<p dir="ltr">This thesis advances the methodologies of quantum state tomography (QST) to validate and optimize quantum processing on Noisy Intermediate-Scale Quantum (NISQ) devices, crucial for the transition to practical quantum systems. Inspired by recent advancements in the field, we propose a novel QST method based on the maximal entropy formalism, specifically addressing scenarios with incomplete measurement sets to provide a robust framework for state reconstruction. We extend this formalism to an informationally complete (IC) set of observables and introduce a variational approach for quantum state preparation, easily implementable on near-term quantum devices. Our developed maximal entropy-based QST protocol is applied to ultrafast molecular dynamics specifically for studying photoexcited ammonia molecule, enabling direct measurement and manipulation of electronic quantum coherences and exploring entanglement effects in molecular systems. Through this approach, we achieve a groundbreaking milestone by, for the first time, constructing the entanglement entropy of the electronic subsystem - an otherwise inaccessible metric. In doing so it also provides the first physical interpretation of the maximal entropy parameters in an experimental setting and highlights the potential for feedback between time-resolved quantum dynamics and quantum information science. Furthermore, building upon our advancements in state tomography, we propose a variational quantum algorithm for Hamiltonian learning that leverages the time dynamics of observables. Additionally, we reverse engineer the maximal entropy approach and demonstrate the use of entropy to refine the traditional geometric Brownian motion (GBM) method for better capturing real system complexities by addressing its log-normality restrictions, which opens new avenues for quantum sampling techniques. Through these contributions, this thesis showcases the Maximal Entropy formalism’s efficacy in QST and set the stage for future innovations and applications in cutting-edge quantum research.</p>

We consider the purity-based entanglement detection scheme introduced in [C. Moura Alves and D. Jaksch, Phys. Rev. Lett. 93, 110501 (2004)]. We describe how it could be implemented in an optical lattice using two-atom loss, and prove that in this form it detects all pure entangled states even without any spatial resolution. We then prove that correcting for certain reasonable types of experimental error is possible, and practical for error rates up to the order of one over the number of lattice sites considered. Limited spatial resolution similarly becomes a significant improvement over no spatial resolution only at nearly single site level. We also show how to use this process for state parameter estimation and collapse-revival evidence of entanglement, for which it remains useful even when the error rate is too high to permit unambiguous entanglement detection. We also consider an optical lattice bosonic analogue of the fractional quantum Hall (FQH) effect. This system can reach high “magnetic fields” very difficult to attain in the solid state FQH system, where the discrete nature of the lattice becomes important. Near simple rational numbers l/n of flux quanta per lattice cell, we find that the single particle states become nearly periodic with period n lattice sites, and have an n fold degeneracy which leads to FQH states resembling those of n-internal-state particles. Standard time of flight expansion would reveal this periodicity and be able to distinguish FQH states from vortex lattice or Mott insulator states. Shot noise correlation would provide further information on the nature of the FQH states.

Nesta tese de doutoramento nós tivemos como principal objetivo desenvolver novos métodos para proteção da informação e computação quântica. Começamos, de forma introdutória, ilustrando os conceitos básicos e fundamentais da teoria da informação e computação quântica, como os bits quânticos (qubits), o operador densidade, o emaranhamento e as operações lógicas quânticas. Na seqüência, apresentamos os formalismos utilizados para tratar sistemas abertos, ou seja, sujeitos a erros, além das principais técnicas existentes a fim de proteger a informação quântica, como os códigos de correção de erros, os subespaços livres de erros e o desacoplamento dinâmico. Finalmente, baseando-nos na técnica de desacoplamento dinâmico, introduzimos um esquema de proteção para operações lógicas quânticas e o emaranhamentos entre qubits utilizando campos de alta freqüência. Ilustramos em detalhes a proteção da operação lógica quântica de Hadamard e do emaranhamento entre dois qubits, além de apresentarmos as principais diferenças e vantagens de nosso método quando comparado às técnicas tradicionais de desacoplamento dinâmico. / The main objective of this thesis is the development of a new procedure for quantum information and computation protection. We begin by briefly illustrating the basic concepts of quantum information and computation theory, such as quantum bits (qubits), density matrix operator, entanglement, and quantum logical operations. Subsequently, we present the formalism utilized to treat quantum open systems, i.e., systems subjected to errors, and the main strategies to protect quantum information, such as quantum error correction codes, decoherence-free subspaces, and dynamical decoupling. Finally, based on the dynamical decoupling strategies, we introduce a procedure to protect quantum logical operations and entanglement utilizing high-frequency continuous fields. We illustrate, in details, the protection of a Hadamard quantum gate and of entanglement between two qubits, and present the differences and advantages of our procedure when compared with traditional techniques of dynamical decoupling.

Any computation is facilitated by some physical process, and the observable quantities of any physical process can be viewed as a computation. These close ties suggest that the study of what universal computers are capable of may lead to additional insight about the physical universe, and vice versa. In his thesis, we explore three lines of research that are linked to this central theme. The first partition shows how notions of non-computability and undecidability eventually led to evidence of emergence, the concept that even if a ‘theory of everything’ governing all microscopic interactions were discovered, the understanding of macroscopic order is likely to require additional insights. The second partition proposes a physically motivated model of computation that relates quantum complexity, quantum optimal control, and Riemannian geometry. Thus insights in any one of these disciplines could also lead to insights in the others. The remainder of this partition explores a simple application of these relations. The final partition proposes a model of quantum computation that generalizes measurement based computation to continuous variables. We outline its optical implementation, whereby any computation can be performed by single mode measurements on a resource state that can be prepared by passing a collection of squeezed states through a beamsplitter network.

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.

This dissertation focuses on two connected areas: quantum computation and quantum control. Two proposals to construct a quantum computer, using nuclear magnetic resonance (NMR) and superconductivity, are introduced. We give details about the modeling, qubit realization, one and two qubit gates and measurement in the language that mathematicians can understand and fill gaps in the original literatures. Two experimental examples using liquid NMR are also presented. Then we proceed to investigate an example of quantum control, that of a magnetometer using quantum feedback. Previous research has shown that feedback makes the measurement robust to an unknown parameter, the number of atoms involved, with the assumption that the feedback is noise free. To evaluate the effect of the feedback noise, we extend the original model by an input noise term. We then compute the steady state performance of the Kalman filter for both the closed-loop and open-loop cases and retrieve the estimation error variances. The results are compared and criteria for evaluating the effects of input noise are obtained. Computations and simulations show that the level of input noise affects the measurement by changing the region where closed loop feedback is beneficial.

This thesis is a contribution to the debate on the implications of quantum information theory for the foundational problems of quantum mechanics. In Part I an attempt is made to shed some light on the nature of information and quantum information theory. It is emphasized that the everyday notion of information is to be firmly distinguished from the technical notions arising in information theory; however it is maintained that in both settings ‘information’ functions as an abstract noun, hence does not refer to a particular or substance. The popular claim ‘Information is Physical’ is assessed and it is argued that this proposition faces a destructive dilemma. Accordingly, the slogan may not be understood as an ontological claim, but at best, as a methodological one. A novel argument is provided against Dretske’s (1981) attempt to base a semantic notion of information on ideas from information theory. The function of various measures of information content for quantum systems is explored and the applicability of the Shannon information in the quantum context maintained against the challenge of Brukner and Zeilinger (2001). The phenomenon of quantum teleportation is then explored as a case study serving to emphasize the value of recognising the logical status of ‘information’ as an abstract noun: it is argued that the conceptual puzzles often associated with this phenomenon result from the familiar error of hypostatizing an abstract noun. The approach of Deutsch and Hayden (2000) to the questions of locality and information flow in entangled quantum systems is assessed. It is suggested that the approach suffers from an equivocation between a conservative and an ontological reading; and the differing implications of each is examined. Some results are presented on the characterization of entanglement in the Deutsch-Hayden formalism. Part I closes with a discussion of some philosophical aspects of quantum computation. In particular, it is argued against Deutsch that the Church-Turing hypothesis is not underwritten by a physical principle, the Turing Principle. Some general morals are drawn concerning the nature of quantum information theory. In Part II, attention turns to the question of the implications of quantum information theory for our understanding of the meaning of the quantum formalism. Following some preliminary remarks, two particular information-theoretic approaches to the foundations of quantum mechanics are assessed in detail. It is argued that Zeilinger’s (1999) Foundational Principle is unsuccessful as a foundational principle for quantum mechanics. The information-theoretic characterization theorem of Clifton, Bub and Halvorson (2003) is assessed more favourably, but the generality of the approach is questioned and it is argued that the implications of the theorem for the traditional foundational problems in quantum mechanics remains obscure.

Nesta tese de doutoramento nós tivemos como principal objetivo desenvolver novos métodos para proteção da informação e computação quântica. Começamos, de forma introdutória, ilustrando os conceitos básicos e fundamentais da teoria da informação e computação quântica, como os bits quânticos (qubits), o operador densidade, o emaranhamento e as operações lógicas quânticas. Na seqüência, apresentamos os formalismos utilizados para tratar sistemas abertos, ou seja, sujeitos a erros, além das principais técnicas existentes a fim de proteger a informação quântica, como os códigos de correção de erros, os subespaços livres de erros e o desacoplamento dinâmico. Finalmente, baseando-nos na técnica de desacoplamento dinâmico, introduzimos um esquema de proteção para operações lógicas quânticas e o emaranhamentos entre qubits utilizando campos de alta freqüência. Ilustramos em detalhes a proteção da operação lógica quântica de Hadamard e do emaranhamento entre dois qubits, além de apresentarmos as principais diferenças e vantagens de nosso método quando comparado às técnicas tradicionais de desacoplamento dinâmico. / The main objective of this thesis is the development of a new procedure for quantum information and computation protection. We begin by briefly illustrating the basic concepts of quantum information and computation theory, such as quantum bits (qubits), density matrix operator, entanglement, and quantum logical operations. Subsequently, we present the formalism utilized to treat quantum open systems, i.e., systems subjected to errors, and the main strategies to protect quantum information, such as quantum error correction codes, decoherence-free subspaces, and dynamical decoupling. Finally, based on the dynamical decoupling strategies, we introduce a procedure to protect quantum logical operations and entanglement utilizing high-frequency continuous fields. We illustrate, in details, the protection of a Hadamard quantum gate and of entanglement between two qubits, and present the differences and advantages of our procedure when compared with traditional techniques of dynamical decoupling.

Orientadores: Reginaldo Palazzo Júnior, Clarice Dias de Albuquerque / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação / Made available in DSpace on 2018-08-20T02:14:58Z (GMT). No. of bitstreams: 1 Cunha_MaiconHenrique_M.pdf: 692136 bytes, checksum: 3fe313a507d63bb6531e79d113a8cf55 (MD5) Previous issue date: 2012 / Resumo: Neste trabalho, apresentamos um estudo sobre Computação Quântica Topológica, uma área de pesquisa inserida na computação quântica que busca resolver o problema da decoerência na construção do computador quântico de uma maneira inovadora. Essa computação envolve aspectos de áreas distintas relacionadas a mecânica quântica: teoria de grupos, representação de grupo, anyons e outras. Por isso, uma fundamentação teórica básica nesses tópicos é necessária e será apresentada para embasar o modelo geral de Computação Quântica Topológica. O modelo de Fibonacci é um caso específico que será tratado com ênfase por ser o mais difundido e o único universal conhecido até o momento. Com o modelo de Fibonacci, construímos novas portas quânticas, cuja análise possibilitou conclusões e um refinamento no algoritmo existente para encontrar tais portas / Abstract: In this work, we present a study about Topological Quantum Computation, a research area included in quantum computation that seeks to solve the problem of decoherence in building a quantum computer according to an innovative way. This involves computing aspects of different areas related to quantum mechanics: group theory, group representation, anyons and others. Thus a basic theoretical foundation in these topics is necessary and will be presented to support the general model of Topological Quantum Computation. The Fibonacci model is a particular case, which will be discussed with emphasis, being the most widespread and the only universally known until this moment. With the Fibonacci model, we construct new quantum gates, whose analysis allowed a number of conclusions to be draw, as well as a refinement of the existing algorithm to find such ports / Mestrado / Telecomunicações e Telemática / Mestre em Engenharia Elétrica

The advent of quantum computing has challenged classical conceptions of which problems are efficiently solvable in our physical world. This raises the general question of what broad relationships exist between physical principles and computation. The current thesis explores this question within the operationally-defined framework of generalised probabilistic theories. In particular, we investigate the limits on computational power imposed by simple physical principles. At present, the best known upper bound on the power of quantum computers is that <b>BQP</b> is contained in <b>AWPP</b>, where <b>AWPP</b> is a classical complexity class contained in PP. We define a circuit-based model of computation in the above mentioned operational framework and show that in theories where local measurements suffice for tomography, efficient computations are also contained in <b>AWPP</b>. Moreover, we explicitly construct a theory in which the class of efficiently solvable problems exactly equals <b>AWPP</b>, showing this containment to be tight. We also investigate how simple physical principles bound the power of computational paradigms which combine computation and communication in a non-trivial fashion, such as interactive proof systems. Additionally, we show how some of the essential components of computational algorithms arise from certain natural physical principles. We use these results to investigate the relationship between interference behaviour and computational power, demonstrating that non-trivial interference behaviour is a general resource for post-classical computation. We then investigate whether post-quantum interference is a resource for post-quantum computation. Sorkin has defined a hierarchy of possible post-quantum interference behaviours where, informally, the order in the hierarchy corresponds to the number of paths that have an irreducible interaction in a multi-slit experiment. In quantum theory, at most pairs of paths can ever interact in a fundamental way. We consider how Grover's speed-up depends on the order of interference in a theory, and show that, surprisingly, the quadratic lower bound holds regardless of the order of interference.