Prof Dieter Jaksch

Abstract:

Contextuality is a nonclassical feature of quantum systems---exhibited by data that is produced empirically or theoretically---which in the realm of sheaf theory is characterised by local consistency but global inconsistency. A large part of this thesis is concerned with studying how this signature of nonclassicality is apparent also when measurements are embedded in some causal structure, and so motivating the study of causal contextuality. We begin with temporal correlations, which occur when measurements are performed sequentially on the system, and in which the definition of nonclassicality becomes sensitive to memory resources with which the classical system is equipped. For certain types of such memory, we show that there exists a map from the temporal setup to a (appropriately defined) contextuality setup, such that every nonclassical temporal empirical model satisfying no-signalling constraints consistent with the memory function corresponds to a contextual empirical model on this constructed scenario---one can view this also as a simulation of a subset of the temporal correlations by the contextuality setup. The existence of such a map allows us to apply a result from Vorob'ev in order to say, for any temporal setup and choice memory function, whether nonclassical correlations can arise. We then study causal setups by employing the notion of strategy from game semantics. We in particular show how `playing off' Nature strategies, corresponding to adaptive hidden variables, against Experimenter strategies, which may also be adaptive, realises the classical correlations of certain causal setups from the literature. We show that adaptivity on the side of the Experimenter, by reducing the sets of measurements empirical data is obtained over, can remove the inconsistencies that are imperative for the observation of contextuality. In the second part of the thesis, we study spin Hamiltonians on random graphs, focusing on exact descriptions in the thermodynamic limit. By utilising the graphon, which is the limit object of a dense random graphs sequence, we are able to derive analytical results for certain graphons and certain choice of Hamiltonian. Our overarching result is that the equilibrium physics in the thermodynamic limit is described by a set of coupled equations containing the graphon, and which describes product, \ie unentangled, states

Abstract:

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

Abstract:

Many complex systems can spontaneously oscillate under nonperiodic forcing. Such self-oscillators are commonplace in biological and technological assemblies where temporal periodicity is needed, such as the beating of a human heart or the vibration of a cello string. While self-oscillation is well understood in classical nonlinear systems and their quantized counterparts, the spontaneous emergence of periodicity in quantum systems is more elusive. Here, we show that this behavior can emerge within the repeated-interaction description of open quantum systems. Specifically, we consider a many-body quantum system that undergoes dissipation due to sequential coupling with auxiliary systems at random times. We develop dynamical symmetry conditions that guarantee an oscillatory long-time state in this setting. Our rigorous results are illustrated with specific spin models, which could be implemented in trapped-ion quantum simulators.