This project was motivated by the following question: what information do the properties of a random graph contain about the performance of a quantum search acting on it? To investigate this problem, we define a notion of search time to quantify the behavior of a quantum search, and find strong evidence of a relation between its distribution and the model of random graph on which the search was performed. Surprisingly, we also find strong evidence that the return time of a classical random walk initialized at the marked vertex is closely related to its search time, and that the distribution of degrees over the graph vertices may play a significant role in this relation. / Mathematics

Equilibration, thermalization and scrambling appear to be intimately related, however their exact relation is unknown. As presented in the introduction these fields are at their current stages still quite popular and new directions are appearing regularly. We first put the fields into focus in the introduction and then the following chapters present three manuscripts as my contributions to these fields during my Master of Science project. Further introductions and background are presented in the manuscripts and my contributions are summarized at the beginning of each chapter. / Thesis / Master of Science (MSc)

No / We consider two distant mesoscopic SQUID rings, approximated with two-level systems, interacting with two-mode microwaves. The Hamiltonian of the system is used to calculate its time evolution. The cases with microwaves which at t = 0 are in separable states (classically correlated) or entangled states (quantum mechanically correlated) are studied. It is shown that the Josephson currents in the two SQUID rings are also correlated.

Quantum information technologies offer significantly more computational power for certain tasks and secure communication lines compared to the available classical machines. In recent years there have been numerous proposals for the implementation of quantum computers in several different systems that each come with their own advantages and challenges. This dissertation primarily focuses on challenges, specifically interactions with the environment, and applications of two of such systems: Semiconductor quantum dots and topological insulators. The first part of the dissertation is devoted to the study of semiconductor quantum dots as candidates for quantum information storage and sources of single-photon emission. The spin of the electron trapped in a self-assembled quantum dot can be used as a quantum bit of information for quantum technology applications. This system possesses desirable photon emission properties, including efficiency and tunability, which make it one of the most advanced single-photon emitters. This interface is also actively explored for the generation of complex entangled photonic states with applications in quantum computing, networks, and sensing. First, an overview of the relevant developments in the field will be discussed and our recent contributions, including protocols for the control of the spin and a scheme for the generation of entangled photon states from coupled quantum dots, will be presented. We then look at the interaction between the electron and the surrounding nuclear spins and describe how its interplay with optical driving can lead to dynamic nuclear polarization. The second part of the dissertation follows a similar study in topological insulators: The role of time-reversal breaking magnetic impurities in topological materials and how spinful impurities enable backscattering mechanisms by lifting the topological protection of edge modes. I will present a model that allows for an analytical study of the effects of magnetic impurities within an experimental framework. It will be discussed how the same platform also enables a novel approach for applications of spintronics and quantum information, such as studying the entanglement entropy between the impurities and chiral modes of the system. / Doctor of Philosophy / Quantum information science has received special attention in recent years due to its promising advantages compared to classical machines. Building a functional quantum processor is an ongoing effort that has enjoyed enormous advancements over the past few years. Several different condensed matter platforms have been considered as potential candidates for this purpose. This dissertation addresses some of the major challenges in two of the candidate platforms: Quantum dots and topological insulators. We look at methods for achieving high-performance optical control of quantum dots. We further utilize quantum dots special ability to emit photons for specific quantum technology applications. We also address the nuclear spin problem in these systems which is the main source of destruction of quantum information and one of the main obstacles in building a quantum computer. This is followed by the study of a similar problem in topological insulators: Addressing the interaction with magnetic impurities of topological insulators. Included with each of these topics is a description of relevant experimental setups. As such, the studies presented in this dissertation pave the way for a better understanding of the two major obstacles of hyperfine interactions and the optical controllability of these platforms.

In this thesis we study the subset of quantum operations that can be implemented using only local quantum operations and classical communication (LOCC). This restricted paradigm serves as a tool to study not only quantum correlations and other nonlocal quantum effects, but also resource transformations such as channel capacities. The mathematical structure of LOCC is complex and difficult to characterize. In the first part of this thesis we provide a precise description of LOCC and related operational classes in terms of quantum instruments. Our formalism captures both finite round protocols as well as those that utilize an unbounded number of communication rounds. This perspective allows us to measure the distance between two LOCC instruments and hence discuss the closure of LOCC in a rigorous way. While the set of LOCC is not topologically closed, we show that the operations that can be implemented using some fixed number rounds of communication constitute a compact subset of all quantum operations. We also exhibit a subset of LOCC measurements that is closed. Additionally we establish the existence of an open ball around the completely depolarizing map consisting entirely of LOCC implementable maps. In the second part of this thesis we focus on the task of discriminating states from some known set S by LOCC. Building on the work in the paper "Quantum nonlocality without entanglement", we provide a framework for lower bounding the error probability of any LOCC protocol aiming at discriminating the states from S. We apply our framework to an orthonormal product basis known as the domino states. This gives an alternative and simplified bound quantifying how well these states can be discriminated using LOCC. We generalize this result for similar bases in larger dimensions, as well as the "rotated" domino states, resolving a long-standing open question. These results give new examples of quantitative gaps between the classes of separable and LOCC operations. In the last part of this thesis, we ask what differentiates separable from LOCC operations. Both of these classes play a key role in the study of entanglement. Separable operations are known to be strictly more powerful than LOCC ones, but no simple explanation of this phenomenon is known. We show that, in the case of bipartite von Neumann measurements, the ability to interpolate is an operational principle that separates LOCC and separable operations.

In this thesis we study the subset of quantum operations that can be implemented using only local quantum operations and classical communication (LOCC). This restricted paradigm serves as a tool to study not only quantum correlations and other nonlocal quantum effects, but also resource transformations such as channel capacities. The mathematical structure of LOCC is complex and difficult to characterize. In the first part of this thesis we provide a precise description of LOCC and related operational classes in terms of quantum instruments. Our formalism captures both finite round protocols as well as those that utilize an unbounded number of communication rounds. This perspective allows us to measure the distance between two LOCC instruments and hence discuss the closure of LOCC in a rigorous way. While the set of LOCC is not topologically closed, we show that the operations that can be implemented using some fixed number rounds of communication constitute a compact subset of all quantum operations. We also exhibit a subset of LOCC measurements that is closed. Additionally we establish the existence of an open ball around the completely depolarizing map consisting entirely of LOCC implementable maps. In the second part of this thesis we focus on the task of discriminating states from some known set S by LOCC. Building on the work in the paper "Quantum nonlocality without entanglement", we provide a framework for lower bounding the error probability of any LOCC protocol aiming at discriminating the states from S. We apply our framework to an orthonormal product basis known as the domino states. This gives an alternative and simplified bound quantifying how well these states can be discriminated using LOCC. We generalize this result for similar bases in larger dimensions, as well as the "rotated" domino states, resolving a long-standing open question. These results give new examples of quantitative gaps between the classes of separable and LOCC operations. In the last part of this thesis, we ask what differentiates separable from LOCC operations. Both of these classes play a key role in the study of entanglement. Separable operations are known to be strictly more powerful than LOCC ones, but no simple explanation of this phenomenon is known. We show that, in the case of bipartite von Neumann measurements, the ability to interpolate is an operational principle that separates LOCC and separable operations.

<p>Although trapped ion technology is well-suited for quantum information science, scalability of the system remains one of the main challenges. One of the challenges associated with scaling the ion trap quantum computer is the ability to individually manipulate the increasing number of qubits. Using micro-mirrors fabricated with micro-electromechanical systems (MEMS) technology, laser beams are focused on individual ions in a linear chain and steer the focal point in two dimensions. Multiple single qubit gates are demonstrated on trapped 171Yb+ qubits and the gate performance is characterized using quantum state tomography. The system features negligible crosstalk to neighboring ions (< 3e-4), and switching speeds comparable to typical single qubit gate times (< 2 us). In a separate experiment, photons scattered from the 171Yb+ ion are coupled into an optical fiber with 63% efficiency using a high numerical aperture lens (0.6 NA). The coupled photons are directed to superconducting nanowire single photon detectors (SNSPD), which provide a higher detector efficiency (69%) compared to traditional photomultiplier tubes (35%). The total system photon collection efficiency is increased from 2.2% to 3.4%, which allows for fast state detection of the qubit. For a detection beam intensity of 11 mW/cm2, the average detection time is 23.7 us with 99.885(7)% detection fidelity. The technologies demonstrated in this thesis can be integrated to form a single quantum register with all of the necessary resources to perform local gates as well as high fidelity readout and provide a photon link to other systems.</p> / Dissertation

Information processors that harness quantum mechanics may be able to outperform their classical counterparts at certain tasks. Quantum information processing (QIP) can utilize the quantum mechanical phenomenon of entanglement to implement quantum algorithms. Endohedral fullerenes, where atoms, ions or clusters are trapped in a carbon cage, are a class of nanomaterials that show great promise as the basis for a solid-state QIP architecture. Some endohedral fullerenes are spin–active, and offer the potential to encode information in their spin-states. This thesis addresses the challenges of how to engineer the components of a scalable QIP architecture based on endohedral fullerenes. It focuses on the synthesis and characterization of molecules which may, in the future, permit the demonstration of entanglement; the optical read-out of quantum states; and the creation of quasi-one-dimensional molecular arrays. Due to its long spin decoherence time, N@C₆₀ is the selected as the basic molecular unit for ‘coupled’ fullerene pairs, molecular systems for which it may be possible to demonstrate entanglement. To this end, isolated fullerene pairs, in the form of spin-bearing fullerene dimers, are created. This begins with the processing of N@C₆₀ at the macroscale and leads towards the synthesis of ¹⁵N@C₆₀-¹⁵N@C₆₀ dimers at the microscale. High throughput processing is introduced as the most efficient technique to obtain high purity N@C₆₀ on a reasonable timescale. A scheme to produce symmetric and asymmetric fullerene dimers is also demonstrated. EPR spectroscopy of the dimers in the solid-state confirms derivatization, whilst permitting the modelling of spin–spin interactions for 'coupled' fullerene pairs. This suggests that the optimum inter–spin separation for which to observe spin–spin coupling in powders is circa 3 nm. Motivated by the properties of the trivalent erbium ion for the optical detection of quantum states, optically–active erbium–doped fullerenes are also investigated. These erbium metallofullerenes are synthesized and isolated as individual isomers. They are characterized by low temperature photoluminescence spectroscopy, emitting in the infra- red at a wavelength of 1.5 &mu;m. The luminescence is markedly different where a C₂ cluster is trapped alongside the erbium ions in the fullerene cage. Er₂C₂@C₈₂ (isomer I) exhibits emission linewidths that are comparable to those observed for Er³⁺ in crystals. Finally, the discovery of a novel praseodymium-doped fullerene is reported. The balance of evidence favours the structure being assigned as Pr₂@C₇₂. This novel endohedral fullerene forms quasi-one-dimensional arrays in carbon nanotubes, which is a useful proof-of-principle of how a scaled fullerene-based architecture may be achieved.

Entanglement is one of the most technologically important quantum phenomena and its con-trolled creation brings us a step closer to the realisation of a quantum computer. Hybrid electron and nuclear spin systems which combine long nuclear decoherence times with the high polarisation and rapid processing times of electron spins are considered reliable candidates for the representation of the fundamental building block of a quantum computer, the qubit. In the literature electron spins quite often play the role of a mediator which can access, manipulate and couple states with long coherence times, beneficial for storing quantum information. Despite the fact that an electron spin can be a useful resource for nuclear spin systems, its permanent presence can be a source of decoherence. The use of transient photoexcited electron spins provide an additional advantage and once the operations which involve the electron spin are completed, the electron spin can decay and not interfere further with the evolution of the system. In this thesis we report magnetic resonance experiments and density functional theory calculations for the demonstration of nuclear - nuclear entanglement using photoexcited triplet states. We study homonuclear and heteronuclear fullerene derivatives and we identify in each case the relevant parameters that can lead to high fidelity entangling operations. The hyperfine interaction in a homonuclear system is the key parameter which determines the degree of entanglement between the nucelar spins according to a recent theoretical proposal. We measure and calculate the hyperfine interaction in homonuclear systems with ¹³C nuclear spins in order to prove the feasibility of this scheme. Further experiments on a fullerene system with two nuclear spins a ³¹P and a ¹H show that entangling operations of high fidelity which involve the demonstration of CNOT gates, are possible within the lifetime of the triplet state.

The ability to address and manipulate individual information carriers in a deterministic, coherent, and scalable manner is a central theme in quantum information processing. Neutral atoms trapped by laser light are amongst the most promising candidates for storing and processing information in a quantum computer or simulator, so a scalable and flexible scheme for their control and manipulation is paramount. This thesis demonstrates a fast and versatile method to address and dynamically control the position (the motional degrees of freedom) of neutral atoms trapped in optical tweezers. The tweezers are generated by using the direct image of a Spatial Light Modulator (SLM) which can control and shape a large number of optical dipole-force traps. Trapped atoms adapt to any change in the potential landscape, such that one can re-arrange and randomly access individual sites within atom-trap arrays. A diffraction limited imaging system is used to map the intensity distribution of the SLM onto a cloud of cold atoms captured and cooled using a Magneto Optical Surface Trap (MOST).