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We investigate cold bosonic impurity atoms trapped in a vortex lattice formed by condensed bosons of another species. We describe the dynamics of the impurities by a bosonic Hubbard model containing occupation-dependent parameters to capture the effects of strong impurity-impurity interactions. These include both a repulsive direct interaction and an attractive effective interaction mediated by the Bose-Einstein condensate. The occupation dependence of these two competing interactions drastically affects the Hubbard model phase diagram, including causing the disappearance of some Mott lobes.

Identifying the types of algorithms and applications that can be solved on today's quantum computers is one of the fundamental goals of modern quantum algorithms research. Underpinning this effort is the potential leap in human technology should it be shown that quantum computers can perform useful classically-intractable calculations in the absence of quantum error correction. In this thesis we study the new paradigm of variational quantum algorithms (VQAs), designed specifically to solve optimisation problems on near-term hardware. We develop several novel algorithms spanning a range of application areas and provide estimates of the physical resources required to match classical methods. Thereafter, we attempt to run these algorithms on current quantum computers, probing their capabilities in the process. In this endeavour, we also develop our own algorithmic solution to device noise in the form of a new approach to approximate quantum circuit recompilation. We begin with studying the feasibility of simulating the time evolution of quantum systems on current quantum computers, an important problem due to its classical hardness. We build an algorithm specifically focused on the electron-phonon Hamiltonian and subsequently realise the first demonstration of obtaining its dynamics on real quantum hardware. Subsequently, we test whether a similar result is possible for time-evolution based VQAs. For this we construct a near-term quantum computing approach to dynamical mean-field theory and find that whilst not possible currently, such an algorithm could be realistically evaluated on the next generation of quantum computers in the immediate future. We then show how quantum circuits can be used as machine learning models and apply them to self-supervised learning, one of the most demanding tasks in deep learning. Through our experiments we observe a numerical advantage for the learning of visual representations using small-scale quantum neural networks over equivalently structured classical networks, a first step on the ladder towards general quantum advantage. Through this, we also highlight the potential of near-term quantum computing in problems with only empirically established classical complexity.

Ultracold dipolar systems, featuring strong long-range dipole-dipole interactions, have been brought under increasing quantum control over recent years. In this thesis, we study the use of these properties for quantum information processing and simulation of many-body Bose-Hubbard models with off-site interactions. We first investigate robust entangling protocols for polar molecules, finding that shaped microwave pulses provide two-qubit entangling gates based on the dipole-dipole interaction with robustness to experimentally-relevant errors. We then numerically study the ground state properties of hard-core dipolar bosons confined in cylindrical optical lattices, where the curved lattice surface causes the anisotropic dipole-dipole interactions to vary around the ring of the cylinder. This expands the physics of analogous square lattice models due to the cooperation and competition of the interactions in different directions. Finally, we investigate the applicability of the lowest-band dipolar Bose-Hubbard model itself by comparing lattice and continuum methods for small systems of strongly interacting dipolar bosons in optical lattice potentials.

From the motion of hurricanes to the cooling of a cup of tea, dissipative dynamics are ubiquitous in nature. Unfortunately, when studying lattice-based strongly correlated quantum systems, a common focus of experiments and highly relevant for material science, the curse of dimensionality prevents us from directly solving the exact equations describing the dissipative motion. We instead often use approximations and numerical analysis, as few analytical techniques currently exist which can provide useful insights. This thesis studies the constraints that non-abelian algebraic structures place on the dynamics of such non-equilibrium many-body quantum systems. In doing so, we develop novel algebraic methods that provide previously absent analytic insight into a range of problems. We apply these new techniques to study dynamical phases of quantum matter that are of both theoretical and experimental interest. We study how simple algebraic structures imply persistent non-stationarity in dissipative many-body quantum systems. These results enable us to develop a theory of quantum synchronisation based on symmetries and to explore a novel class of time crystals where order is induced by noise. These dynamical phases of matter have been gaining recent interest for their insights into thermalisation and their potential application in quantum technologies. We therefore use our results to explore experimentally realisable models. We also use the general principles of prethermalisation to propose a new theory to potentially explain recent experiments studying light-induced superconductivity in terms of eta-pairing and approximate symmetries. We discuss the merits and weaknesses of our proposal in relation to the experimental evidence. We also indicate techniques that, within our theory, stabilise superconducting states by harnessing the quantum Zeno effect For certain systems, known as integrable, there are enough symmetries to allow for exact solutions using the Bethe ansatz. We demonstrate that Bethe ansatz techniques can be extended and applied to non-equilibrium problems. As an example, we use this method to analytically study the dynamical phase transitions of an XXZ spin chain with boundary loss.

Turbulence is among the most important unsolved problems of physics. Its numerical solution is hindered by the extreme number of computational variables needed to accurately resolve the broad range of length-scales that become relevant during turbulent flow. It turns out that a similar challenge appears in a completely different branch of physics. Quantum many-body systems are described by elements of a vector space whose dimension grows exponentially with the number of particles, making direct simulation infeasible. However, the actual information contained in realistic quantum states is typically only polynomially large in the number of particles, which in principle makes it possible to represent them using a polynomial number of parameters. Tensor network methods do precisely this for quantum systems with local interactions by removing unrealised long-distance correlations from the solution space, which in turn enables accurate simulation of otherwise intractable quantum systems. In this thesis, a simple tensor network formalism known as the matrix product state (MPS) ansatz is transferred from quantum physics onto fluid dynamics and used to numerically examine two paradigmatic turbulent flows: the 2D temporally decaying jet and 3D collapse of the Taylor-Green vortex. We find both flows to be structured according to the classical, scale-local view of turbulence, where flow features of disparate scales are largely uncorrelated. We eliminate these unrealistic interscale correlations from the solution space through our MPS encoding of the velocity field, and then formulate a MPS algorithm for simulating turbulence. With this algorithm, we find that the incompressible Navier-Stokes equations can be accurately solved even when reducing the number of computational parameters by more than one order of magnitude compared to traditional direct numerical simulation. The outlook is threefold. Further work towards harnessing the power of tensor networks for turbulence simulation holds the promise of computational fluid dynamics calculations that are yet inconceivable in scale. Moreover, the close connection that our MPS algorithm has to quantum physics points towards the exciting prospect of solving the Navier-Stokes equations on a quantum computer. Finally, from a theoretical standpoint, this work also lays the foundations for studying the structures of turbulence using tensor network theory. One topic of particular interest here is what tensor network geometry is most appropriate for turbulence simulations and why. Answering this question will illuminate the structure of turbulence from a completely new angle, and perhaps help unravel the old riddle that is predicting the dynamics of turbulent flows.

Controlling the states of quantum systems is a subject of major interest for both fundamental science and technological development. In this thesis, we carry out theoretical studies on how this can be achieved using light, both of classical and of quantum mechanical nature. We study a driven cavity system, where the driving field is classical and the cavity field is quantum mechanical. We demonstrate how virtual scattering of driving photons inside the cavity via two-photon processes can induce controllable long-range electron interactions in two-dimensional materials. We show that both the strength and the sign of the interactions can be tuned by the driving parameters and that the interactions are not screened effectively except at very low frequencies. For realistic cavity parameters, driving-induced heating of the electrons by inelastic photon scattering is suppressed and coherent electron interactions dominate. When the interactions are attractive, they cause an instability in the Cooper channel at a temperature proportional to the square root of the driving intensity. Our results open up a new avenue for engineering quantum materials on-demand and the possibility of studying the effect of long-range interactions in condensed matter systems. Following that, we take a more general approach to investigate the effect of long-range density-density electron attractive interactions on a superconductor. We show that the low-lying excitations of a 2D conventional superconductor can be significantly altered by these long-range interactions. Using BCS theory, we find that these interactions combine non-linearly with the intrinsic local attractions of the superconductor to increase the Bogoliubov quasiparticle excitation energies, thus enlarging the superconducting gap. Moreover, we show how the long-range nature of the driven-cavity-induced attractions qualitatively changes the collective excitations of the superconductor. Specifically, they lead to the appearance of additional collective excitations of the excitonic modes. They further push the Higgs mode "in gap": it lies below the Bogoliubov quasiparticle continuum such that it cannot decay through the quasiparticle excitations.

Identifying mechanisms which can guide many-body quantum systems into regimes where properties such as entanglement and long-range coherence are manifest is a fundamental goal for those working in the field of strongly correlated systems. With this comes the potential to realise and exploit states of matter such as superconductors and superfluids, where quantum behaviour is observable at the macroscopic level. In this thesis we study how symmetries and heating can, counterintuively, be used to realise phases of matter with such desirable properties. We prove how heating a many-body system whilst preserving certain symmetries - such as those of the special unitary group - result in the formation of maximum entropy states which are confined to a subspace of the total Hilbert space and are capable of possessing finite, completely uniform off-diagonal correlations. This mechanism is termed heating-induced order and is independent of any microscopic details. We use the Hubbard model as a central example where this mechanism can be observed. Heating is introduced to the system via periodic driving or local dissipation and we study the various ordered steady states which emerge in this setup. We discuss the applicability of this mechanism to the thermodynamic limit and its relevance to recent solid-state experiments observing photo-induced superconductivity in irradiated compounds. We then show how, in an open quantum system, the satisfaction of a set of simple symmetry-based conditions guarantees an absence of stationarity and the formation of coherent oscillations in the long-time limit. We prove how this result subsumes and goes beyond the established notion of a Decoherence Free Subspace. When these conditions are satisfied in the presence of heating-induced order we observe the formation of an entangled, correlated state undergoing identical limit cycles at all positions in space. This leads us to formulate a novel process for quantum synchronisation which is based on the combination of these symmetry-based conditions and the mechanism of heating-induced order.

In this thesis, I present my research on the numerical simulation of finite-size strongly-correlated lattice systems using the density matrix renormalisation group (DMRG). Ultracold gases in optical lattices have become the experimental setup of choice to simulate models from condensed-matter physics, because of their high degree of tunability and control. Their inherently finite size and alternative implementation call for a more in-detail study of how to define and characterise the phases of the models that they simulate. I use DMRG to implement lattice Hamiltonians of highly-correlated systems with long-range interactions and identify their ground states on a finite system geometry. I describe in detail the process of constructing long-range Hamiltonians for their use in DMRG. I then apply these numerical methods to two fundamental models. I study the ground state of a fractional quantum Hall system and identify it as the well-known Laughlin state in a still unexplored parameter regime, by calculating its topological entanglement entropy and by using a set of physical observables that are available in a finite cylindrical geometry. I then study a dipolar Bose-Hubbard model and perform a systematic study of the order parameters that can best be used to characterise its phases in an ultracold gas setting by comparing how different types of observables are sensitive to finite-size and boundary effects. My results are meant to provide guidance for future experimental realisations of bosonic lattice models of small sizes.

Controlling and stabilising collective phases of many-body quantum systems is a problem of deep fundamental and technological interest. In this thesis, we perform a theoretical investigation on how useful quantum phases may be engineered in strongly correlated fermionic lattice systems, especially through periodic driving. We first compute the phase diagram of the one-dimensional t–J model with the addition of non-standard pair hopping terms. We show that at dilute fillings these terms enhance superconductivity while, counter- intuitively, suppressing it at large fillings. We argue that this is due to dynamical constraints originating from the fact that local pairs cannot overlap. We conjecture that these constraints may play a more significant role in the physics of two-dimensional systems where the t–J model is studied as a candidate model of high-Tc superconductivity. We begin to investigate these dynamical constraints on a ladder geometry. We then study the fermionic Hubbard model under periodic driving. We show that the driving induces a strong and robust singlet-pairing effect consistent with a superconducting state. This could provide a new mechanism for light-induced superconductivity in some classes of strongly cor- related materials. We show using Floquet theory that the dynamics of the driven Hubbard model are described precisely by the t–J model studied in the previous section. As the driven Hubbard model is also implementable with ultracold fermions in an optical lattice, our results could lead to their use as a quantum simulator for a broad class of candidate models for high-Tc superconductivity.

In this dissertation, we theoretically demonstrate laser light induced control and cooling over the order parameter of superconducting cuprates. We predict a non-equilibrium state with enhanced robustness against thermal fluctuations when pumped with coherent pulses. This might ultimately increase the critical temperature of the cuprate. In particular, we propose a parametric cooling scheme for a bilayer cuprate modelled phenomenologically as a stack of long intrinsic Josephson junctions. We identify a parameter regime, which according to the switching current distribution allows a reduction of the effective system temperature by approximately 25%. This cooling scheme suppresses the thermal excitation of Josephson vortices. This might lead to an indirect mechanism for controlling topological excitations in the phase of the superconducting order parameter. We further find that the strong driving of the cuprates can lead to parametric instabilities. To investigate the effect strong driving can have on the dissipative dynamics, we derive a FloquetMarkov master equation. We find that strong driving is a measure for probing the bath spectral density of otherwise inaccessible condensed matter systems.

We study the dynamics of nonlinear photonic lattices driven by two-photon parametric processes. By means of matrix-product-state based calculations, we show that a quantum many-body state with long-range hidden order can be generated from the vacuum. This order resembles that characterizing the Haldane insulator. A possible explanation highlighting the role of the symmetry of the drive, and the effect of photon loss are discussed. An implementation based in superconducting circuits is proposed and analyzed, 10 pages, 5 figures

The past three decades have seen extraordinary progress in the manipulation of neutral atoms with laser light, to the point where it is now routine to trap and cool both individual atoms and entire atomic clouds to temperatures of only a few tens of nanoKelvin in a controlled and repeatable fashion. In this thesis we study several applications of Rydberg atoms – atoms with an electron in a highly excited state – within such ultracold atomic systems. Due to their highly-excited electron, Rydberg atoms have a number of exaggerated properties: in addition to being physically large, they have long radiative lifetimes, and interact strongly both with one another and with applied external fields. Rydberg atoms consequently find many interesting applications within ultracold atomic physics. We begin this thesis by analysing the way in which a rubidium atom prepared in an excited Rydberg state decays to the ground state. Using quantum defect theory to model the wavefunction of the excited electron, we compute branching ratios for the various decay channels that lead out of the Rydberg states of rubidium. By using these results to carry out detailed simulations of the radiative cascade process, we show that the dynamics of spontaneous emission from Rydberg states cannot be adequately described by a truncated atomic level structure. We then investigate the stability of ultra-large diatomic molecules formed by pairs of Rydberg atoms. Using quantum defect theory to model the electronic wavefunctions, we apply molecular integral techniques to calculate the equilibrium distance and binding energy of these molecular Rydberg states. Our results indicate that these Ryberg macro-dimers are predicted to show a potential minimum, with equilibrium distances of up to several hundred nanometres. In the second half of this thesis, we present a new method of symbolically evaluating functions of matrices. This method, which we term the method of path-sums, has applications to the simulation of strongly-correlated many-body Rydberg systems, and is based on the combination of a mapping between matrix multiplications and walks on weighted directed graphs with a universal result on the structure of such walks. After presenting and proving this universal graph theoretic result, we develop the path-sum approach to matrix functions. We discuss the application of path-sums to the simulation of strongly-correlated many-body quantum systems, and indicate future directions for the method.

This work is a theoretical study of fermionic models. We focus on problems where highly controllable quantum simulators of these models have an important role, and we utilise both the analogue and the digital paradigm of quantum simulation. In the part on analogue quantum simulation, we focus on the proposed 'spin-asymmetric' Josephson effect where Cooper-paired spins display frequency synchronized Josephson oscillations with spin-dependent amplitudes. We consider different scenarios where the phenomenon could manifest in ultracold atomic Fermi gases. We study a Fermi gas Josephson junction in the recently realized Josephson plasma oscillation regime with an additional spin-dependent potential and show that the asymmetry in the resulting spin-dependent plasma oscillation amplitudes is on the order of a couple of per cent. We also demonstrate numerically that spin-asymmetric Josephson-like currents occur in a one-dimensional spin-dependent optical superlattice, with amplitude asymmetries up to 39%. Finally, we show that at zero temperature the tunable critical current in ferromagnetic Josephson junctions can be explained by the spin-asymmetric Josephson effect. In the part where digital quantum simulation is used, we propose a hybrid quantum-classical approach to studying strongly correlated fermion models. In this approach, a digital quantum simulator works in conjunction with a classical feedback loop to solve the infinite-dimensional Hubbard model directly in the thermodynamic limit. The scheme implements the well-established dynamical mean-field theory (DMFT) method, such that the digital quantum simulator solves the classically hard DMFT impurity problem and self-consistency is taken care of in a classical computer. We first present a few-qubit proof-of-principle setup for equilibrium systems that implements the simplified 'two-site' DMFT. This few-qubit setup is used for a qualitative description of the Mott transition in the half-filled infinite-dimensional Hubbard model. We then describe a scalable setup for simulating non-equilibrium many-body quantum dynamics by proposing the implementation of the non-equilibrium extension of DMFT with the hybrid device.