The efficiency of compiling high-level quantum algorithms into instruction sets native to quantum computers defines the moment in the future when we will be able to solve interesting and important problems on quantum computers. In my work I focus on the new methods for compiling single qubit operations that appear in many quantum algorithms into single qubit operations natively supported by several popular architectures. In addition, I study several questions related to synthesis and optimization of multiqubit operations. When studying the single qubit case, I consider two native instruction sets. The first one is Clifford+T; it is supported by conventional quantum computers implementing fault tolerance protocols based on concatenated and surface codes, and by topological quantum computers based on Ising anyons. The second instruction set is the one supported by topological quantum computers based on Fibonacci anyons. I show that in both cases one can use the number theoretic structure of the problem and methods of computational algebraic number theory to achieve improvements over the previous state of the art by factors ranging from 10 to 1000 for instances of the problem interesting in practice. This order of improvement might make certain interesting quantum computations possible several years earlier. The work related to multiqubit operations is on exact synthesis and optimization of Clifford+T and Clifford circuits. I show an exact synthesis algorithm for unitaries generated by Clifford+T circuits requiring exponentially less number of gates than previous state of the art. For Clifford circuits two directions are studied: the algorithm for finding optimal circuits acting on a small number of qubits and heuristics for larger circuits optimization. The techniques developed allows one to reduce the size of encoding and decoding circuits for quantum error correcting codes by 40-50\% and also finds their applications in randomized benchmarking protocols.

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Doctor of Philosophy / Electrical and Computer Engineering / Sanjoy Das / Quantum circuit synthesis is one of the major areas of current research in the field of quantum computing. Analogous to its Boolean counterpart, the task involves constructing arbitrary quantum gates using only those available within a small set of universal gates that can be realized physically. However, unlike the latter, there are an infinite number of single qubit quantum gates, all of which constitute the special unitary group SU(2). Realizing any given single qubit gate using a given universal gate family is a complex task. Although gates can be synthesized to arbitrary degree of precision as long as the set of finite strings of the gate family is a dense subset of SU(2), it is desirable to accomplish the highest level of precision using only the minimum number of universal gates within the string approximation. Almost all algorithms that have been proposed for this purpose are based on the Solovay-Kitaev algorithm. The crux of the Solovay-Kitaev algorithm is the use of a procedure to decompose a given quantum gate into a pair of group commutators with the pair being synthesized separately. The Solovay-Kitaev algorithm involves group commutator decomposition in a recursive manner, with a direct approximation of a gate into a string of universal gates being performed only at the last level, i.e. in the leaf nodes of the search tree representing the execution of the Solovay-Kitaev algorithm. The main contribution of this research is in integrating conventional optimization procedures within the Solovay-Kitaev algorithm. Two specific directions of research have been studied. Firstly, optimization is incorporated within the group commutator decomposition, so that a more optimal pair of group commutators are obtained. As the degree of precision of the synthesized gate is explicitly minimized by means of this optimization procedure, the enhanced algorithm allows for more accurate quantum gates to be synthesized than what the original Solovay-Kitaev algorithm achieves. Simulation results with random gates indicate that the obtained accuracy is an order of magnitude better than before. Two versions of the new algorithm are examined, with the optimization in the first version being invoked only at the bottom level of Solovay-Kitaev algorithm and when carried out across all levels of the search tree in the next. Extensive simulations show that the second version yields better results despite equivalent computation times. Theoretical analysis of the proposed algorithm is able to provide a more formal, quantitative explanation underlying the experimentally observed phenomena. The other direction of investigation of this research involves formulating the group commutator decomposition in the form of bi-criteria optimization. This phase of research relaxed the equality constraint in the previous approach and with relaxation, a bi-criteria optimization is proposed. This optimization algorithm is new and has been devised primarily when the objective needs to be relaxed in different stages. This bi-criteria approach is able to provide comparably accurate synthesis as the previous approach.

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Translinear circuits are circuits in which the exponential relationship between the output current and input voltage of a circuit element is exploited to realize various algebraic or differential equations. This thesis is concerned with a subclass of translinear circuits, in which the basic translinear element, called a multiple-input translinear element (MITE), has an output current that is exponentially related to a weighted sum of its input voltages. MITE networks can be used for the implementation of the same class of functions as traditional translinear circuits. The implementation of algebraic or (algebraic) differential equations using MITEs can be reduced to the implementation of the product-of-power-law (POPL) relationships, in which an output is given by the product of inputs raised to different powers. Hence, the synthesis of POPL relationships, and their optimization with respect to the relevant cost functions, is very important in the theory of MITE networks. In this thesis, different constraints on the topology of POPL networks that result in desirable system behavior are explored and different methods of synthesis, subject to these constraints, are developed. The constraints are usually conditions on certain matrices of the network, which characterize the weights in the relevant MITEs. Some of these constraints are related to the uniqueness of the operating point of the network and the stability of the network. Conditions that satisfy these constraints are developed in this work. The cost functions to be minimized are the number of MITEs and the number of input gates in each MITE. A complete solution to POPL network synthesis is presented here that minimizes the number of MITEs first and then minimizes the number of input gates to each MITE. A procedure for synthesizing POPL relationships optimally when the number of gates is minimal, i.e., 2, has also been developed here for the single--output case. A MITE structure that produces the maximum number of functions with minimal reconfigurability is developed for use in MITE field--programmable analog arrays. The extension of these constraints to the synthesis of linear filters is also explored, the constraint here being that the filter network should have a unique operating point in the presence of nonidealities. Synthesis examples presented here include nonlinear functions like the arctangent and the gaussian function which find application in analog implementations of particle filters. Synthesis of dynamical systems is presented here using the examples of a Lorenz system and a sinusoidal oscillator. The procedures developed here provide a structured way to automate the synthesis of nonlinear algebraic functions and differential equations using MITEs.

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El efecto multipactor de ruptura en RF ha sido objeto de numerosos estudios desde hace más de 80 años, a partir del desarrollo de los primeros aceleradores de partículas en la primera mitad del siglo XX. A mediados de ese siglo, con el desarrollo de fuentes de alta potencia para aplicaciones radar y la llegada de los satélites artificiales, la investigación del multipactor cobró una considerable relevancia, al convertirse este fenómeno en un riesgo determinante para costosos proyectos comerciales. Las guías de onda con secciones rectas canónicas, como las rectangulares o las coaxiales, han sido tradicionalmente las más utilizadas en dispositivos de microondas. Sus principales ventajas son que sus campos electromagnéticos pueden resolverse analíticamente, lo que permite su aplicación directa en diseños complejos, y la simplicidad de su fabricación. Pero las capacidades de computación y las prestaciones de los algoritmos se han multiplicado con los años, lo que ha permitido ampliar el espectro de posibles topologías a geometrías casi arbitrarias, ofreciendo al diseñador una mayor libertad creativa. En todo caso, gran parte de los dispositivos de microondas actuales siguen confiando en la madurez y fiabilidad de las tecnologías de guía de onda tradicionales, que no requieren una inversión adicional en equipos de fabricación. La supresión del efecto multipactor es la motivación para arriesgarse a probar topologías de guía de onda innovadoras, como la guía en forma de cuña. Es en este contexto donde este trabajo de doctorado pretende ofrecer una contribuci'on. En primer lugar, se ha desarrollado un modelo numérico para predecir el efecto multipactor de ruptura en guías de onda huecas en forma de cuña. Esta herramienta ha permitido la identificación de criterios óptimos de diseño. Así mismo, se ha adaptado un método de síntesis de filtros paso-banda en guía rectangular para poder realizar un diseño similar pero basado en la nueva topología. Como culminación, las estructuras diseñadas se han fabricado y medido, con el fin de comprobar sus prestaciones electromagnéticas y su sensibilidad al efecto multipactor. Se ha registrado además una patente para proteger estos nuevos filtros. En resumen, el trabajo ha abarcado el ciclo de actividades relacionadas con el desarrollo industrial completo de un dispositivo pasivo de microondas: investigación básica, análisis, diseño, fabricación y calificación con medidas en el laboratorio. Estas medidas han comprobado la mejora prevista en los umbrales de multipactor de los filtros de microondas con topología en forma de cu¿na, y han confirmado que pueden ofrecer respuestas en frecuencia similares a aquellas de filtros basados en una guía de onda rectangular equivalente. Las implicaciones de los resultados han sido evaluadas a fondo y resumidas en este documento. Como observación final, se ha intentado redactar esta investigación de manera que refleje el proceso natural de aprendizaje, mostrando los aciertos y errores experimentados en el camino, todos los cuales han conducido al resultado final. Este reto no hubiera sido posible sin el apoyo y compromiso de varios profesionales de diferentes centros de investigación e industrias europeas (Universidad Politécnica de Valencia, Universidad de Valencia, Agencia Espacial Europea, Thales Alenia Espacio Espa¿na, Technische Universit¿at Darmstadt, 'Ecole Polythecnique F'ed'erale de Lausanne, Tesat, Aurora Software and Testing y Val Space Consortium), a los cuales estoy agradecido. / The multipactor RF breakdown effect has been object of numerous studies for over 80 years, since the development of the first particle accelerators in the beginning of the 20th century. Around the middle of that century, with the development of high power sources for radar applications and with the emergence of the artificial satellites, a new impulse was given to the multipactor research, since it became a risk for expensive commercial projects. Traditionally, waveguides with canonical cross sections, like rectangular or coaxial ones, have been the building blocks of most microwave devices. Their main advantages are that their electromagnetic fields can be solved analytically, enabling their direct application in complex designs, as well as their manufacturing simplicity. But over the years the computation capabilities and algorithms have continuously evolved, which has broadened the spectrum of possible topologies to almost arbitrary geometries, offering the designer more room for creativity. However, most of the current microwave devices still trust on the mature canonical waveguide technologies, which do not require an additional investment in manufacturing equipment. The suppression of the multipactor effect is the motivation for considering an innovative waveguide topology, like the wedge-shaped waveguide. It is within this context where this PhD work aims to offer a contribution. On the one hand, a numerical model for predicting the multipactor breakdown effect in wedge-shaped hollow waveguides has been developed. This tool has aided in the derivation of optimised design criteria. On the other hand, a bandpass filter synthesis method for rectangular waveguide has been adapted in order to calculate a similar design based on the new topology. As a culmination, the designed structures have been manufactured and tested, in order to verify their electromagnetic performance and their multipactor sensibility. A patent was also filed to protect these new filters. In short, this work has comprised the cycle of activities related to the whole industrial development of a passive microwave device: basic research, analysis, design, manufacturing and qualification through testing. These measurements have verified the predicted improvement in the multipactor thresholds of microwave filters with wedge-shaped topology, and have confirmed that they can offer similar frequency responses to the equivalent rectangular waveguide ones. The implications of the results have been thoroughly evaluated and summarised in this document. As a final remark, this research document has been drafted to reflect the natural learning process, and to show the rights and wrongs experienced in the way, which all have led to the final result. Such an endeavour would not have been possible without the support and commitment of several professionals from different European research centres and industries (Universidad Polit'ecnica de Valencia, Universidad de Valencia, European Space Agency, Thales Alenia Espacio Spain, Technische Universit¿at Darmstadt, 'Ecole Polythecnique F'ed'erale de Lausanne, Tesat, Aurora Software and Testing and Val Space Consortium), for which I am grateful. / Hueso González, J. (2013). Analysis of wedge-shaped waveguides and design of multipactor-resistant microwave bandpass filters. Análisis de guías de onda en forma de cuña y diseño de filtros de microondas paso-banda resistentes al efecto multipactor [Tesis doctoral]. Editorial Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/33750

RF and microwave applications represent one of the fastest-growing segments of the high performance electronics market, where ongoing innovation is critical. Manufacturers compete intensively to meet market needs with reduced cost, size, weight and many other performance criteria demands. Under this scenario, transmission lines based on metamaterial concepts can be considered a very interesting alternative to the conventional transmission lines. They are more compact (compatible with planar manufacturing processes) and present higher degrees of design flexibility. Furthermore, metamaterial transmission lines can also provide many other unique properties not achievable with ordinary transmission lines, such as dispersion or impedance engineering. Nevertheless, the impact in the industry is still not relevant, mostly due to the complexity of the related synthesis and design procedures. These procedures are mainly based on the engineer’s experience, with the help of costly full-wave electromagnetic (EM) simulators and parameter extraction methods. The aim of this thesis is to contribute to simplify and speed up the synthesis and design procedures of artificial transmission lines. In particular, the lines obtained by periodically loading a conventional transmission line with electrically small resonators, such as split ring resonators (SSRs) or its complementary particle (CSRR). The design procedure is automated by using Space Mapping techniques. In contrast to other alternative methods, real synthesis is found from the circuit schematic (that provides a given target response) and without need of human intervention. Some efforts to make the method practical and useful have been carried out. Given a certain target response, it is determined whether it can be physically implemented with a chosen technology, and hence proceeding next to find the synthesis, or not. For this purpose, a two-step Aggressive Space Mapping approach is successfully proposed. In contrast to other methods, the real synthesis is found from certain target circuit values (corresponding to the equivalent circuit model that characterizes the structure to be synthesized). Different efforts have been carried out in order to implement a useful and practical method. Some of them were focused to determine if, given certain circuit parameters (which determine the target response) and certain given technology specifications (permittivity and height of the substrate, technology limits), that response is physically realizable (convergence region). This technique was successfully formulated and it is known as “Two-Step Aggressive Space Mapping Approach”. In this work, the latest improvements made till date, from the synthesis of basic unit cells until different applications and kinds of metamaterial-based circuits, are presented. The results are promising and prove the validity of the method, as well as its potential application to other basic cells and more complex designs. The general knowledge gained from these cases of study can be considered a good base for a coming implementation in commercial software tools, which can help to improve its competitiveness in markets, and also contribute to a more general use of this technology. / Rodríguez Pérez, AM. (2014). Synthesis of Planar Microwave Circuits based on Metamaterial Concepts through Aggressive Space Mapping [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/48465