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Author: mhjensen

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-<h1>Quantum Machine Learning</h1>
+<h1>Quantum Machine Learning for Finance</h1>
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 <!-- author(s): Master of Science thesis project -->
@@ -122,14 +122,13 @@ <h4>May 4, 2025</h4>
 <h2 id="quantum-computing-and-machine-learning" class="anchor">Quantum Computing and Machine Learning </h2>
 
 <p><b>Quantum Computing and Machine Learning</b> are two of the most promising
-approaches for studying complex physical systems where several length
-and energy scales are involved.  
+approaches for studying complex systems with many degrees of freedom.
 </p>
 
 <p>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, which use bits to represent data as bits \( 0 \) or \( 1 \), 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
@@ -156,7 +155,7 @@ <h2 id="quantum-computing-and-machine-learning" class="anchor">Quantum Computing
 effectively.
 </p>
 
-<p>Quantum computing and QML hold promise for various applications, including:</p>
+<p>Quantum computing and QML hold promise for many different types of  applications, including:</p>
 
 <ol>
 <li> Drug Discovery: Simulating molecular structures to expedite the development of new medications.</li>
@@ -179,16 +178,16 @@ <h2 id="quantum-computing-and-machine-learning" class="anchor">Quantum Computing
 <li> Classical and quantum Boltzmann machines</li>
 </ol>
 <p>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
+with an emphasis on simulating time series, in particular  of relevance for financial
+problems. The thesis will be done in close collaboration with <b>Norges
 Bank Invenstment Management, Simula Research laboratory and the
-University of Oslo.
+University of Oslo</b>.
 </p>
 <h2 id="support-vector-machines" class="anchor">Support vector machines </h2>
 
 <p>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
+maximal-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
@@ -215,7 +214,7 @@ <h2 id="quantum-neural-networks-and-variational-circuits" class="anchor">Quantum
 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.
+classical networks, suggesting a potential quantum advantage.
 </p>
 <h2 id="boltzmann-machines" class="anchor">Boltzmann machines </h2>
 
@@ -227,12 +226,7 @@ <h2 id="boltzmann-machines" class="anchor">Boltzmann machines </h2>
 to binary spin-systems and grouped into those that determine the
 output, the visible nodes, and those that act as latent variables, the
 hidden nodes.
-</p>
-
-<p>Furthermore, the network structure is linked to an energy function
-which facilitates the definition of a probability distribution over
-the possible node configurations by using a concept from statistical
-mechanics, i.e., Gibbs states.  The aim of BM training is to learn a
+The aim of BM training is to learn a
 set of weights such that the resulting model approximates a target
 probability distribution which is implicitly given by training data.
 This setting can be formulated as discriminative as well as generative
@@ -244,24 +238,36 @@ <h2 id="boltzmann-machines" class="anchor">Boltzmann machines </h2>
 <p>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 reference [1] below.
+network using a Hermitian operator, normally a parameterized Hamiltonian.
 </p>
 <h3 id="specific-tasks-and-milestones" class="anchor">Specific tasks and milestones  </h3>
 
-<p>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:
+<p>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, if time allows. The results will be compared with
+those from their classical counterparts.  The final aim is to study
+data from finance with both classical and quantum Machine Learning
+algorithms in order to assess and test quantum machine learning
+algorithms and their potential for the analysis of data from finance.
+In setting up the algorithms, existing software libraries like
+Scikit-Learn, PennyLane, Qiskit and other will be used. This will allow for an efficient development and study of both classical and quantum machine learning algorithms.
 </p>
 
+<p>The thesis consists of three basic steps:</p>
+
 <ol>
-<li> Develop a classical machine code for studies of classification and regression problems.</li>
-<li> Compare the results from the classical Boltzmann machine with other deep learning methods.</li>
-<li> 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.</li>
+<li> Develop a classical machine framework for studies of supervised classification and regression problems, with an emphasis on data from finance. The main emphasis rests on deep learning methods (neural networks, Boltzmann machines and recurrent neural networks) and support vector machines.</li>
+<li> Compare and evaluate the results from the classical machine learning methods and assess their relevance for financial data.</li>
+<li> Develop and implement  codes for quantum machine learning algorithms (quantum support vector machines, quantum neural networks and possibly quantum Boltzmann machines) to be run on existing quantum computers and classical computers. Compare the performance of the quantum machine learning  with the abovementioned classical methods with an emphasis on  financial data.</li>
 </ol>
 <p>The milestones are:</p>
 <ol>
-<li> 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.</li> 
-<li> 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.</li>
-<li> Spring 2026: The final part is to use the variational Quantum Boltzmann machines to study quantum mechanical systems. Finalize thesis.</li> 
+<li> Spring 2025: Study basic quantum machine learning algorithms (quantum support vector machines, quantum neural networks) for simpler supervised problems from finance and/or other fields.</li>
+<li> Spring 2025: Compare the results of the simpler data sets with classical machine learning methods</li>
+<li> Fall 2025: Set uo final data from finance to be analyzed with classical and quantum machine learning algorithms</li>
+<li> Fall 2025: Develop a software framework which includes quantum support vector machines and quantum neural networks.</li>
+<li> Spring 2026: The final part is to include Quantum Boltzmann machines, if time allows,  and analyze the results from the diffirent methods. Finalize thesis.</li> 
 </ol>
 <p>The thesis is expected to be handed in May/June 2026.</p>
 <h3 id="literature" class="anchor">Literature </h3>
@@ -270,6 +276,7 @@ <h3 id="literature" class="anchor">Literature </h3>
 <li> Amin et al., <b>Quantum Boltzmann Machines</b>, Physical Review X <b>8</b>, 021050 (2018).</li>
 <li> Maria Schuld and Francesco Petruccione, <b>Supervised Learning with Quantum Computers</b>, Springer, 2018.</li>
 <li> Claudio Conti, Quantum Machine Learning (Springer), sections 1.5-1.12 and chapter 2, see <a href="https://link.springer.com/book/10.1007/978-3-031-44226-1" target="_self"><tt>https://link.springer.com/book/10.1007/978-3-031-44226-1</tt></a>.</li>
+<li> Morten Hjorth-Jensen, Quantum Computing and Quantum Machine Learning, lecture notes with extensive codes at <a href="https://github.com/CompPhysics/QuantumComputingMachineLearning" target="_self"><tt>https://github.com/CompPhysics/QuantumComputingMachineLearning</tt></a>, in particular the last five sets of lectures.</li>
 </ol>
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