In this thesis we investigate geometric and topological structures in the context of entanglement and quantum computation. A parallel transport condition is introduced in the context of Franson interferometry based on the maximization of two-particle coincidence intensity. The dependence on correlations is investigated and it is found that the holonomy group is in general non-Abelian, but Abelian for uncorrelated systems. It is found that this framework contains a parallel transport condition developed by Levay in the case of two-qubit systems undergoing local SU(2) evolutions. Global phase factors of topological origin, resulting from cyclic local SU(2) evolution, called topological phases, are investigated in the context of multi-qubit systems. These phases originate from the topological structure of the local SU(2)-orbits and are an attribute of most entangled multi-qubit systems. The relation between topological phases and SLOCC-invariant polynomials is discussed. A general method to find the values of the topological phases in an n-qubit system is described. A non-adiabatic generalization of holonomic quantum computation is developed in which high-speed universal quantum gates can be realized by using non-Abelian geometric phases. It is shown how a set of non-adiabatic holonomic one- and two-qubit gates can be implemented by utilizing transitions in a generic three-level Λ configuration. The robustness of the proposed scheme to different sources of error is investigated through numerical simulation. It is found that the gates can be made robust to a variety of errors if the operation time of the gate can be made sufficiently short. This scheme opens up for universal holonomic quantum computation on qubits characterized by short coherence times.

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The topological aspects of superconductivity on doped Weyl semimetals are investigated. / Topological phases of matter have sparked significant experimental and theoretical interest due to the topologically robust edge modes they host, as well as their classification through rich mathematics. An interesting example of a topological phase in three dimensions, the Weyl semimetal, can exhibit topological ordering through the existence of Fermi arcs on the surfaces of the material. For the doped Weyl semimetal, we investigate possible resulting Weyl superconducting states --- both the inter-Fermi surface pairing state following Li and Haldane, and the intra-Fermi surface pairing state following Burkov --- in this thesis, and study their topological properties by computing the gapless Weyl-Majorana edge modes they host. The results obtained in Ref. \cite{LH} for the inter-Fermi surface superconducting state are reproduced, and the bulk and edge properties of the intra-Fermi surface pairing superconducting state are studied in detail. / Thesis / Master of Science (MSc) / In this thesis, we study an interesting class of topological materials called the Weyl semimetal as well as its associated superconducting phases. A description of the Fermi arcs on Weyl semimetals are given, and the topological properties of the inter-Fermi surface and intra-Fermi surface pairing states are studied in detail.

The discovery of the fractional quantum Hall effect for two-dimensional electron gases immersed in a strong orthogonal magnetic field represents a cornerstone of modern physics. The states responsible for the appearance of the fractional quantum Hall effect have been found to be part of a whole new class of phases of matter, characterized by an internal order with unprecedented properties and known as topological order. This fact opened up a completely new territory for physical studies, paving the way towards many of the current hot topics in physics, such as topological phases of matter, topological order and topological quantum computing. As it happens for most topologically-ordered phases, fractional quantum Hall states are breeding ground for the observation of many exotic physical phenomena. Important examples include the appearance of degenerate ground states when the system in placed on a space with non-trivial topology, the existence of chiral gapless edge excitations which unidirectionally propagate without suffering of back-scattering processes, and the possibility of hosting elementary excitations, known as quasiparticles and quasiholes, carrying fractional charge and anyonic statistics. Even though for years since their discovery fractional quantum Hall states have been studied only in electronic systems, the recent advances made in the domains of quantum simulators and artificial gauge fields opened the possibility to realize bosonic analogs of these states in platforms based on ultracold atoms and photons. Reaching the appropriate conditions for the simulation of the fractional quantum Hall effect with neutral particles (such as atoms and photons) has required decades of both theoretical and experimental efforts and passed through the implementation of many topological models at the single-particle level. However, we strongly believe that the stage is set finally and that bosonic fractional quantum Hall states will be realized soon in different set-ups. Motivated by this fact, we dedicate this Thesis to the study of the edge and quasihole excitations of bosonic fractional quantum Hall states with the goal of guiding near future experiments towards exciting discoveries such as the observation of anyons. In the first part of the Thesis we focus our attention on the behavior of the edge excitations of the bosonic $ u=1/2$ Laughlin state (a paradigmatic wave function for the fractional quantum Hall effect) in the presence of cylindrically symmetric hard-wall confining potentials. With respect to electronic devices, atomic and photonic platforms offers indeed a more precise control on the external potential confining the systems, as confirmed by the recent realization of flat-bottomed traps for ultracold atoms and by the flexibility in designing optical cavities. At the same time, most of the theoretical works in this direction have considered harmonic confinements, for which the edge states have been found to display the standard chiral Luttinger liquid behavior, leaving the field open for our analysis of new physics beyond the Luttinger paradigm. In the second part we propose a novel method to probe the statistical properties of the quasihole excitations on top of a fractional quantum Hall state. As compared to the previous proposals, it does not rely on any form of interference and it has the undeniable advantage of requiring only the measurements of density-related observables. As we have already mentioned, although the existence of anyons have been theoretically predicted long time ago, it still lacks a clear-cut experimental evidence and this motivated people working with ultracold atoms and photons to push their systems into the fractional quantum Hall regime. However, while there exist plenty of proposals for the detection of anyons in solid-state systems (mostly based on interferometric schemes in which currents are injected into the system and anyons travel along its edges), in the present literature the number of detection schemes applicable in ultracold atomic and/or photonic set-ups is much smaller and they are typically as demanding as those proposed in the electronic context. Finally, in the last part of the Thesis we move to the lattice counterparts of the fractional quantum Hall states, the so-called fractional Chern insulators. Still with the purpose of paving the way for future experimental studies with quantum simulators, we focus our attention of the simplest bosonic version of these states and, in particular, on the properties of its quasihole excitations. Although this topic has already been the subject of intense studies, most of the previous works were limited either to system sizes which are too small to host anyonic excitations, or to unphysical conditions, such as periodic geometries and non-local Hamiltonians. Our study investigates for the first time the properties of genuine quasihole excitations in experimentally relevant situations.

<p>Quantum phenomena related to geometric and topological phases are investigated. The first results presented are theoretical extensions of these phases and related effects. Also experimental proposals to measure some of the described effects are outlined. Thereafter, applications of geometric and topological phases in quantum computation are discussed.</p><p>The notion of geometric phases is extended to cover mixed states undergoing unitary evolutions in interferometry. A comparison with a previously proposed definition of a mixed state geometric phase is made. In addition, an experimental test distinguishing these two phase concepts is proposed. Furthermore, an interferometry based geometric phase is introduced for systems undergoing evolutions described by completely positive maps.</p><p>The dynamics of an Aharonov-Bohm system is investigated within the adiabatic approximation. It is shown that the time-reversal symmetry for a semi-fluxon, a particle with an associated magnetic flux which carries half a flux unit, is unexpectedly broken due to the Aharonov-Casher modification in the adiabatic approximation.</p><p>The Aharonov-Casher Hamiltonian is used to determine the energy quantisation of neutral magnetic dipoles in electric fields. It is shown that for specific electric field configurations, one may acquire energy quantisation similar to the Landau effect for a charged particle in a homogeneous magnetic field.</p><p>We furthermore show how the geometric phase can be used to implement fault tolerant quantum computations. Such computations are robust to area preserving perturbations from the environment. Topological fault-tolerant quantum computations based on the Aharonov-Casher set up are also investigated.</p>

Quantum phenomena related to geometric and topological phases are investigated. The first results presented are theoretical extensions of these phases and related effects. Also experimental proposals to measure some of the described effects are outlined. Thereafter, applications of geometric and topological phases in quantum computation are discussed. The notion of geometric phases is extended to cover mixed states undergoing unitary evolutions in interferometry. A comparison with a previously proposed definition of a mixed state geometric phase is made. In addition, an experimental test distinguishing these two phase concepts is proposed. Furthermore, an interferometry based geometric phase is introduced for systems undergoing evolutions described by completely positive maps. The dynamics of an Aharonov-Bohm system is investigated within the adiabatic approximation. It is shown that the time-reversal symmetry for a semi-fluxon, a particle with an associated magnetic flux which carries half a flux unit, is unexpectedly broken due to the Aharonov-Casher modification in the adiabatic approximation. The Aharonov-Casher Hamiltonian is used to determine the energy quantisation of neutral magnetic dipoles in electric fields. It is shown that for specific electric field configurations, one may acquire energy quantisation similar to the Landau effect for a charged particle in a homogeneous magnetic field. We furthermore show how the geometric phase can be used to implement fault tolerant quantum computations. Such computations are robust to area preserving perturbations from the environment. Topological fault-tolerant quantum computations based on the Aharonov-Casher set up are also investigated.

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The investigation into topological phases on non-periodic lattices has recently gained wide interest because of the discovery of never-before seen phenomena lacking a counterpart in periodic lattices. In this thesis, I present the results of my work on the lattice Laughlin state on fractal lattices and that of the BHZ model on quasicrystals. I show that the entanglement spectrum has the same topological fingerprint as in periodic lattices, and thus can be used as a probe of topological order in these new environments, where such probes are severely lacking, especially for interacting topological phases. I also show how the entanglement entropy displays precise oscillations as a function of lattice filling in fractal lattices, and is smooth for periodic lattices. I study the on-site particle densities, and anyonic excitations on different kinds of fractal lattices and show how radically different they are from the 2D case. Finally, I study the BHZ model on the Amman-Beenker tiling and show the different kinds of Bulk Localized Transport(BLT) states, the edge states, and how the latter can be used to pump charge between different kind of BLT states. I couple two layers of the half-BHZ, which are time-reversed partners of each other, with a simple time-reversal symmetric hopping, and show that the BLT and edge states still survive.