Optimal control theory for a unitary operation under dissipative evolution

We show that optimizing a quantum gate for an open quantum system requires the time evolution of only three states irrespective of the dimension of Hilbert space. This represents a significant reduction in computational resources compared to the complete basis of Liouville space that is commonly believed necessary for this task. The reduction is based on two observations: the target is not a general dynamical map but a unitary operation; and the time evolution of two properly chosen states is sufficient to distinguish any two unitaries. We illustrate gate optimization employing a reduced set of states for a controlled phasegate with trapped atoms as qubit carriers and a iSWAP gate with superconducting qubits.

Optimizing Robust Quantum Gates in Open Quantum Systems

We are currently at the cusp of a revolution in quantum technology that relies not just on the passive use of quantum effects, but on their active control. At the forefront of this revolution is the implementation of a quantum computer. Encoding information in quantum states as “qubits” allows to use entanglement and quantum superposition to perform calculations that are infeasible on classical computers. The fundamental challenge in the realization of quantum computers is to avoid decoherence – the loss of quantum properties – due to unwanted interaction with the environment. This thesis addresses the problem of implementing entangling two-qubit quantum gates that are robust with respect to both decoherence and classical noise. It covers three aspects: the use of efficient numerical tools for the simulation and optimal control of open and closed quantum systems, the role of advanced optimization functionals in facilitating robustness, and the application of these techniques to two of the leading implementations of quantum computation, trapped atoms and superconducting circuits. After a review of the theoretical and numerical foundations, the central part of the thesis starts with the idea of using ensemble optimization to achieve robustness with respect to both classical fluctuations in the system parameters, and decoherence. For the example of a controlled phasegate implemented with trapped Rydberg atoms, this approach is demonstrated to yield a gate that is at least one order of magnitude more robust than the best known analytic scheme. Moreover this robustness is maintained even for gate durations significantly shorter than those obtained in the analytic scheme. Superconducting circuits are a particularly promising architecture for the implementation of a quantum computer. Their flexibility is demonstrated by performing optimizations for both diagonal and non-diagonal quantum gates. In order to achieve robustness with respect to decoherence, it is essential to implement quantum gates in the shortest possible amount of time. This may be facilitated by using an optimization functional that targets an arbitrary perfect entangler, based on a geometric theory of two-qubit gates. For the example of superconducting qubits, it is shown that this approach leads to significantly shorter gate durations, higher fidelities, and faster convergence than the optimization towards specific two-qubit gates. Performing optimization in Liouville space in order to properly take into account decoherence poses significant numerical challenges, as the dimension scales quadratically compared to Hilbert space. However, it can be shown that for a unitary target, the optimization only requires propagation of at most three states, instead of a full basis of Liouville space. Both for the example of trapped Rydberg atoms, and for superconducting qubits, the successful optimization of quantum gates is demonstrated, at a significantly reduced numerical cost than was previously thought possible. Together, the results of this thesis point towards a comprehensive framework for the optimization of robust quantum gates, paving the way for the future realization of quantum computers.

The Unsupervised Acquisition of a Lexicon from Continuous Speech

We present an unsupervised learning algorithm that acquires a natural-language lexicon from raw speech. The algorithm is based on the optimal encoding of symbol sequences in an MDL framework, and uses a hierarchical representation of language that overcomes many of the problems that have stymied previous grammar-induction procedures. The forward mapping from symbol sequences to the speech stream is modeled using features based on articulatory gestures. We present results on the acquisition of lexicons and language models from raw speech, text, and phonetic transcripts, and demonstrate that our algorithm compares very favorably to other reported results with respect to segmentation performance and statistical efficiency.

An Equivalence Between Sparse Approximation and Support Vector Machines

In the first part of this paper we show a similarity between the principle of Structural Risk Minimization Principle (SRM) (Vapnik, 1982) and the idea of Sparse Approximation, as defined in (Chen, Donoho and Saunders, 1995) and Olshausen and Field (1996). Then we focus on two specific (approximate) implementations of SRM and Sparse Approximation, which have been used to solve the problem of function approximation. For SRM we consider the Support Vector Machine technique proposed by V. Vapnik and his team at AT&T Bell Labs, and for Sparse Approximation we consider a modification of the Basis Pursuit De-Noising algorithm proposed by Chen, Donoho and Saunders (1995). We show that, under certain conditions, these two techniques are equivalent: they give the same solution and they require the solution of the same quadratic programming problem.

Aitchison Geometry for Probability and Likelihood as a new approach to mathematical statistics

The Aitchison vector space structure for the simplex is generalized to a Hilbert space structure A2(P) for distributions and likelihoods on arbitrary spaces. Central notations of statistics, such as Information or Likelihood, can be identified in the algebraical structure of A2(P) and their corresponding notions in compositional data analysis, such as Aitchison distance or centered log ratio transform. In this way very elaborated aspects of mathematical statistics can be understood easily in the light of a simple vector space structure and of compositional data analysis. E.g. combination of statistical information such as Bayesian updating, combination of likelihood and robust M-estimation functions are simple additions/ perturbations in A2(Pprior). Weighting observations corresponds to a weighted addition of the corresponding evidence. Likelihood based statistics for general exponential families turns out to have a particularly easy interpretation in terms of A2(P). Regular exponential families form finite dimensional linear subspaces of A2(P) and they correspond to finite dimensional subspaces formed by their posterior in the dual information space A2(Pprior). The Aitchison norm can identified with mean Fisher information. The closing constant itself is identified with a generalization of the cummulant function and shown to be Kullback Leiblers directed information. Fisher information is the local geometry of the manifold induced by the A2(P) derivative of the Kullback Leibler information and the space A2(P) can therefore be seen as the tangential geometry of statistical inference at the distribution P. The discussion of A2(P) valued random variables, such as estimation functions or likelihoods, give a further interpretation of Fisher information as the expected squared norm of evidence and a scale free understanding of unbiased reasoning

Statistical treatment of grain-size curves and empirical distributions: densities as compositions?

The preceding two editions of CoDaWork included talks on the possible consideration of densities as infinite compositions: Egozcue and D´ıaz-Barrero (2003) extended the Euclidean structure of the simplex to a Hilbert space structure of the set of densities within a bounded interval, and van den Boogaart (2005) generalized this to the set of densities bounded by an arbitrary reference density. From the many variations of the Hilbert structures available, we work with three cases. For bounded variables, a basis derived from Legendre polynomials is used. For variables with a lower bound, we standardize them with respect to an exponential distribution and express their densities as coordinates in a basis derived from Laguerre polynomials. Finally, for unbounded variables, a normal distribution is used as reference, and coordinates are obtained with respect to a Hermite-polynomials-based basis. To get the coordinates, several approaches can be considered. A numerical accuracy problem occurs if one estimates the coordinates directly by using discretized scalar products. Thus we propose to use a weighted linear regression approach, where all k- order polynomials are used as predictand variables and weights are proportional to the reference density. Finally, for the case of 2-order Hermite polinomials (normal reference) and 1-order Laguerre polinomials (exponential), one can also derive the coordinates from their relationships to the classical mean and variance. Apart of these theoretical issues, this contribution focuses on the application of this theory to two main problems in sedimentary geology: the comparison of several grain size distributions, and the comparison among different rocks of the empirical distribution of a property measured on a batch of individual grains from the same rock or sediment, like their composition

Periodic Table of Elements

An essential tool for chemistry students. Presentation of the table in which elements are assigned a symbol and shows the atomic number of each element.

Exercise sheet 7 pdf

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Exam 1999 pdf

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Exam 2000

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Exercise sheet 7

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Exam 1999

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Joyería Hefestos vida y muerte empresarial : estudios empíricos de perdurabilidad de la morbilidad a la perdurabilidad

Este trabajo de grado es un acercamiento a las lecciones empresariales de perdurabilidad basado en el caso de estudio de Joyería Hefestos. Esta empresa ha sido símbolo de la reproducción de piezas de arte precolombino y artesanías de nuestro país, alcanzando reconocimiento nacional e internacional por su originalidad, diseño y calidad. El objetivo de esta investigación es identificar la ruta de éxito que Joyería Hefestos implementó para salir de la crisis que enfrentó durante la década de los noventa. Por tal motivo, se identificarán las causas de la crisis, evaluando las decisiones que la empresa tomó para sobrevivir y hoy en día permanecer en el mercado de la joyería en Colombia. Este proyecto de investigación trabajó de la mano con el Sr. Javier Parra,gerente general; empleados y académicos que nos permitieron evaluar la situación de la empresa antes, durante y después de la crisis. Mediante diferentes herramientas como entrevistas, encuestas y visitas empresariales se evaluó la pertinencia las decisiones estratégicas que se tomaron en el momento preciso para sortear este periodo de caos y alcanzar solvencia financiera y operativa. El trabajo se estructuró bajo los lineamientos de la línea de Estrategia de la Universidad del Rosario, donde se analizaron las variables de Dirección y Gerencia para explicar las causas y consecuencias antes, durante y después de la crisis de la gestión administrativa. Esto permitirá identificar la ruta del éxito mediante la cual la empresa superó este periodo gracias a la incorporación bajo los estatutos del acuerdo de reestructuración Ley 550/1999 que permitió en gran medida el resurgimiento de la empresa junto con el compromiso fiel y constante del personal de Joyería Hefestos.

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A Modest Proposal to Clarify the Status of Coca in the United Nations Conventions

The implementation of anti-drug policies that focus on illicit crops in the Andean countries faces many significant obstacles, one of which is the cultural clash it generates between the main stakeholders. On the one hand one finds the governments and agencies that attempt to implement crop substitution and eradication policies and on the other the peasant and natives communities that have traditionally grown and used coca or those peasants who have found in coca an instrument of power and political leverage that they never had before. The confrontation about coca eradication, alternative development and other anti-drug policies in coca growing areas transcends drug related issues and is part of a wider and deeper confrontation that reflects the long-term unsolved conflicts of the Andean societies. All Andean countries have stratified and fragmented societies in which peasants and Indians have been excluded from power. In Bolivia, Ecuador and Peru most peasants belong to native communities many of which have remained segregated from “white” society. The mixing of the races (mestizaje) in Colombia occurred early during the Conquest and Colony. Those of Indian descent became subservient to the Spanish and Creoles. The society that evolved was (and still is) highly hierarchical, authoritarian, and has subjacent racist values. The resulting political system has been exclusionary of large portions of the population. Among Indian communities coca has been used for millennia and its use has become an identity symbol of their resistance against what may be looked at as foreign invasion. “The Andean Indian chews coca because that way he affirms his identity as son and owner of the land that yesterday the Spaniard took away and today the landowner keeps away from him. To chew coca is to be Indian...and to quietly and obstinately challenge the contemporary lords that descend from the old encomenderos and the older conquistadors” (Vidart, 1991: 61, author’s translation). In Andean literature on illegal drugs as well as in seminars, colloquia and other meetings where drug policies are debated, complaints are frequently expressed about the treatment of coca in the same category as cocaine, heroin, morphine amphetamines and other “hard” drugs. The complainants assert that “coca is not cocaine” and that it is unfair to classify coca, a nature given plant which has been used for millennia in the Andes without significant negative effects on users, in the same category as man made psychotropic drugs. They also argue that coca has manifold social and religious meanings in indigenous cultures, that coca is sacred and that the requirement of the1961 Single Convention demanding that Bolivia and Peru completely eradicate coca within 25 years is limiting Indigenous communities in their freedom to practice their religions. In most debates about drug interdiction, the views of those who oppose that approach are not accepted as legitimate. Indeed, “prohibitionists” demonize drugs and those who oppose drug policies in Latin America frequently demonize the United States as the imperialist power that imposes them. This dual polarization is a main obstacle to establish a meaningful policy debate aimed at broadening the policy consensus necessary for successful policy implementation. This essay surveys the status of coca in the United Nations Conventions, explains why it is confusing, and how a few changes would eliminate some of the sources of conflict and help organize and control licit coca markets in the Andes. The current disorganized and weakly controlled legal coca market in Peru has been analyzed to demonstrate its deficiencies and to illustrate possible improvements in international drug control policies.