Quantum Gate and Quantum State Preparation through Neighboring Optimal Control

Successful implementation of fault-tolerant quantum computation on a system of qubits places severe demands on the hardware used to control the many-qubit state. It is known that an accuracy threshold Pa exists for any quantum gate that is to be used for such a computation to be able to continue for an unlimited number of steps. Specifically, the error probability Pe for such a gate must fall below the accuracy threshold: Pe < Pa. Estimates of Pa vary widely, though Pa ∼ 10−4 has emerged as a challenging target for hardware designers. I present a theoretical framework based on neighboring optimal control that takes as input a good quantum gate and returns a new gate with better performance. I illustrate this approach by applying it to a universal set of quantum gates produced using non-adiabatic rapid passage. Performance improvements are substantial comparing to the original (unimproved) gates, both for ideal and non-ideal controls. Under suitable conditions detailed below, all gate error probabilities fall by 1 to 4 orders of magnitude below the target threshold of 10−4. After applying the neighboring optimal control theory to improve the performance of quantum gates in a universal set, I further apply the general control theory in a two-step procedure for fault-tolerant logical state preparation, and I illustrate this procedure by preparing a logical Bell state fault-tolerantly. The two-step preparation procedure is as follow: Step 1 provides a one-shot procedure using neighboring optimal control theory to prepare a physical qubit state which is a high-fidelity approximation to the Bell state |β01⟩ = 1/√2(|01⟩ + |10⟩). I show that for ideal (non-ideal) control, an approximate |β01⟩ state could be prepared with error probability ϵ ∼ 10−6 (10−5) with one-shot local operations. Step 2 then takes a block of p pairs of physical qubits, each prepared in |β01⟩ state using Step 1, and fault-tolerantly prepares the logical Bell state for the C4 quantum error detection code.

Identificación de riesgos de error material en auditorías externas de estados financieros, aplicadas por los auditores independientes del municipio de San Salvador.

La presente investigación tiene sus orígenes desde el siglo XV, cuando se presumía que la auditoría fue incentivada por la necesidad del hombre de llevar a cabo actividades de observación, investigación, comprobación y verificación de la información financiera generada por las empresas; para ser más concretos nació en el seno de algunas familias pudientes de Inglaterra, otorgando a la palabra “Auditor” el significado de “Persona que Oye”. Desde hace ya algunos años, el ejercicio de los contadores públicos es vital en el proceso de auditorías de estados financieros, lo cual conlleva a la necesidad de estructurar a profundidad aspectos relevantes para el desarrollo de las auditorias, entre los cuales el riesgo de error material es fundamental, no solo por la importancia del tema, sino además porque son ellos quienes expresan una opinión acerca de la razonabilidad de las cifras de la información financiera sujeta a revisión y estudio. Y es ahí donde surge el IAASB (International Auditing and Assurance Standards Board), que es el ente encargado de emitir normas y lineamientos de auditoría y atestiguamiento para uso de todos los profesionales, en virtud de un proceso de auditoría de establecimiento de normas, el cual proporciona información de interés público sobre el desarrollo de auditorías, incluyendo en este como identificar riesgos de error material mediante el entendimiento de la entidad y su entorno. La investigación de esta problemática, surgió con la necesidad de los contadores de herramientas de apoyo para ejercer la profesión, que se apegan a la normativa técnica contable de nuestro país; por lo que objetivo principal que persigue esta investigación es diseñar cuestionarios de control interno para la identificación de riesgos de error material en auditorías externas de Estados Financieros, aplicadas por auditores independientes del municipio de San Salvador. Durante la investigación de campo, se notó que la mayoría de los auditores utiliza diferentes técnicas para identificar riesgos de error material, lo que da la pauta que es indispensable contar con una herramienta de apoyo, que incorpore el entendimiento de la entidad y su entorno, así como el control interno, además de la aplicación de las políticas para el logro de sus objetivos, estos aspectos ayudan a detectar los fraudes y errores, no olvidando que las investigaciones con la administración son de mucha utilidad. Uno de los obstáculos en la entrevista fue la falta de recopilación de información sobre cuáles son las técnicas que utilizan estos auditores para identificar riesgos. Finalmente, se concluye que los conocimientos técnicos sobre el riesgo de negocio son primordiales para los auditores, que el entendimiento de riesgos de errores materiales es fundamental al momento de ejecutar auditorías externas de estados financieros y que la evaluación del control interno y los riesgos del negocio son importantes para la identificación de errores materiales. Para lo cual se recomienda, establecer una planificación de auditoría con base al análisis de la entidad y de los posibles riesgos de negocio, mediante el análisis del entorno. Es indispensable que los profesionales que ejercen la auditoría, cuenten con una base técnica que les permita identificar estratégicamente aquellos riesgos de error material que afecten adversamente el logro de los objetivos que persigue la entidad, sin dejar atrás el estudio y evaluación del control interno de la entidad auditada, ya que se consideran puntos clave para identificar riesgos.

Autonomous formation flying: unified control and collision avoidance methods for close manoeuvring spacecraft

The idea of spacecraft formations, flying in tight configurations with maximum baselines of a few hundred meters in low-Earth orbits, has generated widespread interest over the last several years. Nevertheless, controlling the movement of spacecraft in formation poses difficulties, such as in-orbit high-computing demand and collision avoidance capabilities, which escalate as the number of units in the formation is increased and complicated nonlinear effects are imposed to the dynamics, together with uncertainty which may arise from the lack of knowledge of system parameters. These requirements have led to the need of reliable linear and nonlinear controllers in terms of relative and absolute dynamics. The objective of this thesis is, therefore, to introduce new control methods to allow spacecraft in formation, with circular/elliptical reference orbits, to efficiently execute safe autonomous manoeuvres. These controllers distinguish from the bulk of literature in that they merge guidance laws never applied before to spacecraft formation flying and collision avoidance capacities into a single control strategy. For this purpose, three control schemes are presented: linear optimal regulation, linear optimal estimation and adaptive nonlinear control. In general terms, the proposed control approaches command the dynamical performance of one or several followers with respect to a leader to asymptotically track a time-varying nominal trajectory (TVNT), while the threat of collision between the followers is reduced by repelling accelerations obtained from the collision avoidance scheme during the periods of closest proximity. Linear optimal regulation is achieved through a Riccati-based tracking controller. Within this control strategy, the controller provides guidance and tracking toward a desired TVNT, optimizing fuel consumption by Riccati procedure using a non-infinite cost function defined in terms of the desired TVNT, while repelling accelerations generated from the CAS will ensure evasive actions between the elements of the formation. The relative dynamics model, suitable for circular and eccentric low-Earth reference orbits, is based on the Tschauner and Hempel equations, and includes a control input and a nonlinear term corresponding to the CAS repelling accelerations. Linear optimal estimation is built on the forward-in-time separation principle. This controller encompasses two stages: regulation and estimation. The first stage requires the design of a full state feedback controller using the state vector reconstructed by means of the estimator. The second stage requires the design of an additional dynamical system, the estimator, to obtain the states which cannot be measured in order to approximately reconstruct the full state vector. Then, the separation principle states that an observer built for a known input can also be used to estimate the state of the system and to generate the control input. This allows the design of the observer and the feedback independently, by exploiting the advantages of linear quadratic regulator theory, in order to estimate the states of a dynamical system with model and sensor uncertainty. The relative dynamics is described with the linear system used in the previous controller, with a control input and nonlinearities entering via the repelling accelerations from the CAS during collision avoidance events. Moreover, sensor uncertainty is added to the control process by considering carrier-phase differential GPS (CDGPS) velocity measurement error. An adaptive control law capable of delivering superior closed-loop performance when compared to the certainty-equivalence (CE) adaptive controllers is finally presented. A novel noncertainty-equivalence controller based on the Immersion and Invariance paradigm for close-manoeuvring spacecraft formation flying in both circular and elliptical low-Earth reference orbits is introduced. The proposed control scheme achieves stabilization by immersing the plant dynamics into a target dynamical system (or manifold) that captures the desired dynamical behaviour. They key feature of this methodology is the addition of a new term to the classical certainty-equivalence control approach that, in conjunction with the parameter update law, is designed to achieve adaptive stabilization. This parameter has the ultimate task of shaping the manifold into which the adaptive system is immersed. The performance of the controller is proven stable via a Lyapunov-based analysis and Barbalat’s lemma. In order to evaluate the design of the controllers, test cases based on the physical and orbital features of the Prototype Research Instruments and Space Mission Technology Advancement (PRISMA) are implemented, extending the number of elements in the formation into scenarios with reconfigurations and on-orbit position switching in elliptical low-Earth reference orbits. An extensive analysis and comparison of the performance of the controllers in terms of total Δv and fuel consumption, with and without the effects of the CAS, is presented. These results show that the three proposed controllers allow the followers to asymptotically track the desired nominal trajectory and, additionally, those simulations including CAS show an effective decrease of collision risk during the performance of the manoeuvre.