Quantum walks: a comprehensive review

Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists, mathematicians and engineers. In this paper we review theoretical advances on the foundations of both discrete- and continuous-time quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discrete-time quantum walks. Furthermore, we have reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete- time quantum walks.Comment: Paper accepted for publication in Quantum Information Processing Journa

Simulation of open quantum dynamics and investigation of quantum correlations in finite systems

This thesis reports a series of theoretical studies regarding the dynamics of fewbody controllable quantum systems. Generally speaking, the main focus is on the behavior of correlations in open quantum systems and how these could be used both for applications to quantum technologies and investigations of more fundamental phenomena. The general physical setting for most of the results presented is trappedion systems. These have been proven to be an almost prefect practical platform for realizing a quantum computer. Furthermore, thanks to their exceptional degree of controllability, trapped ions have been lately employed to also simulate basic physics, ranging from condensed-matter to high-energy physics. Although the ndings in this manuscript are theoretical, real experimental parameters have been taken into account in order to provide a more realistic modeling. To this aim, a mixed of analytical and numerical methods have been extensively utilized. Concluding, we do believe that the theory developed in this thesis could be experimentally tested to give a more insightful view on open quantum system dynamics, both from a foundational and applicative point of view

Quantum phase transitions in transverse field spin models: from statistical physics to quantum information

We review quantum phase transitions of spin systems in transverse magnetic fields taking the examples of the spin-1/2 Ising and XY models in a transverse field. Beginning with an overview of quantum phase transitions, we introduce a number of model Hamiltonians. We provide exact solutions in one spatial dimension connecting them to conformal field theoretical studies. We also discuss Kitaev models and some other exactly solvable spin systems. Studies of quantum phase transitions in the presence of quenched randomness and with frustrating interactions are presented in detail. We discuss novel phenomena like Griffiths-McCoy singularities. We then turn to more recent topics like information theoretic measures of the quantum phase transitions in these models such as concurrence, entanglement entropy, quantum discord and quantum fidelity. We then focus on non-equilibrium dynamics of a variety of transverse field systems across quantum critical points and lines. After mentioning rapid quenching studies, we dwell on slow dynamics and discuss the Kibble-Zurek scaling for the defect density following a quench across critical points and its modifications for quenching across critical lines, gapless regions and multicritical points. Topics like the role of different quenching schemes, local quenching, quenching of models with random interactions and quenching of a spin chain coupled to a heat bath are touched upon. The connection between non-equilibrium dynamics and quantum information theoretic measures is presented at some length. We indicate the connection between Kibble-Zurek scaling and adiabatic evolution of a state as well as the application of adiabatic dynamics as a tool of a quantum optimization technique known as quantum annealing. The final section is dedicated to a detailed discussion on recent experimental studies of transverse Ising-like systems.Comment: 106 pages, 38 figures; an expanded version has been published as a book (330 pages, 72 figures, 874 references) as A. Dutta, G. Aeppli, B. K. Chakrabarti, U. Divakaran, T. F. Rosenbaum and D. Sen, Quantum Phase Transitions in Transverse Field Spin Models: From Statistical Physics to Quantum Information (Cambridge University Press, Cambridge, 2015

Collective quantum effects in field theory and gravity

Collective quantum effects have traditionally not received much attention in high energy physics. Recently, however, a model for black hole physics was put forward, in which black holes are described as Bose-condensates of gravitons close to a critical point. In a different line of research, estimates of high-energy collisions in the electro-weak theory have hinted that scattering processes with multiple Higgs or vector Bosons in the final state might be in reach for future particle colliders. In both scenarios, collective quantum effects may be crucial for understanding the physics. In the first part of this thesis, we address the black hole condensate picture of Dvali and Gomez. We study a Bosonic many-body system (attractive Lieb-Liniger) which exhibits a quantum phase transition and was proposed as a model for the graviton condensate. We demonstrate that, even for macroscopic particle number, quantum effects are prominent at the critical point. This becomes especially clear in the entanglement of different momentum modes and in the quantum discord between two successive density measurements. We point out that the leading contribution to these phenomena arises from long-wavelength modes and is therefore insensitive to ultra-violet physics. For black holes in the graviton condensate picture, these findings imply a breakdown of the semiclassical description and may be the key to resolving the long-standing information problem. We then turn our attention to the question of information processing in black holes. Inspired by the properties of three-dimensional attractive Bose condensates, we propose a concrete mechanism for fast scrambling in graviton-condensate black holes. To bolster our claims, we perform simulations of the Lieb-Liniger model in an appropriate regime that reveal entanglement-generation in logarithmic time. We also point out that the idea of instability and possibly chaos as the origin of fast quantum breaking and scrambling may also be relevant for other models of black holes. In the second part of this thesis, we use techniques of integrability (Bethe ansatz) to address the phase transition of the attractive Lieb-Liniger model analytically. We derive the continuum limit of the Bethe equations and solve it for the ground state at arbitrary coupling. We establish an exact equivalence between the Bethe-ansatz description in the large-particle-number limit and the large-N saddle point of Euclidean two dimensional U(N) Yang-Mills theory quantized on a sphere. The transition between the homogeneous and solitonic phases of the Lieb-Liniger model is thus dual to the Douglas-Kazakov confinement-deconfinement transition. In the last part, we consider scattering amplitudes involving many particles. In a simple integral-model, we study in detail the breakdown of perturbation theory and emphasize that the pure tree-level approximation fails earlier, parametrically. We then demonstrate, in three different (integral and quantum mechanical) model systems, that the physical high multiplicity amplitudes can be predicted on the basis of leading-order information from (non-perturbative) saddle points. In the non-Borel summable cases, one non-perturbative saddle contribution alone dominates the amplitudes. We highlight that high-multiplicity amplitudes may thus be a fruitful application for the methods of resurgence theory.Kollektive Quanteneffekte haben traditionell keine große Aufmerksamkeit in der Hochenergiephysik erfahren. Vor kurzem ist jedoch ein Modell für die Physik schwarzer Löcher vorgeschlagen worden, in dem diese als Bose-Kondensate von Gravitonen nahe an einem kritischen Punkt beschrieben werden. In einer anderen Forschungsrichtung haben Abschätzungen der Hochenergiekollisionen in der elektroschwachen Theorie Hinweise darauf geliefert, dass Streuprozesse mit mehreren Higgs- oder Vektorbosonen im Endzustand in Reichweite künftiger Teilchenbeschleuniger sein könnten. In beiden Fällen dürften kollektive Quanteneffekte zentral für das Verständnis der Physik sein. Im ersten Teil dieser Arbeit behandeln wir das Gravitonkondensat-Bild für schwarze Löcher von Dvali und Gomez. Wir untersuchen ein Bosonisches Vielteilchensystem (attraktives Lieb-Liniger), das einen Quantenphasenübergang zeigt und als Modell für Gravitonkondensate vorgeschlagen worden ist. Wir zeigen, dass - selbst für makroskopische Teilchenzahlen - Quanteneffekte am kritischen Punkt wichtig sind. Das wird an der Verschränkung unterschiedlicher Impulsmoden und dem Quantenmissklang zwischen zwei aufeinanderfolgenden Dichtemessungen besonders klar. Wir heben hervor, dass der führende Beitrag zu diesen Phänomenen aus langwelligen Moden hervorgeht und daher von der ultravioletten Physik unabhängig ist. Diese Ergebnisse implizieren für schwarze Löcher im Gravitonkondensat-Bild, dass die semiklassische Beschreibung zusammenbricht, und sie könnten der Schlüssel dazu sein, das lange bestehende Informationsproblem zu lösen. Dann wenden wir uns der Frage der Informationsverarbeitung in schwarzen Löchern zu. Inspiriert von den Eigenschaften dreidimensionaler attraktiver Bose-Kondensate schlagen wir einen konkreten Mechanismus für das schnelle Scrambling in Gravitonkondensat-schwarzen-Löchern vor. Um diese Behauptung zu stützen führen wir Simulationen am Lieb-Liniger Modell in einem geeigneten Regime durch, die Verschränkungs-Erzeugung in logarithmischer Zeit offenbaren. Wir weisen auch darauf hin, dass die Idee, Instabilität und gegebenenfalls Chaos als Ursache für schnelles Quantenbrechen und Scrambling zu betrachten, relevant für andere Modelle von schwarzen Löchern sein kann. Im zweiten Teil dieser Arbeit verwenden wir Integrabilitäts-Techniken (Bethe-Ansatz) um den Phasenübergang des attraktiven Lieb-Liniger Modells analytisch zu analysieren. Wir leiten den Kontinuumslimes der Bethe-Gleichungen her und lösen ihn für den Grundzustand bei beliebiger Kopplungsstärke. Wir stellen eine genaue Äquivalenz zwischen der Bethe-Ansatz Beschreibung im Vielteilchen-Limes und dem groß-N Sattelpunkt von Euklidischer zweidimensionaler U(N) Yang-Mills Theorie, auf der Sphäre quantisiert, her. Der Übergang zwischen der homogenen und solitonischen Phase des Lieb-Liniger Modells ist dadurch dual zum Douglas-Kazakov Übergang zwischen Confinement und Deconfinement. Im letzten Teil widmen wir uns Streuamplituden von vielen Teilchen. In einem einfachen Integralmodell untersuchen wir im Detail den Zusammenbruch der Störungstheorie und betonen, dass reine Baum-Näherungen parametrisch noch früher versagen. Wir demonstrieren dann, dass sich die Streuamplituden hoher Multiplizität, in drei verschiedenen (Integral- und quantenmechanischen) Modellsystemen, auf Basis der führenden Ordnung von (nicht-perturbativen) Sattelpunkten, vorhersagen lassen. In den nicht Borel-summierbaren Fällen dominiert allein der Beitrag eines nicht-perturbativen Sattelpunkts. Wir zeigen auf, dass die Amplituden hoher Multiplizität daher wohl eine lohnenswerte Anwendung für die Techniken der Resurgenztheorie sind.Deutsche Übersetzung des Titels: Kollektive Quanteneffekte in Feldtheorie und Gravitatio

Exploring the non-equilibrium dynamics of kinetically constrained spin systems: Rydberg quantum simulation and artificial dissipation

This thesis discusses the non-equilibrium dynamics of one-dimensional quantum many-body systems. In particular, we investigate two distinct situations in which interesting dynamical properties arise, i.e., when the quantum evolution is subject to kinetic constraints or competes with an artificial dissipation through stochastic resets. Both topics have attracted considerable interest in the last decade, as they offer a playground to theoretically investigate the long-standing question of how isolated quantum systems evolve under non-equilibrium conditions. From the experimental point of view, the recent technological progress in the control and manipulation of ultracold atomic gases has led to new breakthroughs in the domains of quantum simulation and quantum computation. Key for the latter applications is the utilization of atomic Rydberg states in which atoms, trapped in optical tweezers, interact via state-dependent electrostatic dipolar forces. These strong interactions make Rydberg systems ideal for the realization of kinetic constraints, which cause a restriction of the connectivity between many-body states in the Hilbert space. A prominent example of a kinetic constraint is the Rydberg blockade, in which an excited Rydberg atom prevents the surrounding atoms to be excited to the Rydberg state. This effect has been largely exploited to implement controlled gates and complex many-body dynamics. Much less explored is the opposite situation, called the facilitation (or anti-blockade) constraint, where the interactions shift the otherwise detuned laser in resonance. In this case only atoms at the correct distance to an already excited atom are resonantly driven by the laser, thereby creating an “avalanche” of excitations. The first part of the thesis is devoted to the study of the facilitation dynamics in Rydberg chains. The facilitation constraint favours the dynamical creation of contiguous Rydberg excitations. We find that the resulting Rydberg excitation “cluster” develops long-range interactions that cause the onset of Bloch oscillations, preventing the system from reaching an ergodic stationary state. Contrary to the blockade constraint, facilitation is more challenging to implement in current Rydberg quantum simulators. The reason for this difficulty is that facilitation is particularly affected by mechanical effects and position disorder. These two problems originate respectively from the mechanical forces that displace the atoms from their initial positions and the spreading of the atomic wave functions in the optical traps. The interplay between the electronic degrees of freedom and the vibrational ones leads to a coupling between the (internal) Rydberg dynamics and the (external) atomic motion. We find that such spin-phonon coupling inhibits the facilitation mechanism, suppressing the expansion of the excitation cluster. This vibronic interaction can be also exploited to explore molecular physics in Rydberg atom arrays. We show this by considering a system composed of three atoms trapped in optical tweezers that form an equilateral triangle. We find that the atomic vibrations in the traps break the electronic degeneracy and generate a structural Jahn-Teller distortion, paving the way towards the exploration of molecular physics at the exaggerated length scales typical of Rydberg systems. The second part of the thesis investigates the effects of stochastic resetting on the stationary properties of quantum many-body spin systems. Stochastic resetting is a process that interrupts the dynamics of a system at random times and resets it to a certain state. Then the dynamics restarts again. This process leads very generally to a non-equilibrium stationary state. When the choice of the reset state is determined by the outcome of a measurement taken immediately before resetting, we find that resetting induces an emergent non-Markovian open dynamics, described by a generalized Lindblad equation. We also show that stochastic resetting can generate quantum correlation and collective behaviour even in a non-interacting system, showing its potential for quantum sensing applications. The structure of the thesis is as follows. In the first chapter we introduce the topics covered in the thesis and provide useful references for the reader. In the second chapter we review the physics of Rydberg systems, including their single-body properties and their interactions. We also explain how Rydberg quantum simulators are used for the implementation of kinetic constraints. In the third chapter we review the physics of stochastic resetting and the main mathematical techniques used in the thesis. In the fourth chapter we summarize the original results contained in the thesis. The fifth chapter is dedicated to the conclusions and an outlook on possible future research directions

Astrophysics in 2006

The fastest pulsar and the slowest nova; the oldest galaxies and the youngest stars; the weirdest life forms and the commonest dwarfs; the highest energy particles and the lowest energy photons. These were some of the extremes of Astrophysics 2006. We attempt also to bring you updates on things of which there is currently only one (habitable planets, the Sun, and the universe) and others of which there are always many, like meteors and molecules, black holes and binaries.Comment: 244 pages, no figure

Quantum Entanglement and Networking with Spin-Optomechanics

Non-relativistic quantum mechanics have proven to be a significant framework to understand the non-classical behaviour of light and the microcosmos. Perhaps, one of the first technological revolutions within quantum theory came with the invention of the transistor, whereby a purely quantum mechanical description was required. Currently, another outstanding revolution is taking place in a crossroad where information science meets quantum mechanics (this being the quantum information field). Such an area of work contemplates both the fascinating theoretical aspect of quantum correlations, as well as implementations towards quantum tasks performed by a universal quantum computer; tasks that cannot be realised (or they are hard to implement) within the classical domain. This Thesis is devoted to study the dynamics of quantum entanglement in spin-optomechanics systems. In particular, we explore the quantum stabilization of quantum entanglement, a quantum concentration scheme in opto-mechanics and an interfacing of matter and light towards quantum networking applications. Additionally, we also investigate theoretical aspects of quantum correlations within thermal environments, as well as the topical area of quantum sudden transitions. In Chapter 1, we provide a brief summary of quantum information and of the quantum optics framework to cover elementary concepts and techniques used subsequently in this work. Subsequently, in Chapter 2 we present the stabilization of quantum entanglement in a non-linear qubit-oscillator system. The inclusion of a modest nonlinearity gives three results, i) the loss of periodicity of the system, ii) the occurrence of quadrature squeezing appearing for a short time, and iii) the quantum entanglement reaches higher values in contrast to the case without non-linearity. In Chapter 3, a technique to concentrate/distill a two-mode vacuum state in optomechanics via unsharp measurements is presented. Here, one of the optical modes is injected into a cavity at first, and thereafter, it is nonlinearly coupled to a mechanical oscillator. Afterwards, the position of the oscillator is measured using pulsed optomechanics and homodyne detection. The results show that this measurement can conditionally increase the initial entanglement. Next, in Chapter 4, stimulated by optomechanical transducers and quantum networking, a light-matter system is constructed where a qubit is coupled to a cavity mode mediated through a mechanical oscillator. The qubit-oscillator conditionally displaced Hamiltonian and the oscillator-cavity radiation-pressure interaction generate a maximal qubit-cavity entanglement. Additionally, we consider the case in which the cavity mode is coupled to a waveguide, numerical calculations show a promising qubit-fibre entanglement under a weak matter-light coupling. For the quantum network case, we coupled a generic qubit in the first node to a second qubit-cavity distant Jaynes-Cummings system coupled through an optical fibre, where qubit-qubit correlations can be achieved in the quantum open systems scenario. In Chapter 5, we study the evolution of an open quantum system within the Born-Markov microscopic master equation (MME). Essentially, two distant two-level atoms are trapped in fibre-coupled cavities. Under the approximation of one-excitation allowed in the atom-cavity-fibre basis, we can obtain quantum correlations induced by thermal fluctuations from the environments. Lastly, in Chapter 6, we bring together previously elements explored in this Thesis. The system is a hybrid atomic-mechanical system formed from two remote qubits interacting with individual harmonic oscillators. This system, as in Chapter 4, explores interesting applications in quantum networking schemes. The two qubits are initially prepared in a Bell-diagonal state, and consequently the two-qubit correlations exhibit few interesting effects such as freezing, sudden changes and revivals in the evolution of the quantum entropic discord. To conclude, I summarize my findings in Chapter 7

Quantum Algorithms for Matrix Problems and Machine Learning

This dissertation presents a study of quantum algorithms for problems that can be posed as matrix function tasks. In Chapter 1 we demonstrate a simple unifying framework for implementing of smooth functions of matrices on a quantum computer. This framework captures a variety of problems that can be solved by evaluating properties of some function of a matrix, and we identify speedups over classical algorithms for some problem classes. The analysis combines ideas from the classical theory of function approximation with the quantum algorithmic primitive of implementing linear combinations of unitary operators. In Chapter 2 we continue this study by looking at the role of sparsity of input matrices in constructing efficient quantum algorithms. We show that classically pre-processing an input matrix by spectral sparsification can be profitable for quantum Hamiltonian simulation algorithms, without compromising the simulation error or complexity. Such preprocessing incurs a one time cost linear in the size of the matrix, but can be exploited to exponentially speed up subsequent subroutines such as inversion. In Chapter 3, we give an application of this theory of matrix functions to the problem of estimating the Renyi entropy of an unknown quantum state. We combine matrix function techniques with mixed state quantum computation in the one-clean qubit model, and are able to bound of the expected runtime of our algorithm in terms of the unknown target quantity. In addition to the theme of analysing the complexity of our algorithms, we also identify instances that are of practical relevance, leading us to some problems of machine learning. In Chapter 4 we investigate kernel based learning methods using random features. We work with the QRAM input model suitable for big data, and show how matrix functions and the quantum Fourier transform can be used to devise a quantum algorithm for sampling random features that are optimised for given input data and choice of kernel. We obtain a potential exponential speedup over the best known classical algorithm even without explicit assumptions of sparsity or low rank. Finally in Chapter 5 we consider the technique of beamsearch decoding used in natural language processing. We work in the query model, and show how quantum search with advice can be used to construct a quantum search decoder that can find the optimal parse (which may for instance be a best translation, or text-to-speech transcript) at least quadratically faster than the best known classical algorithms, and obtain super-quadratic speedups in the expected runtime.Science and Engineering Research Board (Department of Science and Technology), Government of Indi