Quantum computing has emerged as a new field that may have the potential to revolutionize the landscape of information processing and computational power, although physically constructing quantum hardware has proven difficult,and quantum computers in the current Noisy Intermediate Scale Quantum (NISQ) era are error prone and limited in the number of qubits they contain.A sub-field within quantum algorithms research which holds potential for the NISQ era, and which has seen increasing activity in recent years, is quantum machine learning, where researchers apply approaches from classical machine learning to quantum computing algorithms and explore the interplay between the two. This master thesis investigates feature selection and autoencoding algorithms for quantum computers. Our review of the prior art led us to focus on contributing to three sub-problems: A) Embedded feature selection on quantum annealers, B) short depth quantum autoencoder circuits, and C)embedded compressed feature representation for quantum classifier circuits.For problem A, we demonstrate a working example by converting ridge regression to the Quadratic Unconstrained Binary Optimization (QUBO) problem formalism native to quantum annealers, and solving it on a simulated backend. For problem B we develop a novel quantum convolutional autoencoder architecture and successfully run simulation experiments to study its performance.For problem C, we choose a classifier quantum circuit ansatz based on theoretical considerations from the prior art, and experimentally study it in parallel with a classical benchmark method for the same classification task,then show a method from embedding compressed feature representation onto that quantum circuit. / Kvantberäkning är ett framväxande område som potentiellt kan revolutionera informationsbehandling och beräkningskraft. Dock är praktisk konstruktion av kvantdatorer svårt, och nuvarande kvantdatorer i den s.k. NISQ-eran lider av fel och begränsningar i antal kvantbitar de kan hantera. Ett lovande delområde inom kvantalgoritmer är kvantmaskininlärning, där forskare tillämpar klassiska maskininlärningsmetoder på kvantalgoritmer och utforskar samspelet mellande två områdena.. Denna avhandling fokuserar på kvantalgoritmer för funktionsval,och datakompression (i form av s.k. “autoencoders”). Vi undersöker tre delproblem: A) Inbäddat funktionsval på en kvantannealer, B) autoencoder-kvantkretsar för datakompression, och C) inbyggt funktionsval för kvantkretsar för klassificering. För problem A demonstrerar vi ett fungerande exempel genom att omvandla ridge regression till problemformuleringen "Quadratic Unconstrained Binary Optimization"(QUBO) som är nativ för kvantannealers,och löser det på en simulerad backend. För problem B utvecklar vi en ny konvolutionerande autoencoder-kvantkrets-arkitektur och utför simuleringsexperimentför att studera dess prestanda. För problem C väljer vi en kvantkrets-ansats för klassificering baserad på teoretiska överväganden från tidigare forskning och studerar den experimentellt parallellt med en klassisk benchmark-metod församma klassificeringsuppgift, samt visar en metod för inbyggt funktionsval (i form av datakompression) i denna kvantkrets.

Le sujet de thèse est une étude théorique des modes normaux d’oscillations plasma dans des chaînes de jonctions Josephson supra-conductrices. Les propriétés de ces modes normaux peuvent être contrôlés en choisissant une modulation spatiale appropriée de paramètres des jonctions le long de la chaîne et/ou un couplage approprié à l'environnement extérieur. Le travail théorique au sein du LPMMC se fait en étroite collaboration avec l'équipe expérimentale"Cohérence Quantique" à l'Institut Néel. Les problèmes spécifiques étudiés dans la thèse sont : modélisation détaillée du couplage des modes normaux à l'environnement pour leur caractérisation dans une expérience de transmission de micro-ondes, dissipation intrinsèque des oscillations du plasma à cause de quasi-particules hors équilibre, l'optimisation de la structure spatiale de la chaîne de jonctions Josephson pour son utilisation en tant qu'une super-inductance. / The subject of thesis is a theorerical study of normal modes of plasma oscillations in superconducting Josephson junction chains. The properties of these normal modes can be controlled by choosing an appropriate spatial modulation of the junction parameters along the chain and/or an appropriate coupling to the external environment. The theoretical work at LPMMC is performed in a close collaboration with the experimental Quantum Coherence group at Néel Institute. The specific problems studied in this thesis are : detailed modeling of the normal mode coupling to the environment for probing them in a microwave transmission experiment, intrinsic dissipation of plasma oscillations due to the presence of non-equilibrium quasi-particles, optimization of the spatial structure of the Josephson junction chain for its use as a super-inductance.

L'informatique quantique exploite les phénomènes de la théorie quantique pour le traitement de l'information, tandis que l'apprentissage automatique s'intéresse aux algorithmes qui peuvent s'améliorer en fonction des expériences passées. L'informatique quantique a produit des algorithmes qui dépassent de loin les capacités des ordinateurs classiques que nous utilisons tous les jours. Cependant, l'identification de nouveaux algorithmes quantiques fut moins prolifique que dans le cas classique. Ces dernières années, on a cherché à combiner l'informatique quantique et l'apprentissage automatique. Le cadre de l'apprentissage automatique a servi à apprendre les paramètres de circuits quantiques paramétrés dans l'espoir d'apprendre à résoudre des problèmes où les phénomènes quantiques peuvent aider grâce au traitement de l'information quantique. L'objectif principal de ce mémoire est de pousser plus loin cette idée d'apprentissage de circuits quantiques et de fonder solidement ses capacités en développant une architecture universelle de circuit quantique paramétré. La première contribution est une évaluation d'algorithmes d'optimisation itératifs actuels pour les circuits quantiques paramétrés en tant que modèles d'apprentissage automatique, ainsi que la présentation d'un algorithme d'optimisation itératif simple, mais robuste. La deuxième contribution est une architecture de circuit quantique dans laquelle une famille de petits circuits avec des connexions arbitraires peut être intégrée. / Quantum information processing leverages the phenomena of quantum theory for information processing, while machine learning concerns itself with algorithms that can improve based on past experiences. Quantum information processing has produced algorithms that go far past the capabilities of the classical computers we use every day. However, the identification of new quantum algorithms has been much slower than during the early days of classical computing. In recent years, there has been a push to combine quantum information processing and machine learning. The framework of machine learning has been used to learn quantum circuits in the hopes of learning to solve problems where quantum phenomena can help through the use of quantum information processing. The main goal of this thesis is to further push this idea of learning quantum circuits and to solidly ground its capabilities by developing a learnable parametrized universal quantum circuit. The first contribution is an assessment of current optimization methods for parametrized quantum circuits as machine learning models. The second contribution is a quantum circuit architecture in which a family of smaller circuits with arbitrary connections can be embedded.

Les amplificateurs paramétriques Josephson (JPA) se sont révélés être un outil indispensablepour l’étude expérimentale de dispositifs quantiques dans le régime micro-onde ; car ilsrajoutent uniquement le minimum de bruit imposé par la mécanique quantique[1]. Cependant,ces amplificateurs sont beaucoup plus difficiles à utiliser et optimiser que leurs homologuesclassiques. Récemment, plusieurs expériences réalisées avec des circuits supraconducteurs,composés d’une jonction Josephson polarisée en tension en série avec un résonateur microonde,ont montré qu’une paire de Cooper peut traverser la barrière de la jonction par effettunnel en émettant un ou plusieurs photons avec une énergie totale de 2e fois la tensionappliquée. Dans cette thèse, nous montrerons qu’un tel circuit permet de mettre en place unamplificateur micro-onde préservant la phase que nous appelons « Amplificateur basé sur letunneling inélastique de paires de Cooper » (ICTA). Il est alimenté par une tension continueet peut fonctionner avec un bruit très proche de la limite quantique.Nous commencerons en présentant le principe du fonctionnement de l’ICTA. Par analogieavec la théorie quantique des JPAs[2], nous avons étudié les performances de cet amplificateurcomme le gain, la bande passante et le bruit. Ensuite, nous présenterons la premièrepreuve expérimentale d’une amplification proche de la limite quantique sans utilisation d’unepompe micro-onde externe, mais simplement d’une tension continue dans une configurationextrêmement simple. Ces mesures ont été faites sur des échantillons avec des jonctionsen aluminium, dénommés ICTA de première génération. Selon nos résultats théoriques etexpérimentaux, nous avons conçu des circuits hyperfréquences où l’impédance présentéeà la jonction dépend de fréquences spécifiques afin d’optimiser les performances de notreamplificateur. Ces échantillons, dénommés ICTA de seconde génération, ont été fabriquésavec du nitrure de niobium. Une amélioration significative du gain et du bruit a été prouvée.Un tel amplificateur, alimenté par une simple tension continue, pourrait rendre la mesurede signaux micro-ondes au niveau du photon unique beaucoup plus faciles et permettred’intégrer plusieurs amplificateurs sur une seule puce. Il pourrait donc être un élémentimportant pour la lecture de qubit dans les processeurs quantiques à grande échelle. / Josephson parametric amplifiers (JPA), have proven to be an indispensable tool for awide range of experiments on quantum devices in the microwave frequency regime, becausethey provide the lowest possible noise. However, JPAs remain much more difficult to use andoptimize than conventional microwave amplifiers. Recent experiments with superconductingcircuits consisting of a DC voltage-biased Josephson junction in series with a resonator,showed that a tunneling Cooper pair can emit one or several photons with a total energyof 2e times the applied voltage. In this thesis we show that such q circuit can be used toimplement a new type of phase preserving microwave amplifier, which we call InelasticCooper pair Tunneling Amplifier (ICTA). It is powered by a simple DC bias and offers nearquantum-limited noise performance.We start this work by presenting a brief and simple picture of the basic ICTA operatingprinciples. In analogy with the quantum theory of JPAs, we calculate the performances ofthis amplifier such as the gain, bandwidth and noise. Then, we present the first experimentalproof that amplification close to the quantum limit is possible without microwave drive inan extremely simple setup. These measurements are made on a first generation of samplesbased on aluminium junctions. According to our theoretical and experimental results, wehave designed microwave circuits presenting specific frequency-dependent impedances tothe junction in order to optimize the performances of our amplifier. This second generationof ICTA samples is fabricated from niobium nitride and provide significantly lower noiseand higher gain.We expect that once fully optimized, such an amplifier, powered by simple DC voltagescould then make measuring microwave signals at the single photon level much easier andallow to deploy many amplifiers on a chip. It could therefore be an important ingredient forqubit readout in large-scale quantum processors.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-08-06T19:17:03Z No. of bitstreams: 1 HERON ARAGÃO MONTEIRO - DISSERTAÇÃO PPGCC 2012..pdf: 15948168 bytes, checksum: e445512265f530700a45c3924f68aa02 (MD5) / Made available in DSpace on 2018-08-06T19:17:03Z (GMT). No. of bitstreams: 1 HERON ARAGÃO MONTEIRO - DISSERTAÇÃO PPGCC 2012..pdf: 15948168 bytes, checksum: e445512265f530700a45c3924f68aa02 (MD5) Previous issue date: 2012-05-31 / Com o avanço da nanotecnologia, a computação quântica tem recebido grande destaque no meio científico. Utilizando os fundamentos da mecânica quântica, têm sido propostos diversos algoritmos quânticos. E, até então, os mesmos têm apresentado ganhos significativos com relação às suas versões clássicas. Na intenção de poder ser verificada a eficiência dos algoritmos quânticos, diversos simuladores vêm sendo desenvolvidos, visto que a confecção de um computador quântico ainda não foi possível. Há duas grandes vertentes de simuladores: os simuladores por software e os simuladores por hardware, chamados de emuladores. Na primeira classe se encontram os programas desenvolvidos em um computador clássico, procurando implementar os fundamentos da mecânica quântica, fazendo uso das linguagens de programação clássicas. Na segunda, são utilizados recursos que não estejam vinculados à plataforma do computador clássico. Dentre os emuladores, particularmente, estudos têm sido realizados fazendo uso de hardware dedicado (mais especificamente, FPGAV). O presente trabalho propõem a verificação da real utilidade da plataforma FPGA, com a intenção de se desenvolver um emulador universal, que permita a emulação de qualquer classe de circuitos, e que os mesmos possam ser implementados com um maior número de q-bits em relação aos circuitos tratados nos trabalhos anteriores. / With the progress of nanotechnology, quantum computing has received great emphasis in scientific circles. Using the basis of quantum mechanics, different quantum algorithms have been proposed. And so far, they have presented significant gains with respect to its classic versions. In order to verify the efficiency of quantum algorithms, several simulators have been developed, since the construction of a quantum computer is not yet possible. There are two major classes of simulators, simulators via software and via hardware. The latter being also called emulators. In the first class, programs are developed in a classical computer, attempting to implement the fundamentais of quantum mechanics, making use of classic programming languages. In the second, resources are used that are not related to the classic computer platform. Among the emulators, in particular, studies have been made using dedicated hardware (more specifically, FPGA's2). The present work proposes the use of the FPGA boards in emulation of quantum circuits aiming a gain scale in relation to the alternatives presented so far. The present work proposes checking the usefulness of the FPGA with the intention of developing an universal emulator that is able to emulate any type of circuit, and that they can be implemented with a larger number of q-bit in respect to the circuits treated in the previous works.

Many quantum computation algorithms, and processes like measurement based quantum computing, require the initial state of the quantum computer to be an eigenstate of a specific unitary operator. Here we study how quantum states that are eigenstates of finite dimensional irreducible representations of the special unitary (SU(d)) and the permutation (S_n) groups can be efficiently constructed in the computational basis formed by tensor products of the qudit states. The procedure is a unitary transform, which first uses Schur-Weyl duality to map every eigenstate to a unique Schur basis state, and then recursively uses the Clebsch - Gordan transform to rotate the Schur basis state to the computational basis. We explicitly provide an efficient quantum algorithm, and the corresponding quantum logic circuit, to generate any desired eigenstate of SU(2) and SU(3) irreducible representations in the computational basis.

<p>The recent groundbreaking demonstrations of quantum supremacy in noisy intermediate scale quantum (NISQ) computing era has triggered an intense activity in establishing finer boundaries between classical and quantum computing. In this dissertation, we use established techniques based on quantum Monte Carlo (QMC) to map quantum problems into probabilistic networks where the fundamental unit of computation, p-bit, is inherently probabilistic and can be tuned to fluctuate between ‘0’ and ‘1’ with desired probability. We can view this mapped network as a Boltzmann machine whose states each represent a Feynman path leading from an initial configuration of q-bits to a final configuration. Each such path, in general, has a complex amplitude, ψ which can be associated with a complex energy. The real part of this energy can be used to generate samples of Feynman paths in the usual way, while the imaginary part is accounted for by treating the samples as complex entities, unlike ordinary Boltzmann machines where samples are positive. This mapping of a quantum circuit onto a Boltzmann machine with complex energies should be particularly useful in view of the advent of special-purpose hardware accelerators known as Ising Machines which can obtain a very large number of samples per second through massively parallel operation. We also demonstrate this acceleration using a recently used quantum problem and speeding its QMC simulation by a factor of ∼ 1000× compared to a highly optimized CPU program. Although this speed-up has been demonstrated using a graph colored architecture in FPGA, we project another ∼ 100× improvement with an architecture that utilizes clockless analog circuits. We believe that this will contribute significantly to the growing efforts to push the boundaries of the simulability of quantum circuits with classical/probabilistic resources and comparing them with NISQ-era quantum computers. </p>

Tesis por compendio / [ES] En esta tesis doctoral se propone y evalúa un sistema para detectar señales débiles (weak signals) relacionadas con cambios futuros trascendentales. Si bien la mayoría de las soluciones conocidas se basan en el uso de datos estructurados, el sistema propuesto detecta cuantitativamente estas señales utilizando información heterogénea y no estructurada de fuentes científicas, periodísticas y de redes sociales. La predicción de nuevas tendencias en un medio tiene muchas aplicaciones. Por ejemplo, empresas y startups se enfrentan a cambios constantes en sus mercados que son muy difíciles de predecir. Por esta razón, el desarrollo de sistemas para detectar automáticamente cambios futuros significativos en una etapa temprana es relevante para que cualquier organización tome decisiones acertadas a tiempo. Este trabajo ha sido diseñado para obtener señales débiles del futuro en cualquier campo dependiendo únicamente del conjunto de datos de entrada de documentos. Se aplican técnicas de minería de textos y procesamiento del lenguaje natural para procesar todos estos documentos. Como resultado, se obtiene un mapa con un ranking de términos, una lista de palabras clave clasificadas automáticamente y una lista de expresiones formadas por múltiples palabras. El sistema completo se ha probado en cuatro sectores diferentes: paneles solares, inteligencia artificial, sensores remotos e imágenes médicas. Este trabajo ha obtenido resultados prometedores, evaluados con dos metodologías diferentes. Como resultado, el sistema ha sido capaz de detectar de forma satisfactoria nuevas tendencias en etapas muy tempranas que se han vuelto cada vez más importantes en la actualidad. La computación cuántica es un nuevo paradigma para una multitud de aplicaciones informáticas. En esta tesis doctoral también se presenta un estudio de las tecnologías disponibles en la actualidad para la implementación física de qubits y puertas cuánticas, estableciendo sus principales ventajas y desventajas, y los marcos disponibles para la programación e implementación de circuitos cuánticos. Con el fin de mejorar la efectividad del sistema, se describe un diseño de un circuito cuántico basado en máquinas de vectores de soporte (SVM) para la resolución de problemas de clasificación. Este circuito está especialmente diseñado para los ruidosos procesadores cuánticos de escala intermedia (NISQ) que están disponibles actualmente. Como experimento, el circuito ha sido probado en un computador cuántico real basado en qubits superconductores por IBM como una mejora para el subsistema de minería de texto en la detección de señales débiles. Los resultados obtenidos con el experimento cuántico muestran también conclusiones interesantes y una mejora en el rendimiento de cerca del 20% sobre los sistemas convencionales, pero a su vez confirman que aún se requiere un desarrollo tecnológico continuo para aprovechar al máximo la computación cuántica. / [CA] En aquesta tesi doctoral es proposa i avalua un sistema per detectar senyals febles (weak signals) relacionats amb canvis futurs transcendentals. Si bé la majoria de solucions conegudes es basen en l'ús de dades estructurades, el sistema proposat detecta quantitativament aquests senyals utilitzant informació heterogènia i no estructurada de fonts científiques, periodístiques i de xarxes socials. La predicció de noves tendències en un medi té moltes aplicacions. Per exemple, empreses i startups s'enfronten a canvis constants als seus mercats que són molt difícils de predir. Per això, el desenvolupament de sistemes per detectar automàticament canvis futurs significatius en una etapa primerenca és rellevant perquè les organitzacions prenguen decisions encertades a temps. Aquest treball ha estat dissenyat per obtenir senyals febles del futur a qualsevol camp depenent únicament del conjunt de dades d'entrada de documents. S'hi apliquen tècniques de mineria de textos i processament del llenguatge natural per processar tots aquests documents. Com a resultat, s'obté un mapa amb un rànquing de termes, un llistat de paraules clau classificades automàticament i un llistat d'expressions formades per múltiples paraules. El sistema complet s'ha provat en quatre sectors diferents: panells solars, intel·ligència artificial, sensors remots i imatges mèdiques. Aquest treball ha obtingut resultats prometedors, avaluats amb dues metodologies diferents. Com a resultat, el sistema ha estat capaç de detectar de manera satisfactòria noves tendències en etapes molt primerenques que s'han tornat cada cop més importants actualment. La computació quàntica és un paradigma nou per a una multitud d'aplicacions informàtiques. En aquesta tesi doctoral també es presenta un estudi de les tecnologies disponibles actualment per a la implementació física de qubits i portes quàntiques, establint-ne els principals avantatges i desavantatges, i els marcs disponibles per a la programació i implementació de circuits quàntics. Per tal de millorar l'efectivitat del sistema, es descriu un disseny d'un circuit quàntic basat en màquines de vectors de suport (SVM) per resoldre problemes de classificació. Aquest circuit està dissenyat especialment per als sorollosos processadors quàntics d'escala intermèdia (NISQ) que estan disponibles actualment. Com a experiment, el circuit ha estat provat en un ordinador quàntic real basat en qubits superconductors per IBM com una millora per al subsistema de mineria de text. Els resultats obtinguts amb l'experiment quàntic també mostren conclusions interessants i una millora en el rendiment de prop del 20% sobre els sistemes convencionals, però a la vegada confirmen que encara es requereix un desenvolupament tecnològic continu per aprofitar al màxim la computació quàntica. / [EN] In this doctoral thesis, a system to detect weak signals related to future transcendental changes is proposed and tested. While most known solutions are based on the use of structured data, the proposed system quantitatively detects these signals using heterogeneous and unstructured information from scientific, journalistic, and social sources. Predicting new trends in an environment has many applications. For instance, companies and startups face constant changes in their markets that are very difficult to predict. For this reason, developing systems to automatically detect significant future changes at an early stage is relevant for any organization to make right decisions on time. This work has been designed to obtain weak signals of the future in any field depending only on the input dataset of documents. Text mining and natural language processing techniques are applied to process all these documents. As a result, a map of ranked terms, a list of automatically classified keywords and a list of multi-word expressions are obtained. The overall system has been tested in four different sectors: solar panels, artificial intelligence, remote sensing, and medical imaging. This work has obtained promising results that have been evaluated with two different methodologies. As a result, the system was able to successfully detect new trends at a very early stage that have become more and more important today. Quantum computing is a new paradigm for a multitude of computing applications. This doctoral thesis also presents a study of the technologies that are currently available for the physical implementation of qubits and quantum gates, establishing their main advantages and disadvantages and the available frameworks for programming and implementing quantum circuits. In order to improve the effectiveness of the system, a design of a quantum circuit based on support vector machines (SVMs) is described for the resolution of classification problems. This circuit is specially designed for the noisy intermediate-scale quantum (NISQ) computers that are currently available. As an experiment, the circuit has been tested on a real quantum computer based on superconducting qubits by IBM as an improvement for the text mining subsystem in the detection of weak signals. The results obtained with the quantum experiment show interesting outcomes with an improvement of close to 20% better performance than conventional systems, but also confirm that ongoing technological development is still required to take full advantage of quantum computing. / Griol Barres, I. (2022). Modelling of a System for the Detection of Weak Signals Through Text Mining and NLP. Proposal of Improvement by a Quantum Variational Circuit [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/183029 / Compendio

Integrated quantum photonics show monolithic waveguide chips to be a promising platform for realizing the next generation of quantum optical circuits. This work proposes the implementation of quantum page Rank algorithm on a photonic waveguide lattice. Our contributions are as follows: Continuous-time quantum stochastic walk(QSW)-an alternate paradigm of quantum computing, is a hybrid quantum walk that incorporates both unitary and non-unitary effects. We propose the use of QSW which necessitates the hopping of the quantum crawler on a directed graph, for the quantum page Rank problem. We propose the implementation of quantum page Rank on a photonic waveguide lattice, where we allow the density matrix to evolve according to the Lindblad-Kossakowski master equation, the diagonal of which gives the quantum page Rank. We have also shown the use of the metric of positional Kolmogorov Complexity as an efficient tool for determining whether or not the quantum channel has been compromised. We appositionally encode multi-photon decoy pulses within the stream of single photon pulses. This positional encoding is chosen in such a way as to have low Kolmogorov complexity. The PNS attack on the multi-photon decoy pulses causes a dip in the ratio of the transmittance of the decoy pulses to the signal pulses in the conventional analysis.

Nous présentons un nouvel algorithme d’apprentissage de circuits quantiques basé sur la descente de gradient classique. Comme ce sujet unifie deux disciplines, nous expliquons les deux domaines aux gens de l’autre discipline. Conséquemment, nous débutons par une présentation du calcul quantique et des circuits quantiques pour les gens en apprentissage automatique suivi d’une présentation des algorithmes d’apprentissage automatique pour les gens en informatique quantique. Puis, pour motiver et mettre en contexte nos résultats, nous passons à une légère revue de littérature en apprentissage automatique quantique. Ensuite, nous présentons notre modèle, son algorithme, ses variantes et quelques résultats empiriques. Finalement, nous critiquons notre implémentation en montrant des extensions et des nouvelles approches possibles. Les résultats principaux se situent dans ces deux dernières parties, qui sont respectivement les chapitres 4 et 5 de ce mémoire. Le code de l’algorithme et des expériences que nous avons créé pour ce mémoire se trouve sur notre github à l’adresse suivante : https://github.com/AldoLamarre/quantumcircuitlearning. / We present a new learning algorithm for quantum circuits based on gradient descent. Since this subject unifies two areas of research, we explain each field for people working in the other domain. Consequently, we begin by introducing quantum computing and quantum circuits to machine learning specialists, followed by an introduction of machine learning to quantum computing specialists. To give context and motivate our results we then give a light literature review on quantum machine learning. After this, we present our model, its algorithms and its variants, then discuss our currently achieved empirical results. Finally, we criticize our models by giving extensions and future work directions. These last two parts are our main results. They can be found in chapter 4 and 5 respectively. Our code which helped obtain these results can be found on github at this link : https://github.com/ AldoLamarre/quantumcircuitlearning.