The development of a quantum computer presents one of the greatest challenges in science and engineering to date. The promise of more efficient computing based on entangled quantum states and the superposition principle has led to a worldwide explosion of interest in the fields of quantum information and computation. Among the number of hurdles which must first be cleared before we witness a physical realization are problems associated with environment-induced decoherence and noise more generally. However, the discovery of quantum error correction and the establishment of the accuracy threshold theorem provide us with the hope of someday harnessing the potential power a functioning fault-tolerant quantum information processor has to offer. This dissertation contributes to this effort by investigating a particular class of quantum error correcting codes, namely noiseless subsystem encodings. The passive approach to error correction taken by these encodings provides an efficient means of protection from symmetrically coupled system-environment interactions. Here I will present methods for determining the subsystem-preserving evolutions for noiseless subsystem encodings supported by arbitrary-dimensional physical quantum systems. Implications for universal, collective decoherence-free quantum computation using the derived operations are discussed. Moreover, I will present a proposal for an optical device which is capable of preparing a variety of these noiseless subsystem encodings through a postselection strategy.

Within the field of Quantum Information Processing, we study two subjects: For quantum state tomography, one common assumption is that the experimentalist possesses a stationary source of identical states. We challenge this assumption and propose a method to detect and characterize the drift of nonstationary quantum sources. We distinguish diffusive and systematic drifts and examine how quickly one can determine that a source is drifting. Finally, we give an implementation of this proposed measurement for single photons. For quantum computing, fault-tolerant protocols assume that errors are of certain types. But how do we detect errors of the wrong type? The problem is that for large quantum states, a full state description is impossible to analyze, and so one cannot detect all types of errors. We show through a quantum state estimation example (on up to 25 qubits) how to attack this problem using model selection. We use, in particular, the Akaike Information Criterion. Our example indicates that the number of measurements that one has to perform before noticing errors of the wrong type scales polynomially both with the number of qubits and with the error size. This dissertation includes previously published co-authored material.

In this paper, we discuss the need for quantum error correction. We also describe some basic techniques used in quantum error correction which includes decoherence-free subspaces and subsystems. These subspaces and subsystems are described in detail. We also introduce a numerical algorithm that was used previously to search for these decoherence-free subspaces and subsystems under collective error. It is useful to search for them as they can be used to store quantum information. We use this algorithm in some specific examples involving qubits and qutrits. The results of these algorithm are then compared with the error algebra obtained using Young tableaux. We use these results to describe how the specific numerical algorithm can be used for the search of approximate decoherence-free subspaces and subsystems and minimal noise subsystems.

Neste trabalho, estudamos a teoria quântica de correção de erros, um dos principais métodos de prevenção de perda de informação num computador quântico. Este método, porém, normalmente é estudado considerando-se condições ideais em que a atuação das portas lógicas que constituem o algoritmo quântico não interfere com o tipo de erro que o sistema sofre. Além disso, as medidas de síndrome empregadas no método tradicional são consideradas instantâneas. Nossos objetivos neste trabalho serão avaliar como a alteração dessas duas suposições modificaria o processo de correção de erros. Com relação ao primeiro objetivo, verificamos que, para erros causados por ambientes externos, a atuação de uma porta lógica simultânea ao ruído pode gerar erros que, a princípio, podem não ser corrigíveis pelo código empregado. Propomos em seguida um método de correção a pequenos passos que pode ser usado para tornar desprezíveis os erros incorrigíveis, além de poder ser usado para reduzir a probabilidade de erros corrigíveis. Para o segundo objetivo, estudamos primeiro como medidas de tempo finito afetam a descoerência de apenas um qubit, concluindo que esse tipo de medida pode na verdade proteger o estado que está sendo medido. Motivados por isso, mostramos que, em certos casos, medidas de síndrome finitas realizadas conjuntamente ao ruído são capazes de proteger o estado dos qubits contra os erros mais eficientemente do que se as medidas fossem realizadas instantaneamente ao fim do processo. / In this work, we study the theory of quantum error correction, one of the main methods of preventing loss of information in a quantum computer. This method, however, is normally studied under ideal conditions in which the operation of the quantum gates that constitute the quantum algorithm do not interefere with the kind of error the system undergoes. Moreover, the syndrome measurements employed in the traditional method are considered instantaneous. Our aims with this work are to evaluate how altering these two suppositions would modify the quantum error correction process. In respect with the first objective, we verify that, for errors caused by external environments, the action of a logical gate simultaneously to the noise can provoke errors that, in principle, may not be correctable by the code employed. We subsequently propose a short-step correction method that can be used to render negligible the uncorrectable errors, besides being capable of reducing the probability of occurrence of correctable errors. For the second objective, we first study how finite-time measurements affect the decoherence of a single qubit, concluding that this kind of measurement can actually protect the state under scrutiny. Motivated by that, we demonstrate, that, in certain cases, finite syndrome measurements performed concurrently with the noise are capable of protecting more efficiently the state of the qubits against errors than if the measurements had been performed instantaneously at the the end of the process.

Η κατασκευή ενός ολοκληρωμένου κβαντικού υπολογιστή αποτελεί μια πρόκληση για τη σύγχρονη επιστήμη. Ο κβαντικός υπολογιστής μας δίνει την ελπίδα πως κάποια στιγμή στο κοντινό μέλλον, θα είμαστε σε θέση να λύνουμε προβλήματα ταχύτερα και πιο αποδοτικά από ότι κάνει ένας κλασσικός υπολογιστής σήμερα. Για παράδειγμα, ο κβαντικός αλγόριθμος παραγοντοποίησης του Shor [3] πετυχαίνει εκθετική επιτάχυνση έναντι του κλασσικού, κάτι που σημαίνει πως η χρήση του πρωτόκολλου κρυπτογράφησης RSA δεν θα είναι όσο ασφαλής είναι σήμερα. Αυτό θα έχει ως αποτέλεσμα μεγάλες αλλαγές στις επικοινωνίες και στις συναλλαγές στο προσεχές μέλλον. Στην παρούσα διπλωματική εργασία θα περιγράψουμε τις αρχές που πρέπει να πληρεί ένα κβαντικό σύστημα για να θεωρηθεί κβαντικός υπολογιστής, πώς υλοποιούμε ένα qubit που είναι η μονάδα πληροφορίας του και τέλος θα μιλήσουμε για το πώς κωδικοποιούμε την κβαντική πληροφορία ώστε να είμαστε σε θέση να τη διορθώσουμε. Αρχίζουμε με τη διατύπωση των αρχών της κβαντικής μηχανικής , όπως προκύπτουν από την πειραματική διαδικασία. Συνεχίζουμε με την υπεραγωγιμότητα, το φαινόμενο που μας επιτρέπει να χειριζόμαστε μακροσκοπικά της κβαντικές ιδιότητες της ύλης, όπως και κάποια ακόμα φαινόμενα, όπως αυτό του Meissner, που μας δίνουν τη δυνατότητα να δημιουργήσουμε το κυκλώμα που υλοποιεί το qubit. Τέλος, περιγράφουμε θεωρητικά ένα καθολικό σύνολο από κβαντικές πύλες και τα κυκλώματα διόρθωσης λαθών κβαντικού κώδικα. / The construction of an integrated quantum computer is a challenge for modern science. The quantum computer gives us hope that sometime in the near future, we will be able to solve problems faster and more efficiently than does a conventional computer today. For example, the Shor's quantum algorithm for factoring [3] gave exponential acceleration compared to the classical one, which means that the use of RSA encryption protocol will not be safe as it is today. This will result large changes in communications and transactions in the near future. In this paper we describe the principles that must meet a quantum system to be considered as a quantum computer, how do we implement a qubit which is the unit of information, and finally we'll talk about how we encode quantum information in order to be able to fix it . We begin with the formulation of the principles of quantum mechanics, derived from the experimental procedure. We continue with the superconductivity phenomenon that allows us to manipulate the macroscopic quantum properties of matter, and even some phenomena such as the Meissner, who enable us to create a circuit that implements the qubit. Finally, we describe theoretically a universal set of quantum gates and circuits of error correcting quantum code.

Orientador: Reginaldo Palazzo Junior / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-12T20:22:38Z (GMT). No. of bitstreams: 1 Santos_PolyaneAlves_M.pdf: 825808 bytes, checksum: f4b4d556a54cfca0cb0a84dd5e07a7a3 (MD5) Previous issue date: 2008 / Resumo: A proposta de um sistema criptográfico de chave pública que utiliza códigos convolucionais de memória-unitária clássicos e quânticos apresentada neste trabalho, está baseada na utilização de transformações armadilha que, ao serem aplicadas as submatrizes reduzem a capacidade de correção de erros do código. Este processo proporciona um aumento no grau de privacidade da informação a ser enviada devido a dois fatores: para a determinação de códigos ótimos de memória unitária è necessário resolver o Problema da Mochila e a redução da capacidade de correção de erro dos códigos ocasionada pelo embaralhamento das colunas das submatrizes geradoras. São também apresentados neste trabalho, novos códigos convolucionais quânticos concatenados [(4, 1, 3)]. / Abstract: The proposal of a cryptographic system of public key that uses classical and quantum convolutional codes of unit-memory presented in this work, is based on the use of trapdoors functions which when applied to submatrices reduce the capacity of correction of errors of the code. This process gives us an increase in the degree of privacy of information being sent, because of two factors, namely: to establish good unit-memory codes is necessary to solve the knapsack problem, and the reduction of the capacity of correcting errors of codes provided by scrambling the columns of generating submatrices. We also present in this work, news quantum convolutional codes [(4, 1, 3)]. / Mestrado / Telecomunicações e Telemática / Mestre em Engenharia Elétrica

Neste trabalho, estudamos a teoria quântica de correção de erros, um dos principais métodos de prevenção de perda de informação num computador quântico. Este método, porém, normalmente é estudado considerando-se condições ideais em que a atuação das portas lógicas que constituem o algoritmo quântico não interfere com o tipo de erro que o sistema sofre. Além disso, as medidas de síndrome empregadas no método tradicional são consideradas instantâneas. Nossos objetivos neste trabalho serão avaliar como a alteração dessas duas suposições modificaria o processo de correção de erros. Com relação ao primeiro objetivo, verificamos que, para erros causados por ambientes externos, a atuação de uma porta lógica simultânea ao ruído pode gerar erros que, a princípio, podem não ser corrigíveis pelo código empregado. Propomos em seguida um método de correção a pequenos passos que pode ser usado para tornar desprezíveis os erros incorrigíveis, além de poder ser usado para reduzir a probabilidade de erros corrigíveis. Para o segundo objetivo, estudamos primeiro como medidas de tempo finito afetam a descoerência de apenas um qubit, concluindo que esse tipo de medida pode na verdade proteger o estado que está sendo medido. Motivados por isso, mostramos que, em certos casos, medidas de síndrome finitas realizadas conjuntamente ao ruído são capazes de proteger o estado dos qubits contra os erros mais eficientemente do que se as medidas fossem realizadas instantaneamente ao fim do processo. / In this work, we study the theory of quantum error correction, one of the main methods of preventing loss of information in a quantum computer. This method, however, is normally studied under ideal conditions in which the operation of the quantum gates that constitute the quantum algorithm do not interefere with the kind of error the system undergoes. Moreover, the syndrome measurements employed in the traditional method are considered instantaneous. Our aims with this work are to evaluate how altering these two suppositions would modify the quantum error correction process. In respect with the first objective, we verify that, for errors caused by external environments, the action of a logical gate simultaneously to the noise can provoke errors that, in principle, may not be correctable by the code employed. We subsequently propose a short-step correction method that can be used to render negligible the uncorrectable errors, besides being capable of reducing the probability of occurrence of correctable errors. For the second objective, we first study how finite-time measurements affect the decoherence of a single qubit, concluding that this kind of measurement can actually protect the state under scrutiny. Motivated by that, we demonstrate, that, in certain cases, finite syndrome measurements performed concurrently with the noise are capable of protecting more efficiently the state of the qubits against errors than if the measurements had been performed instantaneously at the the end of the process.

In this thesis, we discuss correlations arising between a system and its environment that lead to errors in an open quantum system. Detecting those correlations would be valuable for avoiding and/or correcting those errors. It was studied previously that we can detect correlations by only measuring the system itself if we know the cause of interaction between the two, for example in the case of a dipole-dipole interaction for a spin 1/2-spin 1/2 interaction Hamiltonian. We investigate the unitary, U which is associated with the exchange Hamiltonian and examine the ability to detect initial correlations between a system and its environment for a spin-1/2(qubit) system interacting with a larger higher dimensional environment. We provide bounds for when we can state with certainty that there are initial system-environment correlations given experimental data.

The overarching theme of this thesis is the question of how constraints influence collective behavior. Constraints are crucial in shaping both static and dynamic properties of systems across diverse areas within condensed matter physics and beyond. For example, the simple geometric constraint that hard particles cannot overlap at high density leads to slow dynamics and jamming in glass formers. Constraints also arise effectively at low temperature as a consequence of strong competing interactions in magnetic materials, where they give rise to emergent gauge theories and unconventional magnetic order. Enforcing constraints artificially in turn can be used to protect otherwise fragile quantum information from external noise. This thesis in particular contains progress on the realization of different unconventional phases of matter in constrained systems. The presentation of individual results is organized by the stage of realization of the respective phase. Novel physical phenomena after conceptualization are often exemplified in simple, heuristic models bearing little resemblance of actual matter, but which are interesting enough to motivate efforts with the final goal of realizing them in some way in the lab. One form of progress is then to devise refined models, which retain a degree of simplification while still realizing the same physics and improving the degree of realism in some direction. Finally, direct efforts in realizing either the original models or some refined version in experiment today are mostly two-fold. One route, having grown in importance rapidly during the last two decades, is via the engineering of artificial systems realizing suitable models. The other, more conventional way is to search for realizations of novel phases in materials. The thesis is divided into three parts, where Part I is devoted to the study of two simple models, while artificial systems and real materials are the subject of Part II and Part III respectively. Below, the content of each part is summarized in more detail. After a general introduction to entropic ordering and slow dynamics we present a family of models devised as a lattice analog of hard spheres. These are often studied to explore whether low-dimensional analogues of mean-field glass- and jamming transitions exist, but also serve as the canonical model systems for slow dynamics in granular materials more generally. Arguably the models in this family do not offer a close resemblance of actual granular materials. However, by studying their behavior far from equilibrium, we observe the onset of slow dynamics and a kinetic arrest for which, importantly, we obtain an essentially complete analytical and numerical understanding. Particularly interesting is the fact that this understanding hinges on the (in-)ability to anneal topological defects in the presence of a hardcore constraints, which resonates with some previous proposals for an understanding of the glass transition. As another example of anomalous dynamics arising in a magnetic system, we also present a detailed study of a two-dimensional fracton spin liquid. The model is an Ising system with an energy function designed to give rise to an emergent higher-rank gauge theory at low energy. We show explicitly that the number of zero-energy states in the model scales exponentially with the system size, establishing a finite residual entropy. A purpose-built cluster Monte-Carlo algorithm makes it possible to study the behavior of the model as a function of temperature. We show evidence for a first order transition from a high-temperature paramagnet to a low-temperature phase where correlations match predictions of a higher-rank coulomb phase. Turning away from heuristic models, the second part of the thesis begins with an introduction to quantum error correction, a scheme where constraints are artificially imposed in a quantum system through measurement and feedback. This is done in order to preserve quantum information in the presence of external noise, and is widely believed to be necessary in order to one day harness the full power of quantum computers. Given a certain error-correcting code as well as a noise model, a particularly interesting quantity is the threshold of the code, that is the critical amount of external noise below which quantum error correction becomes possible. For the toric code under independent bit- and phase-flip noise for example, the threshold is well known to map to the paramagnet to ferromagnet transition of the two-dimensional random-bond Ising model along the Nishimori line. Here, we present the first generalization of this mapping to a family of codes with finite rate, that is a family where the number of encoded logical qubits grows linearly with the number of physical qubits. In particular, we show that the threshold of hyperbolic surface codes maps to a paramagnet to ferromagnet transition in what we call the 'dual'' random-bond Ising model on regular tessellations of compact hyperbolic manifolds. This model is related to the usual random-bond Ising model by the Kramers-Wannier duality but distinct from it even on self-dual tessellations. As a corollary, we clarify long-standing issues regarding self-duality of the Ising model in hyperbolic space. The final part of the thesis is devoted to the study of material candidates of quantum spin ice, a three-dimensional quantum spin liquid. The work presented here was done in close collaboration with experiment and focuses on a particular family of materials called dipolar-octupolar pyrochlores. This family of materials is particularly interesting because they might realize novel exotic quantum states such as octupolar spin liquids, while at the same time being described by a relatively simple model Hamiltonian. This thesis contains a detailed study of ground state selection in dipolar-octupolar pyrochlore magnets and its signatures as observable in neutron scattering. First, we present evidence that the two compounds Ce2Zr2O7 and Ce2Sn2O7 despite their similar chemical composition realize an exotic quantum spin liquid state and an ordered state respectively. Then, we also study the ground-state selection in dipolar-octupolar pyrochlores in a magnetic field. Most importantly, we show that the well-known effective one-dimensional physics -- arising when the field is applied along a certain crystallographic axis -- is expected to be stable at experimentally relevant temperatures. Finally, we make predictions for neutron scattering in the large-field phase and compare these to measurements on Ce2Zr2O7.

A universal, scalable quantum computer will require the use of quantum error correction in order to achieve fault tolerance. The assessment and comparison of error-correcting strategies is performed by classical simulation. However, due to the prohibitive exponential scaling of general quantum circuits, simulations are restrained to specific subsets of quantum operations. This creates a gap between accuracy and efficiency which is particularly problematic when modeling noise, because most realistic noise models are not efficiently simulable on a classical computer. We have introduced extensions to the Pauli channel, the traditional error channel employed to model noise in simulations of quantum circuits. These expanded error channels are still computationally tractable to simulate, but result in more accurate approximations to realistic error channels at the single qubit level. Using the Steane [[7,1,3]] code, we have also investigated the behavior of these expanded channels at the logical error-corrected level. We have found that it depends strongly on whether the error is incoherent or coherent. In general, the Pauli channel will be an excellent approximation to incoherent channels, but an unsatisfactory one for coherent channels, especially because it severely underestimates the magnitude of the error. Finally, we also studied the honesty and accuracy of the expanded channels at the logical level. Our results suggest that these measures can be employed to generate lower and upper bounds to a quantum code's threshold under the influence of a specific error channel.