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

Méthodes exactes et approchées pour le problème de gestion de projet à contraintes de ressources

Le problème de gestion de projet à contraintes de ressources est un des problèmesles plus étudiés dans la littérature. Il consiste à planifier des activités soumises à desrelations de précédence, et nécessitant des ressources renouvelables. L objectif est deminimiser la durée du projet, soit le makespan. Nous étudions le problème de gestion deprojet à contraintes de ressources. Nous nous sommes intéressées à la résolution exactedu problème. Dans la première partie de la thèse, nous élaborons une série de bornesinférieures basées sur le raisonnement énergétique et des formulations mathématiques.Les résultats montrent que les bornes proposées surpassent ceux de la littérature. Dansla deuxième partie, nous proposons des procédures par séparation et évaluation utilisantles bornes inférieures dévelopées dans la première partie.Resource Constrained Project Scheduling Problem is one of the most studied schedulingproblems in the literature. It consists in scheduling activities, submitted to precedencerelationship, and requiring renewable resources to be processed. The objective isto minimize the project duration, i.e., the makespan. We study the Resource ConstrainedProject Scheduling Problem. We are interested on the exact resolution of the problem.In the first part of the thesis, we develop a series of lower bounds based on energeticreasoning and mathematical formulations. The computational results show that theproposed lower bounds outperform the ones of the literature. In the second part, wepropose Branch-and-Bound procedures using the lower bounds developed on the firstpart.TOURS-Bibl.électronique (372610011) / SudocSudocFranceF

Mathematical Models and Decomposition Algorithms for Cutting and Packing Problems

In this thesis, we provide (or review) new and effective algorithms based on Mixed-Integer Linear Programming (MILP) models and/or decomposition approaches to solve exactly various cutting and packing problems. The first three contributions deal with the classical bin packing and cutting stock problems. First, we propose a survey on the problems, in which we review more than 150 references, implement and computationally test the most common methods used to solve the problems (including branch-and-price, constraint programming (CP) and MILP), and we successfully propose new instances that are difficult to solve in practice. Then, we introduce the BPPLIB, a collection of codes, benchmarks, and links for the two problems. Finally, we study in details the main MILP formulations that have been proposed for the problems, we provide a clear picture of the dominance and equivalence relations that exist among them, and we introduce reflect, a new pseudo-polynomial formulation that achieves state of the art results for both problems and some variants. The following three contributions deal with two-dimensional packing problems. First, we propose a method using Logic based Benders’ decomposition for the orthogonal stock cutting problem and some extensions. We solve the master problem through an MILP model while CP is used to solve the slave problem. Computational experiments on classical benchmarks from the literature show the effectiveness of the proposed approach. Then, we introduce TwoBinGame, a visual application we developed for students to interactively solve two-dimensional packing problems, and analyze the results obtained by 200 students. Finally, we study a complex optimization problem that originates from the packaging industry, which combines cutting and scheduling decisions. For its solution, we propose mathematical models and heuristic algorithms that involve a non-trivial decomposition method. In the last contribution, we study and strengthen various MILP and CP approaches for three project scheduling problems

Energy dissipation in granular materials in DEM simulations

Soil has generally been treated as a continuum from as early as the eighteenth century. Since then the analysis of soil behaviour in practical engineering analyses and development of constitutive models has depended on a continuum assumption. However, in order to gain a deeper understanding of the behaviour of soils and their particulate nature, there is a need to move from continuum mechanics to discrete models. Such modelling is possible using the Discrete Element Method (DEM). In this thesis an open source DEM particle simulation software, LIGGGHTS is used to study the relationships between grain scale parameters and energy dissipation in granular media in one-dimensional compression. In order to measure the dissipated energy, changes in energy terms are traced at every time step and the principle of energy conservation applied. The influence of particle size distribution, initial void ratio, and inter-particle friction coefficient on energy dissipation are studied and discussed. It is shown that increasing the coefficient of uniformity decreases the energy dissipated; lowering the initial voids ratio results in steeper energy dissipation curves; and a higher inter-particle coefficient of friction yields more energy dissipation. It is hoped that the knowledge gained of the relationship between grain scale parameters and energy dissipation can be built upon to formulate constitutive relationships within the hyperplasticity framework. It is envisioned that relating grain scale parameters to constitutive models will allow the formulation of models that are purely based on the micro-mechanics of granular media

Modelling and prediction of foam structure

The primary objective of this thesis is to explore the relationship between the surface and underlying structure of dry foams. This relationship is important for both research and industry because the surface film size distribution is typically the only information available in opaque foams when referring to the underlying bubble size distribution. This study is carried out by simulating foams with free surfaces. Firstly, the method of simulating 3D dry foams with free surfaces is presented. The simulation method is verified for foams with uniform bubble sizes by comparing their simulation results to experimental values reported in literature. The validity of the method and its ability to accurately model the structure of both surface and internal bubbles is demonstrated by the excellent agreement between the experimental study and the simulation results. Secondly, the simulation results are shown for the relationship between the surface film size distribution and the surface bubble size distribution. The results show that, for a given surface bubble size, there is a distribution of possible surface film sizes. However, for the range of polydispersity used in this thesis, the distribution of the ratio of film size to the size of bubble to which it is attached is found to be independent of the underlying bubble size distribution. A functional form of this relationship is obtained by nonlinear regression. Based on the functions obtained, the surface film size distributions can be computed using the underlying surface bubble size distribution. This is the inverse of what is acquired and therefore a numerical procedure for obtaining the surface bubble size distributions using the corresponding surface film size distribution is developed. This method is demonstrated to accurately reproduce the results from the full structural foam simulations

A survey of variants and extensions of the resource-constrained project scheduling problem

The resource-constrained project scheduling problem (RCPSP) consists of activities that must be scheduled subject to precedence and resource constraints such that the makespan is minimized. It has become a well-known standard problem in the context of project scheduling which has attracted numerous researchers who developed both exact and heuristic scheduling procedures. However, it is a rather basic model with assumptions that are too restrictive for many practical applications. Consequently, various extensions of the basic RCPSP have been developed. This paper gives an overview over these extensions. The extensions are classified according to the structure of the RCPSP. We summarize generalizations of the activity concept, of the precedence relations and of the resource constraints. Alternative objectives and approaches for scheduling multiple projects are discussed as well. In addition to popular variants and extensions such as multiple modes, minimal and maximal time lags, and net present value-based objectives, the paper also provides a survey of many less known concepts. --project scheduling,modeling,resource constraints,temporal constraints,networks