Dissertations / Theses on the topic 'Quantum many-body simulation'

Nuclei are objects made of nucleons, protons and neutrons. Several dynamical processes that occur in nuclei are of great interest for the scientific community and for possible applications. For example, nuclear fusion can help us produce a large amount of energy with a limited use of resources and environmental impact. Few-nucleon scattering is an essential ingredient to understand and describe the physics of the core of a star. The classical computational algorithms that aim to simulate microscopic quantum systems suffer from the exponential growth of the computational time when the number of particles is increased. Even using today's most powerful HPC devices, the simulation of many processes, such as the nuclear scattering and fusion, is out of reach due to the excessive amount of computational time needed. In the 1980s, Feynman suggested that quantum computers might be more efficient than classical devices in simulating many-particle quantum systems. Following Feynman's idea of quantum computing, a complete change in the computation devices and in the simulation protocols has been explored in the recent years, moving towards quantum computations. Recently, the perspective of a realistic implementation of efficient quantum calculations was proved both experimentally and theoretically. Nevertheless, we are not in an era of fully functional quantum devices yet, but rather in the so-called "Noisy Intermediate-Scale Quantum" (NISQ) era. As of today, quantum simulations still suffer from the limitations of imperfect gate implementations and the quantum noise of the machine that impair the performance of the device. In this NISQ era, studies of complex nuclear systems are out of reach. The evolution and improvement of quantum devices will hopefully help us solve hard quantum problems in the coming years. At present quantum machines can be used to produce demonstrations or, at best, preliminary studies of the dynamics of a few nucleons systems (or other equivalent simple quantum systems). These systems are to be considered mostly toy models for developing prospective quantum algorithms. However, in the future, these algorithms may become efficient enough to allow simulating complex quantum systems in a quantum device, proving more efficient than classical devices, and eventually helping us study hard quantum systems. This is the main goal of this work, developing quantum algorithms, potentially useful in studying the quantum many body problem, and attempting to implement such quantum algorithms in different, existing quantum devices. In particular, the simulations made use of the IBM QPU's , of the Advanced Quantum Testbed (AQT) at Lawrence Berkeley National Laboratory (LBNL), and of the quantum testbed recently based at Lawrence Livermore National Laboratory (LLNL) (or using a device-level simulator of this machine). The our research aims are to develop quantum algorithms for general quantum processors. Therefore, the same developed quantum algorithms are implemented in different quantum processors to test their efficiency. Moreover, some uses of quantum processors are also conditioned by their availability during the time span of my PhD. The most common way to implement some quantum algorithms is to combine a discrete set of so-called elementary gates. A quantum operation is then realized in term of a sequence of such gates. This approach suffers from the large number of gates (depth of a quantum circuit) generally needed to describe the dynamics of a complex system. An excessively large circuit depth is problematic, since the presence of quantum noise would effectively erase all the information during the simulation. It is still possible to use error-correction techniques, but they require a huge amount of extra quantum register (ancilla qubits). An alternative technique that can be used to address these problems is the so-called "optimal control technique". Specifically, rather than employing a set of pre-packaged quantum gates, it is possible to optimize the external physical drive (for example, a suitably modulated electromagnetic pulse) that encodes a multi-level complex quantum gate. In this thesis, we start from the work of Holland et al. "Optimal control for the quantum simulation of nuclear dynamics" Physical Review A 101.6 (2020): 062307, where a quantum simulation of real-time neutron-neutron dynamics is proposed, in which the propagation of the system is enacted by a single dense multi-level gate derived from the nuclear spin-interaction at leading order (LO) of chiral effective field theory (EFT) through an optimal control technique. Hence, we will generalize the two neutron spin simulations, re-including spatial degrees of freedom with a hybrid algorithm. The spin dynamics are implemented within the quantum processor and the spatial dynamics are computed applying classical algorithms. We called this method classical-quantum coprocessing. The quantum simulations using optimized optimal control methods and discrete get set approach will be presented. By applying the coprocessing scheme through the optimal control, we have a possible bottleneck due to the requested classical computational time to compute the microwave pulses. A solution to this problem will be presented. Furthermore, an investigation of an improved way to efficiently compile quantum circuits based on the Similarity Renormalization Group will be discussed. This method simplifies the compilation in terms of digital gates. The most important result contained in this thesis is the development of an algorithm for performing an imaginary time propagation on a quantum chip. It belongs to the class of methods for evaluating the ground state of a quantum system, based on operating a Wick rotation of the real time evolution operator. The resulting propagator is not unitary, implementing in some way a dissipation mechanism that naturally leads the system towards its lowest energy state. Evolution in imaginary time is a well-known technique for finding the ground state of quantum many-body systems. It is at the heart of several numerical methods, including Quantum Monte Carlo techniques, that have been used with great success in quantum chemistry, condensed matter and nuclear physics. The classical implementations of imaginary time propagation suffer (with few exceptions) of an exponential increase in the computational cost with the dimension of the system. This fact calls for a generalization of the algorithm to quantum computers. The proposed algorithm is implemented by expanding the Hilbert space of the system under investigation by means of ancillary qubits. The projection is obtained by applying a series of unitary transformations having the effect of dissipating the components of the initial state along excited states of the Hamiltonian into the ancillary space. A measurement of the ancillary qubit(s) will then remove such components, effectively implementing a "cooling" of the system. The theory and testing of this method, along with some proposals for improvements will be thoroughly discussed in the dedicated chapter.

One of the important early applications of Quantum Mechanics was to explain the Van-der-Waal’s 1/R6 potential that is observed experimentally between two neutral species, such as noble gas atoms, in terms of correlated uncertainty between interacting dipoles, an effect that does not occur in the classical limit [London-Eisenschitz,1930]. When many-body correlations and higher-multipole interactions are taken into account they yield additional many-body and higher-multipole dispersion terms. Dispersion energies are closely related to electrostatic interactions and polarisation [Hirschfelder-Curtiss-Bird,1954]. Hydrogen bonding, the dominant force in water, is an example of an electrostatic effect, which is also strongly modified by polarisation effects. The behaviour of ions is also strongly influenced by polarisation. Where hydrogen bonding is disrupted, dispersion tends to act as a more constant cohesive force. It is the only attractive force that exists between hydrophobes, for example. Thus all three are important for understanding the detailed behaviour of water, and effects that happen in water, such as the solvation of ions, hydrophobic de-wetting, and thus biological nano-structures. Current molecular simulation methods rarely go beyond pair-wise potentials, and so lose the rich detail of many-body polarisation and dispersion that would permit a force field to be transferable between different environments. Empirical force-fields fitted in the gas phase, which is dominated by two-body interactions, generally do not perform well in the condensed (many-body) phases. The leading omitted dispersion term is the Axilrod-Teller-Muto 3-body potential, which does not feature in standard biophysical force-fields. Polarization is also usually ommitted, but it is sometimes included in next-generation force-fields following seminal work by Cochran [1971]. In practice, many-body forces are approximated using two-body potentials fitted to reflect bulk behaviour, but these are not transferable because they do not reproduce detailed behaviour well, resulting in spurious results near inhomogeneities, such as solvated hydrophobes and ions, surfaces and interfaces. The Quantum Drude Oscillator model (QDO) unifies many-body, multipole polarisation and dispersion, intrinsically treating them on an equal footing, potentially leading to simpler, more accurate, and more transferable force fields when it is applied in molecular simulations. The Drude Oscillator is simply a model atom wherein a single pseudoelectron is bound harmonically to a single pseudonucleus, that interacts via damped coulomb interactions [Drude,1900]. Path Integral [Feynman-Hibbs,1965] Molecular Dynamics (PIMD) can, in principle, provide an exact treatment for moving molecules at finite temperature on the Born- Oppenheimer surface due to their pseudo-electrons. PIMD can be applied to large systems, as it scales like N log(N), with multiplicative prefactor P that can be effectively parallelized away on modern supercomputers. There are other ways to treat dispersion, but all are computationally intensive and cannot be applied to large systems. These include, for example, Density Functional Theory provides an existence proof that a functional exists to include dispersion, but we dont know the functional. We outline the existing methods, and then present new density matrices to improve the discretisation of the path integral. Diffusion Monte Carlo (DMC), first proposed by Fermi, allows the fast computation of high-accuracy energies for static nuclear configurations, making it a useful method for model development, such as fitting repulsion potentials, but there is no straightforward way to generate forces. We derived new methods and trial wavefunctions for DMC, allowing the computation of energies for much larger systems to high accuracy. A Quantum Drude model of Xenon, fit in the gas-phase, was simulated in the condensed-phase using both DMC and PIMD. The new DMC methods allowed for calculation of the bulk modulus and lattice constant of FCC-solid Xenon. Both were in excellent agreement with experiment even though this model was fitted in the gasphase, demonstrating the power of Quantum Drudes to build transferable models by capturing many-body effects. We also used the Xenon model to test the new PIMD methods. Finally, we present the outline of a new QDO model of water, including QDO parameters fitted to the polarisabilities and dispersion coefficients of water.

Quantum mechanics led us to reconsider the scope of physics and its building principles, such as the notions of realism and locality. More recently, quantum mechanics has changed in an equal dramatic manner our understanding of information processing and computation. On one hand, the fundamental properties of quantum systems can be harnessed to transmit, store, and manipulate information in a much more efficient and secure way than possible in the realm of classical physics. On the other hand, the development of systematic procedures to manipulate systems of a large number of particles in the quantum regime, crucial to the implementation of quantum based information processing, has triggered new possibilities in the exploration of quantum many-body physics and related areas. In this thesis, we present new results relevant to two important problems in quantum information science: the development of a theory of entanglement, intrinsically quantum correlations of key importance in quantum information theory, and the exploration of the use of controlled quantum systems to the computation and simulation of quantum many-body phenomena. In the first part we introduce a new approach to the study of entanglement by considering its manipulation under operations not capable of generating entanglement. In this setting we show how the landscape of entanglement conversion is reduced to the simplest situation possible: one unique measure completely specifying which transformations are achievable. This framework has remarkable connections with the foundations of thermodynamics, which we present and explore. On the way to establish our main result, we develop new techniques that are of interest on their own. First, we extend quantum Stein's Lemma, characterizing optimal rates in state discrimination, to the case where the alternative hypothesis might vary over particular sets of possibly correlated (non-LLd) states. Second, we show how recent advances in quantum de Finetti type theorems can be employed to decide when the entanglement contained in non-LLd. sequences of states is distillable by local operations and classical communication. In the second part we discuss the usefulness of a quantum computer to the determination of properties of many-body systems. Our first result is a new quantum procedure, based on the phase estimation quantum algorithm, to calculate additive approximations to partition functions and spectrum densities of quantum local Hamiltonians. We give convincing evidence that quantum computation is superior to classical in solving both problems by showing that they are complete for the class of problems efficiently solved in the one-c1ean-qubit model of quantum computation, which is believe to contain classically hard problems. We then present a negative result on the usefulness of quantum computers and prove that the determination of the ground state energy of local quantum Hamiltonians, with the promise that the gap is larger than an inverse polynomial in the number of sites, is hard for the class QCMA, which is believed to contain intractable problems even for quantum computation. In the third and last part, we approach the problem of quantum simulating many-body systems from a more pragmatic point of view. Based on recent experimental developments on cavity quantum electrodynamics, more specifically on the fabrication of arrays of interacting micro-cavities and on their coupling to atomic-like structures in several physical set-ups, we propose and analyse the realization of paradigmatic condensed matter models in such systems, such as the Bose-Hubbard and the anisotropic Heisenberg models. We present· promising properties of such coupled-cavity arrays as simulators of quantum many-body physics, such as the full addressability of individual sites and the access to inhomogeneous models, and discuss the feasibility of an experimental realization with state-of-the-art current technology.

Since they were first envisioned, quantum computers have oft been portrayed as devices of limitless power, able to perform calculations in a mere instant that would take current computers years to determine. This is, of course, not the case. A huge amount of effort has been invested in trying to understand the limits of quantum computers---under which circumstances they outperform classical computers, how large a speed-up can be gained, and what draws the distinction between quantum and classical computing. In this Ph.D. thesis, I investigate a few intriguing properties of quantum computers involving quantum oracles and classically-simulatable quantum circuits. In Part I I study the notion of black-box unitary operations, and procedures for effecting the inverse operation. Part II looks at how quantum oracles can be used to test properties of probability distributions, and Part III considers classes of quantum circuits that can be simulated efficiently on a classical computer. In more detail, Part I studies procedures for inverting black-box unitary operations. Known techniques are generally limited in some way, often requiring ancilla systems, working only for restricted sets of operators, or simply being too inefficient. We develop a novel procedure without these limitations, and show how it can be applied to lift a requirement of the Solovay-Kitaev theorem, a landmark theorem of quantum compiling. Part II looks at property testing for probability distributions, and in particular considers a special type of access known as the \textit{conditional oracle}. The classical conditional oracle was developed by Canonne et al. in 2015 and subsequently greatly explored. We develop a quantum version of this oracle, and show that it has advantages over the classical process. We use this oracle to develop an algorithm that decides whether or not a mixed state is fully mixed. In Part III we study classically-simulatable quantum circuits in more depth. Two well-known classes are Clifford circuits and matchgate circuits, which we briefly review. Using these as inspiration, we use the Jordan-Wigner transform to develop new classes of non-trivial quantum circuits that are also classically simulatable.

Le domaine des problèmes quantiques à N-corps à l'équilibre et hors d'équilibre sont des sujets majeurs de la Physique et de la Physique de la matière condensée en particulier. Les propriétés d'équilibre de nombreux systèmes unidimensionnels en interaction sont bien comprises d'un point de vue théorique, des chaînes de spins aux théories quantiques des champs dans le continue. Ces progrès ont été rendus possibles par le développement de nombreuses techniques puissantes, comme, par exemple, l'ansatz de Bethe, le groupe de renormalisation, la bosonisation, les états produits de matrices ou la théorie des champs invariante conforme. Même si les propriétés à l'équilibre de nombreux modèles soient connues, ceci n'est en général pas suffisant pour décrire leurs comportements hors d'équilibre, et ces derniers restent moins explorés et beaucoup moins bien compris. Les modèles d'impuretés quantiques représentent certains des modèles à N-corps les plus simples. Mais malgré leur apparente simplicité ils peuvent capturer plusieurs phénomènes expérimentaux importants, de l'effet Kondo dans les métaux aux propriétés de transports dans les nanostructures, comme les points quantiques. Dans ce travail nous considérons un modèle d'impureté appelé "modèle de niveau résonnant en interaction" (IRLM). Ce modèle décrit des fermions sans spin se propageant dans deux fils semi-infinis qui sont couplés à un niveau résonant -- appelé point ou impureté quantique -- via un terme de saut et une répulsion Coulombienne. Nous nous intéressons aux situations hors d'équilibre où un courant de particules s'écoule à travers le point quantique, et étudions les propriétés de transport telles que le courant stationnaire (en fonction du voltage), la conductance différentielle, le courant réfléchi, le bruit du courant ou encore l'entropie d'intrication. Nous réalisons des simulations numériques de la dynamique du modèle avec la méthode du groupe de renormalisation de la matrice densité dépendent du temps (tDMRG), qui est basée sur une description des fonctions d'onde en terme d'états produits de matrices. Nous obtenons des résultats de grande précision concernant les courbes courant-voltage ou bruit-voltage de l'IRLM, dans un grand domaine de paramètres du modèle (voltage, force de l'interaction, amplitude de saut vers le dot, etc.). Ces résultats numériques sont analysés à la lumière de résultats exacts de théorie des champs hors d'équilibre qui ont été obtenus pour un modèle similaire à l'IRLM, le modèle de Sine-Gordon avec bord (BSG). Cette analyse est en particulier basée sur l'identification d'une échelle d'énergie Kondo et d'exposants décrivant les régimes de petit et grand voltage. Aux deux points particuliers où les modèles sont connus comme étant équivalents, nos résultats sont en accord parfait avec la solution exacte. En dehors de ces deux points particuliers nous trouvons que les courbes de transport de l'IRLM et du modèle BSG demeurent très proches, ce qui était inattendu et qui reste dans une certaine mesure inexpliqué<br>The fields of in- and out-of-equilibrium quantum many-body systems are major topics in Physics, and in condensed-matter Physics in particular. The equilibrium properties of one-dimensional problems are well studied and understood theoretically for a vast amount of interacting models, from lattice spin chains to quantum fields in a continuum. This progress was allowed by the development of diverse powerful techniques, for instance, Bethe ansatz, renormalization group, bosonization, matrix product states and conformal field theory. Although the equilibrium characteristics of many models are known, this is in general not enough to describe their non-equilibrium behaviors, the latter often remain less explored and much less understood. Quantum impurity models represent some of the simplest many-body problems. But despite their apparent simplicity, they can capture several important experimental phenomena, from the Kondo effect in metals to transport in nanostructures such as point contacts or quantum dots. In this thesis consider a classic impurity model - the interacting resonant level model (IRLM). The model describes spinless fermions in two semi-infinite leads that are coupled to a resonant level -- called quantum dot or impurity -- via weak tunneling and Coulomb repulsion. We are interested in out-of-equilibrium situations where some particle current flows through the dot, and study transport characteristics like the steady current (versus voltage), differential conductance, backscattered current, current noise or the entanglement entropy. We perform extensive state-of-the-art computer simulations of model dynamics with the time-dependent density renormalization group method (tDMRG) which is based on a matrix product state description of the wave functions. We obtain highly accurate results concerning the current-voltage and noise-voltage curves of the IRLM in a wide range parameter of the model (voltage bias, interaction strength, tunneling amplitude to the dot, etc.).These numerical results are analyzed in the light of some exact out-of-equilibrium field-theory results that have been obtained for a model similar to the IRLM, the boundary sine-Gordon model (BSG).This analysis is in particular based on identifying an emerging Kondo energy scale and relevant exponents describing the high- and low- voltage regimes. At the two specific points where the models are known to be equivalent our results agree perfectly with the exact solution. Away from these two points, we find that, within the precision of our simulations, the transport curves of the IRLM and BSG remain very similar, which was not expected and which remains somewhat unexplained

La simulation quantique consiste à réaliser expérimentalement des systèmes artificiels équivalent à des modèles proposés par les théoriciens. Pour réaliser ces systèmes, il est possible d'utiliser des atomes dont les états individuels et les interactions sont contrôlés par la lumière. En particulier, une fois excités dans un état de haute énergie (appelé état de Rydberg), les atomes peuvent être contrôlés individuellement et leurs interactions façonnées arbitrairement par des faisceaux laser. Cette thèse s'intéresse à deux types de simulateurs quantiques à base d'atomes de Rydberg, et en particulier à leurs potentielles limitations.Dans l'expérience du Joint Quantum Institute (USA), nous observons la décohérence dans une structure cubique contenant jusqu'à 40000 atomes. A partir d'atomes préparés dans un état de Rydberg bien défini, nous constatons l'apparition spontanée d'états de Rydberg voisins et le déclenchement d'un phénomène d'avalanche. Nous montrons que ce mécanisme émane de l'émission stimulée produite par le rayonnement du corps noir. Ce phénomène s'accompagne d'une diffusion induite par des interactions de type dipole-dipole résonant. Nous complétons ces observations avec un modèle de champ moyen en état stationnaire. Dans un second temps, l'étude de la dynamique du problème nous permet de mesurer les échelles de temps caractéristiques. La décohérence étant globalement néfaste pour la simulation quantique, nous proposons plusieurs solutions pour en atténuer les effets. Nous évaluons notamment la possibilité de travailler dans un environnement cryogénique, lequel permettrait de réduire le rayonnement du corps noir.Dans l'expérience du Laboratoire Charles Fabry à l'Institut d'Optique (France), nous analysons les limites d'un simulateur quantique générant des structures bi- et tridimensionnelles allant jusqu'à 70 atomes de Rydberg piégés individuellement dans des pinces optiques. Le système actuel étant limité par le temps de vie des structures, nous montrons que l'utilisation d'un cryostat permettrait d'atteindre des tailles de structures jusqu'à 300 atomes. Nous présentons les premiers pas d'une nouvelle expérience utilisant un cryostat à 4K, et en particulier les études amont pour le développement de composants optomécaniques placés sous vide et à froid<br>Quantum simulation consists in engineering well-controlled artificial systems that are ruled by the idealized models proposed by the theorists. Such toy models can be produced with individual atoms, where laser beams control individual atomic states and interatomic interactions. In particular, exciting atoms into a highly excited state (called a Rydberg state) allows to control individual atoms and taylor interatomic interactions with light. In this thesis, we investigate experimentally two different types of Rydberg-based quantum simulators and identify some possible limitations.At the Joint Quantum Institute, we observe the decoherence of an ensemble of up to 40000 Rydberg atoms arranged in a cubic geometry. Starting from the atoms prepared in a well-defined Rydberg state, we show that the spontaneous apparition of population in nearby Rydberg states leads to an avalanche process. We identify the origin of the mechanism as stimulated emission induced by black-body radiation followed by a diffusion induced by the resonant dipole-dipole interaction. We describe our observations with a steady-state mean-field analysis. We then study the dynamics of the phenomenon and measure its typical timescales. Since decoherence is overall negative for quantum simulation, we propose several solutions to mitigate the effect. Among them, we discuss the possibility to work at cryogenic temperatures, thus suppressing the black-body induced avalanche.In the experiment at Laboratoire Charles Fabry (Institut d'Optique), we analyze the limitation of a quantum simulator based on 2 and 3 dimensional arrays of up to 70 atoms trapped in optical tweezers and excited to Rydberg states. The current system is limited by the lifetime of the atomic structure. We show that working at cryogenic temperatures could allow to increase the size of the system up to N=300 atoms. In this context, we start a new experiment based on a 4K cryostat. We present the early stage of the new apparatus and some study concerning the optomechanical components to be placed inside the cryostat

Cold-atom quantum simulators offer unique possibilities to prepare, manipulate, and probe quantum many-body systems. However, despite the high level of control in modern experiments, not all observables of interest are easily accessible. This thesis aims at establishing protocols to measure currently elusive static and dynamic properties of quantum systems. The experimental feasibility of these schemes is illustrated by means of numerical simulations for relevant applications in many-body physics and quantum simulation. In particular, we introduce a general method for measuring dynamical correlations based on non-Hermitian linear response. This enables unbiased tests of the famous fluctuation-dissipation relation as a probe of thermalization in isolated quantum systems. Furthermore, we develop ancilla-based techniques for the measurement of currents and current correlations, permitting the characterization of strongly correlated quantum matter. Another application is geared towards revealing signatures of supersolidity in spin-orbit-coupled Bose gases by exciting the relevant Goldstone modes. Finally, we explore a scenario for quantum-simulating post-inflationary reheating dynamics by parametrically driving a Bose gas into the regime of universal far-from-equilibrium dynamics. The presented protocols also apply to other analog quantum simulation platforms and thus open up promising applications in the field of quantum science and technology.<br>I simulatori quantistici ad atomi freddi offrono possibilità uniche per preparare, manipolare e sondare sistemi quantistici a molti corpi. Tuttavia, nonostante l'alto livello di controllo raggiunto negli esperimenti moderni, non tutte le osservabili di interesse sono facilmente accessibili. Lo scopo di questa tesi è quello di stabilire protocolli per misurare delle proprietà statiche e dinamiche dei sistemi quantistici attualmente inaccessibili. La fattibilità sperimentale di questi schemi è illustrata mediante simulazioni numeriche per applicazioni rilevanti nella fisica a molti corpi e nella simulazione quantistica. In particolare, introduciamo un metodo generale per misurare le correlazioni dinamiche basato su una risposta lineare non hermitiana. Ciò consente test imparziali della famosa relazione fluttuazione-dissipazione come sonda di termalizzazione in sistemi quantistici isolati. Inoltre, sviluppiamo tecniche basate su ancilla per la misura di correnti e correlazioni di corrente, consentendo la caratterizzazione della materia quantistica fortemente correlata. Un'altra applicazione è orientata a rivelare l'impronta della supersolidità nei gas Bose con accoppiamento spin-orbita eccitando il corrispondente modo di Goldstone. Infine, esploriamo uno scenario per la simulazione quantistica della dinamica di riscaldamento post-inflazione modulando parametricamente un gas Bose e portandolo nel regime della dinamica universale lontana dall'equilibrio. I protocolli presentati si applicano anche ad altre piattaforme di simulazione quantistica analogica e aprono quindi applicazioni promettenti nel campo della scienza e della tecnologia quantistica.<br>Quantensimulatoren auf Basis ultrakalter Atome eröffnen einzigartige Möglichkeiten zur Präparation, Manipulation und Untersuchung von Quanten-Vielteilchen-Systemen. Trotz des hohen Maßes an Kontrolle in modernen Experimenten sind jedoch nicht alle interessanten Observablen auf einfache Weise zugänglich. Ziel dieser Arbeit ist es, Protokolle zur Messung aktuell nur schwer erfassbarer statischer und dynamischer Eigenschaften von Quantensystemen zu etablieren. Die experimentelle Realisierbarkeit dieser Verfahren wird durch numerische Simulationen anhand relevanter Anwendungen in der Vielteilchenphysik und Quantensimulation veranschaulicht. Insbesondere wird eine allgemeine Methode zur Messung dynamischer Korrelationen basierend auf der linearen Antwort auf nicht-hermitesche Störungen vorgestellt. Diese ermöglicht unabhängige Tests des berühmten Fluktuations-Dissipations-Theorems als Indikator der Thermalisierung isolierter Quantensysteme. Darüber hinaus werden Verfahren zur Messung von Strömen und Strom-Korrelationen mittels Kopplung an einen Hilfszustand entwickelt, welche die Charakterisierung stark korrelierter Quantenmaterie erlauben. Eine weitere Anwendung zielt auf die Enthüllung spezifischer Merkmale von Supersolidität in Spin-Bahn-gekoppelten Bose-Einstein-Kondensaten ab, indem die relevanten Goldstone-Moden angeregt werden. Schließlich wird ein Szenario zur Quantensimulation post-inflationärer Thermalisierungsdynamik durch die parametrische Anregung eines Bose-Gases in das Regime universeller Dynamik fern des Gleichgewichts erschlossen. Die dargestellten Protokolle lassen sich auch auf andere Plattformen für analoge Quantensimulation übertragen und eröffnen damit vielversprechende Anwendungen auf dem Gebiet der Quantentechnologie.

Ce travail de thèse est dédié à l'étude des statistiques du nombre et corrélations en impulsion dans des gaz de Bose sur réseaux interagissants. Le modèle de Bose-Hubbard est simulé en chargeant des condensats de Bose-Einstein (BEC) d'atomes d'Hélium-4 métastables dans un réseau optique tridimensionnel (3D). Ce modèle présente une transition de phase quantique d'un superfluide à un isolant de Mott induite par des fluctuations quantiques provoquées par l'interaction. L'objectif de ce travail est de comprendre le rôle de ces fluctuations quantiques en analysant leurs signatures dans l'espace des impulsions. Le schéma de détection original utilisé à cette fin fournit la distribution d'impulsion résolue à l'échelle de l'atome unique en 3D. À partir de ces jeux de données composés de milliers d'atomes individuels, les statistiques du nombre d'occupation de différents sous-volumes de l'espace des impulsions fournissent des informations sur les propriétés de corrélation ou de cohérence du gaz de Bose interagissant. À impulsions proches, ces probabilités d'occupation permettent l'identification de statistiques d'état pur sous-jacentes dans le cas d'états many-body classiques tels que les superfluides en réseau et les isolants de Mott. Dans le régime faiblement interagissant, des corrélations bien établies entre les paires d'atomes à impulsions opposées sont observées. De plus, on constate que ces corrélations entre paires diminuent en faveur de corrélations plus complexes entre plus de deux particules lorsque les interactions sont augmentées. Une observation directe de corrélations non-Gaussiennes encapsule la nature statistique complexe des superfluides fortement interagissants bien en amont de la transition de phase vers l'isolant de Mott. Enfin, lors de la transition de phase, on constate une augmentation des fluctuations du nombre d'occupation du mode du BEC, constituant une signature directe des fluctuations quantiques induisant la transition. Des quantités indépendantes de la taille du système, telles que le cumulant de Binder, présentent des variations abruptes même dans un système de taille finie et semblent prometteuses pour constituer des observables appropriés permettant de déterminer le comportement universel lorsqu'elles sont mesurées dans un système homogène<br>This thesis work is dedicated to the study of number statistics and momentum correlations in interacting lattice Bose gases. The Bose-Hubbard model is simulated by loading Bose-Einstein condensates (BECs) of metastable Helium-4 atoms into a three-dimensional (3D) optical lattice. This model exhibits a quantum phase transition from a superfluid to a Mott insulator that is driven by interaction-induced quantum fluctuations. The objective of this work is to comprehend the role of these quantum fluctuations by analyzing their signatures in momentum space. The original detection scheme employed towards this aim provides the single-particle resolved momentum distribution of the atoms in 3D. From such datasets made up of thousands of individual atoms, the number statistics of occupation of different sub-volumes of momentum space yield information about correlation or coherence properties of the interacting Bose gas. At close-by momenta these occupation probabilities permit the identification of underlying pure-state statistics in the case of textbook many-body states such as lattice superfluids and Mott insulators. In the weakly-interacting regime, well-established correlations between pairs of atoms at opposite momenta are observed. Furthermore, these pair correlations are found to decrease in favor of more intricate correlations between more than two particles as interactions are increased. A direct observation of non-Gaussian correlations encapsulates the complex statistical nature of strongly-interacting superfluids well before the Mott insulator phase transition. Finally, at the phase transition, fluctuations of the occupation number of the BEC mode are found to be enhanced, constituting a direct signature of the quantum fluctuations driving the transition. System-size independent quantities such as the Binder cumulant are shown to exhibit distinctive sharp features even in a finite-size system, and hold promise for constituting suitable observables for determining universal behavior when measured in a homogeneous system

Many-body physics studies the collective behavior of systems with a large number of microscopic constituents. The interaction between the fundamental particles creates a common behavior within the system with emergent excitations exhibiting uncommon characteristics. In three spatial dimensions it has recently been found that a new kind of particles can exist characterized by a fractionalized mobility, being either immobile or mobile only along sub-dimensional spaces: fractons. In this thesis I explore fracton phases focusing on their topological and thermal properties. Fractons can be explained as a generalization of usual topological particles with some fundamental differences, which make fracton order a new field on its own. Fracton models are studied first from the point of view of exactly solvable lattice spin models, focusing on the similarities and differences with usual topological models. Fracton phases are also described through the use of symmetric tensor gauge theory. This gives a theoretical background which is used to explore some possible phases at finite densities of fractons, like Fermi liquids and quantum Hall states. The thermal properties of such systems are studied in detail through the use of numerical simulations relying on exact-diagonalization. Various correspondences with systems featuring quantum many-body scars are found, in particular with the PXP model. The non-thermal behavior of the models under study is justified by the fragmentation of the Hilbert space in a large number of separated sub-sectors, not related to symmetries of the model. Further, the range of the local Hamiltonian operators is found to be of fundamental relevance in the thermal properties of the system. For certain ranges it is observed that the models are not able to reach the thermal state at long times. Instead, increasing the length of interactions the system becomes ergodic, with the exception of a small number of special eigenstates which remain non-thermal.

In this thesis, we address the problem of solving for the properties of interacting quantum many-body systems in thermal equilibrium. The complexity of this problem increases exponentially with system size, limiting exact numerical simulations to very small systems. To tackle more complex systems, one needs to use heuristic algorithms that approximate solutions to these systems. Belief propagation is one such algorithm that we discuss in chapters 2 and 3. Using belief propagation, we demonstrate that it is possible to solve for static properties of highly correlated quantum many-body systems for certain geometries at all temperatures. In chapter 4, we generalize the multiscale renormalization ansatz to the anyonic setting to solve for the ground state properties of anyonic quantum many-body systems. The algorithms we present in chapters 2, 3, and 4 are very successful in certain settings, but they are not applicable to the most general quantum mechanical systems. For this, we propose using quantum computers as we discuss in chapter 5. The dimension reduction algorithm we consider in chapter 5 enables us to prepare thermal states of any quantum many-body system on a quantum computer faster than any previously known algorithm. Using these thermal states as the initialization of a quantum computer, one can study both static and dynamic properties of quantum systems without any memory overhead.

Topologically ordered phases flamboyance a cornucopia of intriguing phenomena that cannot be perceived in the conventional phases including the most striking property of hosting anyon quasiparticles having fractional charges and fractional statistics. Such phases were discovered with the remarkable experiment of the fractional quantum Hall effect and are drawing a lot of recognition. Realization of these phases on lattice systems and study of the anyon quasiparticles there are important and interesting avenue to research in unraveling new physics, which can not be found in the continuum, and this thesis is an important contribution in that direction. Also such lattice models hosting anyons are particularly important to control the movement of anyons while experimentally implemented with ultra-cold atoms in optical lattices. We construct lattice models by implementing analytical states and parent Hamiltonians on two-dimensional plane hosting non-Abelian anyons, which are proposed candidates for quantum computations. Such lattice models are suitable to create both quasiholes and quasielectrons in the similar way and thereby avoiding the singularity problem for the quasielectrons in continuum. Anyons in these models are found to be well-screened with proper charges and right statistics. Going beyond two dimensions, we unravel the intriguing physics of topologically ordered phases of matter in fractional dimensions such as in the fractal lattices by employing our model constructions of analytical states and parent Hamiltonians there. We find the anyons to be well-screened with right charges and statistics for all dimensions. Our work takes the first step in bridging the gap between two dimensions and one dimension in addressing topological phases which reveal new physics. Our constructions are particularly important in this context since such lattices lack translational symmetry and hence become unsuitable for the fractional Chern insulator implementations. The special features of topologically ordered phases make these difficult to probe and hence the detection of topological quantum phase transitions becomes challenging. The existing probes suffer from shortcomings uo-to a large extent and therefore construction of new type of probes become important and are on high demand. The robustness of anyon properties draw our attention to propose these as detector of topological quantum phase transitions with significant advantages including the facts that these are numerically cheaper probes and are independent of the boundary conditions. We test our probe in three different examples and find that simple properties like anyon charges detect the transitions.<br>Topologisch geordnete Phasen extravagieren ein Füllhorn faszinierender Phänomene, die in den herkömmlichen Phasen nicht wahrgenommen werden können, einschließlich der auffälligsten Eigenschaft, Quasiteilchen mit fraktionierten Ladungen und fraktion- ierten Statistiken aufzunehmen. Solche Phasen wurden mit dem bemerkenswerten Exper- iment des fraktionierten Quanten-Hall-Effekts entdeckt und finden viel Anerkennung. Die Realisierung dieser Phasen auf Gittersystemen und die Untersuchung der Anyon- Quasiteilchen sind wichtige und interessante Wege zur Erforschung der Entschlüsselung neuer Physik, die im Kontinuum nicht zu finden sind, und diese These ist ein wichtiger Beitrag in diese Richtung. Auch solche Gittermodelle, die Anyons enthalten, sind beson- ders wichtig, um die Bewegung von Anyons zu steuern, während sie experimentell mit ultrakalten Atomen in optischen Gittern implementiert werden. Wir konstruieren Gittermodelle, indem wir analytische Zustände und Eltern-Hamiltonianer auf einer zwei- dimensionalen Ebene implementieren, die nicht-abelsche Anyons enthält, die als Kan- didaten für Quantenberechnungen vorgeschlagen werden. Solche Gittermodelle sind geeignet, sowohl Quasi-Löcher als auch Quasielektronen auf ähnliche Weise zu erzeu- gen und dadurch das Singularitätsproblem für die Quasielektronen im Kontinuum zu vermeiden. Jeder in diesen Modellen wird mit angemessenen Gebühren und richtigen Statistiken gut überprüft. Über zwei Dimensionen hinaus enträtseln wir die faszinierende Physik topologisch geordneter Phasen der Materie in fraktionierten Dimensionen wie in den fraktalen Gittern, indem wir dort unsere Modellkonstruktionen von analytischen Zuständen und Eltern-Hamiltonianern verwenden. Wir finden, dass die Anyons mit den richtigen Gebühren und Statistiken für alle Dimensionen gut überprüft werden. Unsere Arbeit macht den ersten Schritt, um die Lücke zwischen zwei Dimensionen und einer Dimension zu schließen und topologische Phasen anzugehen, die neue Physik enthüllen. Unsere Konstruktionen sind in diesem Zusammenhang besonders wichtig, da solche Gitter keine Translationssymmetrie aufweisen und daher für die fraktionierten Chern- Isolatorimplementierungen ungeeignet werden. Die besonderen Merkmale topologisch geordneter Phasen machen es schwierig, diese zu untersuchen, und daher wird die Detek- tion topologischer Quantenphasenübergänge schwierig. Die vorhandenen Sonden leiden in hohem Maße unter Mängeln, weshalb die Konstruktion neuer Sondenarten wichtig wird und eine hohe Nachfrage besteht. Die Robustheit der Anyon-Eigenschaften lenkt unsere Aufmerksamkeit darauf, diese als Detektor für topologische Quantenphasenübergänge mit signifikanten Vorteilen vorzuschlagen, einschließlich der Tatsache, dass dies numerisch billigere Sonden sind und von den Randbedingungen unabhängig sind. Wir testen unsere Sonde in drei verschiedenen Beispielen und stellen fest, dass einfache Eigenschaften wie Ladungen die Übergänge erfassen.