general algorithm to calculate the
-difference of two eigenphases of unitary operators in the several
-cases of the direct calculations of energy gaps between selected many-body states
-states on quantum computers.
-
-\paragraph{Variational Quantum Eigensolvers (Quantum-mechanical many-body Physics).}
-The specific task here is to implenent and study Quantum Computing
-algorithms like the Quantum-Phase Estimation algorithm and Variational
-Quantum Eigensolvers for solving quantum mechanical many-particle
-problems.  Recent scientific articles have shown the reliability of
-these methods on existing and real quantum computers, see for example
-references [1-5] below.
-
-Here the focus is first on tailoring a Hamiltonian like the pairing
-Hamiltonian and/or Anderson Hamiltonian in terms of quantum gates, as
-done in references [3-5].
-
-Reproducing these results will be the first step of this thesis
-project. The next step includes adding more complicated terms to the
-Hamiltonian, like a particle-hole interaction as done in the work of
-\href{{http://iopscience.iop.org/article/10.1088/0954-3899/37/6/064035/meta}}{Hjorth-Jensen et
-al}.
-
-The final step is to implement the action of these Hamiltonians on
-existing quantum computers like \href{{https://www.rigetti.com/}}{Rigetti's Quantum
-Computer}.
-
-The projects can easily be split into several parts and form the basis
-for collaborations among several students.
-
-\textbf{Literature:}
+The thesis is expected to be handed in May/June 2026.
 
+\paragraph{Literature.}
 \begin{enumerate}
-\item Dumitrescu et al, see \href{{https://arxiv.org/abs/1801.03897}}{\nolinkurl{https://arxiv.org/abs/1801.03897}}
-
-\item Yuan et al., \textbf{Theory of Variational Quantum Simulations}, see  \href{{https://arxiv.org/abs/1812.08767}}{\nolinkurl{https://arxiv.org/abs/1812.08767}} 
-
-\item Ovrum and Hjorth-Jensen, see \href{{https://arxiv.org/abs/0705.1928}}{\nolinkurl{https://arxiv.org/abs/0705.1928}}
-
-\item Stian Bilek, Master of Science Thesis, University of Oslo, 2020, see \href{{https://www.duo.uio.no/handle/10852/82489}}{\nolinkurl{https://www.duo.uio.no/handle/10852/82489}}
-
-\item Heine Åbø Olsson, Master of Science Thesis, University of Oslo, 2020, see \href{{https://www.duo.uio.no/handle/10852/81259}}{\nolinkurl{https://www.duo.uio.no/handle/10852/81259}}
-\end{enumerate}
+\item Amin et al., \textbf{Quantum Boltzmann Machines}, Physical Review X \textbf{8}, 021050 (2018).
 
-\noindent
-\paragraph{Analysis of entanglement in quantum computers.}
-The aim of this project is to study various ways of analyzing
-entanglement theoretically by computing for example the von Neumann
-entropy of a quantum mechanical many-body system. The systems we are
-aiming at here are so-called quantum dots systems which are candidates
-from making quantum gates and circuits. The theoretical results are
-planned to be linked with experimental interpretations via Quantum
-state tomography, which is the standard technique for estimating the
-quantum state of small systems.  If possible, this project could be
-linked with studies of quantum tomography from the SpinQ quantum
-computer at OsloMet.
-
-\textbf{Literature:}
+\item Maria Schuld and Francesco Petruccione, \textbf{Supervised Learning with Quantum Computers}, Springer, 2018.
 
-\begin{enumerate}
-\item B. P. Lanyon et al, \textbf{Efficient tomography of a quantum many-body system}, Nature Physics \textbf{13}, 1158 (2017), see \href{{https://www.nature.com/articles/nphys4244}}{\nolinkurl{https://www.nature.com/articles/nphys4244}}
+\item Claudio Conti, Quantum Machine Learning (Springer), sections 1.5-1.12 and chapter 2, see \href{{https://link.springer.com/book/10.1007/978-3-031-44226-1}}{\nolinkurl{https://link.springer.com/book/10.1007/978-3-031-44226-1}}.
 \end{enumerate}
 
 \noindent