The spin structure factor is the dynamical information coming from inelastic neutron scattering. In this work we develop the technology of tensor networks as a numerical tool to be able to compute physical observables reliably for one-dimensional quantum systems. The main technical message of this thesis is that tensor networks provide a controlled way to compute spin structure factors. The algorithms in this thesis are tested on the anisotropic Majumdar--Ghosh model and the results of these simulations are presented and discussed. / Thesis / Master of Science (MSc)

Understanding the collective behavior of a quantum many-body system, a system composed of a large number of interacting microscopic degrees of freedom, is a key aspect in many areas of contemporary physics. However, as a direct consequence of the difficultly of the so-called many-body problem, many exotic quantum phenomena involving extended systems, such as high temperature superconductivity, remain not well understood on a theoretical level. Entanglement renormalization is a recently proposed numerical method for the simulation of many-body systems which draws together ideas from the renormalization group and from the field of quantum information. By taking due care of the quantum entanglement of a system, entanglement renormalization has the potential to go beyond the limitations of previous numerical methods and to provide new insight to quantum collective phenomena. This thesis comprises a significant portion of the research development of ER following its initial proposal. This includes exploratory studies with ER in simple systems of free particles, the development of the optimisation algorithms associated to ER, and the early applications of ER in the study of quantum critical phenomena and frustrated spin systems.

Quantum computers have the potential to solve certain problems in minutes that would otherwise take classical computers thousands of years due to the exponential speed-up certain quantum algorithms have over classical algorithms. In order to leverage such quantum algorithms, it is necessary for them to run on quantum devices. Examples of such devices include, but are not limited to, semiconductor and superconducting qubits, and semiconductor single and entangled photon emitters. The conventional method of constructing a semiconductor qubit is to apply gates on a semiconductor surface to localize electrons, where the electronic spin states are mapped to a qubit basis. Examples of this include the spin qubit where the spin-1/2 states of a single electron is the qubit basis and the gated singlet-triplet qubit where the states of two coupled electrons are mapped to a qubit basis. In general, gated semiconductor spin qubits are subject to decoherence from the environment which alters the electronic wavefunction by entanglement with the nuclear spins and phonons in the lattice compromising the stability of the qubit. Semiconductor nanostructures can also be designed as photon emitters. Self-assembled quantum dots are an example of such nanostructures and have been shown to emit single photons through exciton recombination and entangled photons through biexciton-exciton cascade. The difficulty in designing photon sources using self-assembled quantum dots is that the size and shape varies from dot to dot, implying that the electronic and magnetic properties also vary. In this thesis, I present the design of a single photon emitter using an InAsP quantum dot embedded in an InP nanowire and the design of a singlet-triplet qubit that is topologically protected from decoherence using an array of such quantum dots embedded in an InP nanowire. The advantage of using quantum dot nanowires over self-assembled quantum dots as photon emitters is that the quantum dot thickness, radius and composition can be controlled deterministically using a technique known as vapour-liquid-solid epitaxy which allows the emission spectrum to be engineered. Using a microscopic model, I simulated an InAsP quantum dot embedded in a nanowire with upwards of millions of atoms and showed that the emission spectrum came in agreement with the actual InAsP/InP quantum dot nanowires that were fabricated at the National Research Council of Canada. Moreover, I showed that altering the distribution of As atoms in the quantum dot can cause dramatic change in the emission spectrum. For the design of the topologically protected singlet-triplet qubit, I demonstrated that the ground state of an array of such quantum dots embedded in an InP nanowire, with four electrons in each dot, is four-fold degenerate and is topologically protected from higher energy states, making the ground state robust against perturbations. This state is known as the Haldane phase and can be understood in terms of two spin-1/2 quasiparticles at each edge of the array. Though the spectral gap in my simulation was of the order of 1 meV, this work provides insight into the potential design of a room temperature operating Haldane qubit where the spectral gap is of the order of room temperature.

Les méthodes de réseau de tenseurs (TN) ont été un ingrédient essentiel des progrès de la physique de la matière condensée et ont récemment suscité l'intérêt de la communauté de l'apprentissage automatique pour leur capacité à représenter de manière compacte des objets de très grande dimension. Les méthodes TN peuvent par exemple être utilisées pour apprendre efficacement des modèles linéaires dans des espaces de caractéristiques exponentiellement grands [1]. Dans ce manuscrit, nous dérivons des limites supérieures et inférieures sur la VC-dimension et la pseudo-dimension d'une grande classe de Modèles TN pour la classification, la régression et la complétion . Nos bornes supérieures sont valables pour les modèles linéaires paramétrés par structures TN arbitraires, et nous dérivons des limites inférieures pour les modèles de décomposition tensorielle courants (CP, Tensor Train, Tensor Ring et Tucker) montrant l'étroitesse de notre borne supérieure générale. Ces résultats sont utilisés pour dériver une borne de généralisation qui peut être appliquée à la classification avec des matrices de faible rang ainsi qu'à des classificateurs linéaires basés sur l'un des modèles de décomposition tensorielle couramment utilisés. En corollaire de nos résultats, nous obtenons une borne sur la VC-dimension du classificateur basé sur le matrix product state introduit dans [1] en fonction de la dimension de liaison (i.e. rang de train tensoriel), qui répond à un problème ouvert répertorié par Cirac, Garre-Rubio et Pérez-García [2]. / Tensor network (TN) methods have been a key ingredient of advances in condensed matter physics and have recently sparked interest in the machine learning community for their ability to compactly represent very high-dimensional objects. TN methods can for example be used to efficiently learn linear models in exponentially large feature spaces [1]. In this manuscript, we derive upper and lower bounds on the VC-dimension and pseudo-dimension of a large class of TN models for classification, regression and completion. Our upper bounds hold for linear models parameterized by arbitrary TN structures, and we derive lower bounds for common tensor decomposition models (CP, Tensor Train, Tensor Ring and Tucker) showing the tightness of our general upper bound. These results are used to derive a generalization bound which can be applied to classification with low-rank matrices as well as linear classifiers based on any of the commonly used tensor decomposition models. As a corollary of our results, we obtain a bound on the VC-dimension of the matrix product state classifier introduced in [1] as a function of the so-called bond dimension (i.e. tensor train rank), which answers an open problem listed by Cirac, Garre-Rubio and Pérez-García [2].

Ce manuscrit regroupe différents travaux explorant les interactions entre les tenseurs et l'apprentissage automatique. Le premier chapitre est consacré à l'extension des modèles de séries reconnaissables de chaînes et d'arbres aux graphes. Nous y montrons que les modèles d'automates pondérés de chaînes et d'arbres peuvent être interprétés d'une manière simple et unifiée à l'aide de réseaux de tenseurs, et que cette interprétation s'étend naturellement aux graphes ; nous étudions certaines propriétés de ce modèle et présentons des résultats préliminaires sur leur apprentissage. Le second chapitre porte sur la minimisation approximée d'automates pondérés d'arbres et propose une approche théoriquement fondée à la problématique suivante : étant donné un automate pondéré d'arbres à n états, comment trouver un automate à m<n états calculant une fonction proche de l'originale. Le troisième chapitre traite de la régression de faible rang pour sorties à structure tensorielle. Nous y proposons un algorithme d'apprentissage rapide et efficace pour traiter un problème de régression dans lequel les sorties des tenseurs. Nous montrons que l'algorithme proposé est un algorithme d'approximation pour ce problème NP-difficile et nous donnons une analyse théorique de ses propriétés statistiques et de généralisation. Enfin, le quatrième chapitre introduit le modèle de mélanges algébriques de distributions. Ce modèle considère des combinaisons affines de distributions (où les coefficients somment à un mais ne sont pas nécessairement positifs). Nous proposons une approche pour l'apprentissage de mélanges algébriques qui étend la méthode tensorielle des moments introduite récemment. . / This thesis tackles several problems exploring connections between tensors and machine learning. In the first chapter, we propose an extension of the classical notion of recognizable function on strings and trees to graphs. We first show that the computations of weighted automata on strings and trees can be interpreted in a natural and unifying way using tensor networks, which naturally leads us to define a computational model on graphs: graph weighted models; we then study fundamental properties of this model and present preliminary learning results. The second chapter tackles a model reduction problem for weighted tree automata. We propose a principled approach to the following problem: given a weighted tree automaton with n states, how can we find an automaton with m<n states that is a good approximation of the original one? In the third chapter, we consider a problem of low rank regression for tensor structured outputs. We design a fast and efficient algorithm to address a regression task where the outputs are tensors. We show that this algorithm generalizes the reduced rank regression method and that it offers good approximation, statistical and generalization guarantees. Lastly in the fourth chapter, we introduce the algebraic mixture model. This model considers affine combinations of probability distributions (where the weights sum to one but may be negative). We extend the recently proposed tensor method of moments to algebraic mixtures, which allows us in particular to design a learning algorithm for algebraic mixtures of spherical Gaussian distributions.

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Le traitement des données massives, communément connu sous l’appellation “Big Data”, constitue l’un des principaux défis scientifiques de la communauté STIC.Plusieurs domaines, à savoir économique, industriel ou scientifique, produisent des données hétérogènes acquises selon des protocoles technologiques multi-modales. Traiter indépendamment chaque ensemble de données mesurées est clairement une approche réductrice et insatisfaisante. En faisant cela, des “relations cachées” ou des inter-corrélations entre les données peuvent être totalement ignorées.Les représentations tensorielles ont reçu une attention particulière dans ce sens en raison de leur capacité à extraire de données hétérogènes et volumineuses une information physiquement interprétable confinée à un sous-espace de dimension réduite. Dans ce cas, les données peuvent être organisées selon un tableau à D dimensions, aussi appelé tenseur d’ordre D.Dans ce contexte, le but de ce travail et que certaines propriétés soient présentes : (i) avoir des algorithmes de factorisation stables (ne souffrant pas de probème de convergence), (ii) avoir un faible coût de stockage (c’est-à-dire que le nombre de paramètres libres doit être linéaire en D), et (iii) avoir un formalisme sous forme de graphe permettant une visualisation mentale simple mais rigoureuse des décompositions tensorielles de tenseurs d’ordre élevé, soit pour D > 3.Par conséquent, nous nous appuyons sur la décomposition en train de tenseurs (TT) pour élaborer de nouveaux algorithmes de factorisation TT, et des nouvelles équivalences en termes de modélisation tensorielle, permettant une nouvelle stratégie de réduction de dimensionnalité et d'optimisation de critère des moindres carrés couplés pour l'estimation des paramètres d'intérêts nommé JIRAFE.Ces travaux d'ordre méthodologique ont eu des applications dans le contexte de l'analyse spectrale multidimensionelle et des systèmes de télécommunications à relais. / Massive and heterogeneous data processing and analysis have been clearly identified by the scientific community as key problems in several application areas. It was popularized under the generic terms of "data science" or "big data". Processing large volumes of data, extracting their hidden patterns, while preforming prediction and inference tasks has become crucial in economy, industry and science.Treating independently each set of measured data is clearly a reductiveapproach. By doing that, "hidden relationships" or inter-correlations between thedatasets may be totally missed. Tensor decompositions have received a particular attention recently due to their capability to handle a variety of mining tasks applied to massive datasets, being a pertinent framework taking into account the heterogeneity and multi-modality of the data. In this case, data can be arranged as a D-dimensional array, also referred to as a D-order tensor.In this context, the purpose of this work is that the following properties are present: (i) having a stable factorization algorithms (not suffering from convergence problems), (ii) having a low storage cost (i.e., the number of free parameters must be linear in D), and (iii) having a formalism in the form of a graph allowing a simple but rigorous mental visualization of tensor decompositions of tensors of high order, i.e., for D> 3.Therefore, we rely on the tensor train decomposition (TT) to develop new TT factorization algorithms, and new equivalences in terms of tensor modeling, allowing a new strategy of dimensionality reduction and criterion optimization of coupled least squares for the estimation of parameters named JIRAFE.This methodological work has had applications in the context of multidimensional spectral analysis and relay telecommunications systems.

Supervisor at Deutsches Elektronen-Synchrotron (DESY) in Zeuthen: Dr. Habil. Karl Jansen / Mit Tensor Netzwerken (TN) untersuchen wir auf einem hexagonalen Gitter das Hubbard-Modell mit einem chemischen Potential. Wir zeigen, dass ein TN als Ansatz für die Zustände des Modells benutzt werden kann und präsentieren die berechneten Eigenschaften bei niedrigen Energien. Unser Algorithmus wendet eine imaginäre Zeitentwicklung auf einen fermionischen projected engangled pair state (PEPS) auf einem endlichen Gitter mit offenen Randbedingungen an. Der Ansatz kann auf einen spezifischen fermionischen Paritätssektor beschränkt werden, was es uns ermöglicht, den Grundzustand und den Zustand mit einem Elektron weniger zu simulieren. Mehrere in unserer Arbeit entwickelte Verbesserungen des Algorithmus führen zu einer erheblichen Steigerung der Effizienz und Genauigkeit. Wir messen Erwartungswerte mit Hilfe eines boundary matrix product state. Wir zeigen, dass Observablen in dieser Näherung mit einer weniger starken Trunkierung, als in der Literatur erwartet wird, berechnet werden können. Dies führt zu einer erheblichen Reduzierung der numerischen Kosten des Algorithmus. Für verschiedene Stärken der lokalen Wechselwirkung, sowie für mehrere chemische Potentiale berechnen wir die Energie, die Teilchenzahl und die Magnetisierung mit guter Genauigkeit. Wir zeigen die Abhängigkeit der Teilchenzahl vom chemischen Potential und berechnen die Energielücke. Wir demonstrieren die Skalierbarkeit zu großen Gittern mit bis zu 30 × 15 Gitterpunkten und machen Vorhersagen in einem Teil des Phasenraums, der für Monte-Carlo nicht zugänglich ist. Allerdings finden wir auch Limitierungen des Algorithmus aufgrund von Instabilitäten, die die Berechnungen im Paritätssektor behindern, welcher orthogonal zum Grundzustand ist. Wir diskutieren Ursachen und Indikatoren und schlagen Lösungen vor. Unsere Arbeit bestätigt, dass TN genutzt werden können, um den niederenergetischen Sektors des Modells zu erforschen. Dies eröffnet den Weg zu einem umfassenden Verständnis des Phasendiagramms. / Using tensor network (TN) techniques, we study the Hubbard model on a honeycomb lattice with a chemical potential, which models the electron structure of graphene. In contrast to Monte Carlo methods, TN algorithms do not suffer from the sign problem when a chemical potential is present. We demonstrate that a tensor network state can be used to simulate the model and present the calculated low energy properties of the Hubbard model. Our algorithm applies an imaginary time evolution to a fermionic projected entangled pair state (PEPS) on a finite lattice with open boundary conditions. The ansatz can be restricted to a specific fermionic parity sector which allows us to simulate the ground state and the state with one electron less. Several improvements of the algorithm developed in our work lead to a substantial performance increase of the efficiency and precision. We measure expectation values with a boundary matrix product state and show that observables can be calculated with a lower bond dimension of this approximation than expected from the literature. This decreases the numerical costs of the algorithm significantly. For varying onsite interactions and chemical potentials we calculate the energy, particle number and magnetization with good precision. We show the dependence of the particle number on the chemical potential and compute the single particle gap. We demonstrate the scalability to large lattices of up to 30 × 15 sites and make predictions in a part of the phase space that is not accessible to Monte Carlo methods. However, we also find limitations of the algorithm due to instabilities that spoil the calculations in the parity sector orthogonal to the ground state. We discuss the causes and indicators of such instabilities and propose solutions. Our work validates that TNs can be utilized to study the low energy properties of the Hubbard model on a honeycomb lattice with a chemical potential, thus opening the road to finally understand its phase diagram.