Contributions in coding over the projective space and proposal of quantum subspace codes in the Grassmannian

Orientador: Reginaldo Palazzo JúniorTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de ComputaçãoResumo: Este trabalho de doutorado consiste em apresentar algumas contribuições em codificação no espaço projetivo. Esta é uma área de pesquisa que possui importantes aplicações em codificação de rede. A conexão entre codificação no espaço projetivo e a codificação de rede se dá através do canal de comunicação matricial. Neste contexto, mostramos um estudo de códigos de subespaços n-shot geometricamente uniforme, trazendo uma construção de tais códigos. Evidenciamos um isomorfismo entre reticulado de um grupo abeliano consistindo do grupo das unidades do corpo finito F_p, p primo, e o diagrama de Hasse de espaços projetivos P((F_p)^m). Esse isomorfismo permite trabalhar em ambas as estruturas, os códigos de subespaços n-shot. Por fim, exibimos uma proposta de construção de códigos quânticos de subespaços na grassmanniana. Neste caso, uma possibilidade é a aplicação em codificação de redes quânticas. Dado isso, propomos duas construções de códigos quânticos de subespaços na grassmanniana. A primeira, descrevemos um rotulamento do estado quântico separável arbitrário universal, e a partir desse rotulamento, associamos a códigos de subespaços na grassmanniana, com a máxima distância de subespaços, estados quânticos com o máximo emaranhamento global, também usamos os códigos de subespaços n-shot para descrever estados quânticos de máximo emaranhamento global generalizado. E a segunda, exibimos um rotulamento associado diretamente à uma classe de estados quânticos de máximo emaranhamento global por meio de uma matriz modificada da classe dos códigos Reed-MullerAbstract: This doctoral thesis consists on the introduction of some contributions for the projective space codification. This is an area with important applications on Network coding. The connection between codification on the projective space and network coding rises through the matrix communication channel. In this context, we present a study of geometrically uniform n-shot subspace codes, and a construction for such codes. We make evident an isomorphism between an Abelian group lattice, consisting on the unit group of the field F_p, p is a prime number, with the Hasse diagram of projective spacesP((F_p)^m). This isomorphism allows to work in both structures, the n-shot subspace codes. In this case, one possibilityis the application on the codification of quantum networks. Given that, we propose two proposals for the construction of subspace quantum codes in the Grassmannian. Thefirst, we describe a labeling of the universal arbitrary and separable quantum state, and from this labeling, we associate subspace codes in the Grassmannian, with the maximum subspace distance, to quantum states with the maximum global entanglement, we also use then-shot subspaces codes to describe quantum states of maximum generalized global entanglement. And the second, we exhibit a labeling directly associated to a class of quantum states of maximum global entanglement through a modified matrix from the class of Reed-Muller codesDoutoradoTelecomunicações e TelemáticaDoutor em Engenharia Elétric

Single-shot fault-tolerant quantum error correction

Conventional quantum error correcting codes require multiple rounds of measurements to detect errors with enough confidence in fault-tolerant scenarios. Here I show that for suitable topological codes a single round of local measurements is enough. This feature is generic and is related to self-correction and confinement phenomena in the corresponding quantum Hamiltonian model. 3D gauge color codes exhibit this single-shot feature, which applies also to initialization and gauge-fixing. Assuming the time for efficient classical computations negligible, this yields a topological fault-tolerant quantum computing scheme where all elementary logical operations can be performed in constant time.Comment: Typos corrected after publication in journal, 26 pages, 4 figure

Collapse of Nonlinear Gravitational Waves in Moving-Puncture Coordinates

We study numerical evolutions of nonlinear gravitational waves in moving-puncture coordinates. We adopt two different types of initial data -- Brill and Teukolsky waves -- and evolve them with two independent codes producing consistent results. We find that Brill data fail to produce long-term evolutions for common choices of coordinates and parameters, unless the initial amplitude is small, while Teukolsky wave initial data lead to stable evolutions, at least for amplitudes sufficiently far from criticality. The critical amplitude separates initial data whose evolutions leave behind flat space from those that lead to a black hole. For the latter we follow the interaction of the wave, the formation of a horizon, and the settling down into a time-independent trumpet geometry. We explore the differences between Brill and Teukolsky data and show that for less common choices of the parameters -- in particular negative amplitudes -- Brill data can be evolved with moving-puncture coordinates, and behave similarly to Teukolsky waves

Quantum states cannot be transmitted efficiently classically

We show that any classical two-way communication protocol with shared randomness that can approximately simulate the result of applying an arbitrary measurement (held by one party) to a quantum state of $n$ qubits (held by another), up to constant accuracy, must transmit at least $\Omega(2^n)$ bits. This lower bound is optimal and matches the complexity of a simple protocol based on discretisation using an $\epsilon$-net. The proof is based on a lower bound on the classical communication complexity of a distributed variant of the Fourier sampling problem. We obtain two optimal quantum-classical separations as easy corollaries. First, a sampling problem which can be solved with one quantum query to the input, but which requires $\Omega(N)$ classical queries for an input of size $N$. Second, a nonlocal task which can be solved using $n$ Bell pairs, but for which any approximate classical solution must communicate $\Omega(2^n)$ bits.Comment: 24 pages; v3: accepted version incorporating many minor corrections and clarification

Second-Order Asymptotics for the Classical Capacity of Image-Additive Quantum Channels

We study non-asymptotic fundamental limits for transmitting classical information over memoryless quantum channels, i.e. we investigate the amount of classical information that can be transmitted when a quantum channel is used a finite number of times and a fixed, non-vanishing average error is permissible. We consider the classical capacity of quantum channels that are image-additive, including all classical to quantum channels, as well as the product state capacity of arbitrary quantum channels. In both cases we show that the non-asymptotic fundamental limit admits a second-order approximation that illustrates the speed at which the rate of optimal codes converges to the Holevo capacity as the blocklength tends to infinity. The behavior is governed by a new channel parameter, called channel dispersion, for which we provide a geometrical interpretation.Comment: v2: main results significantly generalized and improved; v3: extended to image-additive channels, change of title, journal versio

Performance and structure of single-mode bosonic codes

The early Gottesman, Kitaev, and Preskill (GKP) proposal for encoding a qubit in an oscillator has recently been followed by cat- and binomial-code proposals. Numerically optimized codes have also been proposed, and we introduce new codes of this type here. These codes have yet to be compared using the same error model; we provide such a comparison by determining the entanglement fidelity of all codes with respect to the bosonic pure-loss channel (i.e., photon loss) after the optimal recovery operation. We then compare achievable communication rates of the combined encoding-error-recovery channel by calculating the channel's hashing bound for each code. Cat and binomial codes perform similarly, with binomial codes outperforming cat codes at small loss rates. Despite not being designed to protect against the pure-loss channel, GKP codes significantly outperform all other codes for most values of the loss rate. We show that the performance of GKP and some binomial codes increases monotonically with increasing average photon number of the codes. In order to corroborate our numerical evidence of the cat/binomial/GKP order of performance occurring at small loss rates, we analytically evaluate the quantum error-correction conditions of those codes. For GKP codes, we find an essential singularity in the entanglement fidelity in the limit of vanishing loss rate. In addition to comparing the codes, we draw parallels between binomial codes and discrete-variable systems. First, we characterize one- and two-mode binomial as well as multi-qubit permutation-invariant codes in terms of spin-coherent states. Such a characterization allows us to introduce check operators and error-correction procedures for binomial codes. Second, we introduce a generalization of spin-coherent states, extending our characterization to qudit binomial codes and yielding a new multi-qudit code.Comment: 34 pages, 11 figures, 4 tables. v3: published version. See related talk at https://absuploads.aps.org/presentation.cfm?pid=1351