Hybrid algorithms to solve linear systems of equations with limited qubit resources

The solution of linear systems of equations is a very frequent operation and thus important in many fields. The complexity using classical methods increases linearly with the size of equations. The HHL algorithm proposed by Harrow et al. achieves exponential acceleration compared with the best classical algorithm. However, it has a relatively high demand for qubit resources and the solution $\left| x \right\rangle$ is in a normalized form. Assuming that the eigenvalues of the coefficient matrix of the linear systems of equations can be represented perfectly by finite binary number strings, three hybrid iterative phase estimation algorithms (HIPEA) are designed based on the iterative phase estimation algorithm in this paper. The complexity is transferred to the measurement operation in an iterative way, and thus the demand of qubit resources is reduced in our hybrid algorithms. Moreover, the solution is stored in a classical register instead of a quantum register, so the exact unnormalized solution can be obtained. The required qubit resources in the three HIPEA algorithms are different. HIPEA-1 only needs one single ancillary qubit. The number of ancillary qubits in HIPEA-2 is equal to the number of nondegenerate eigenvalues of the coefficient matrix of linear systems of equations. HIPEA-3 is designed with a flexible number of ancillary qubits. The HIPEA algorithms proposed in this paper broadens the application range of quantum computation in solving linear systems of equations by avoiding the problem that quantum programs may not be used to solve linear systems of equations due to the lack of qubit resources.Comment: 22 pages, 6 figures, 6 tables, 48 equation

QudCom: Towards Quantum Compilation for Qudit Systems

Qudit-based quantum computation offers unique advantages over qubit-based systems in terms of noise mitigation capabilities as well as algorithmic complexity improvements. However, the software ecosystem for multi-state quantum systems is severely limited. In this paper, we highlight a quantum workflow for describing and compiling qudit systems. We investigate the design and implementation of a quantum compiler for qudit systems. We also explore several key theoretical properties of qudit computing as well as efficient optimization techniques. Finally, we provide demonstrations using physical quantum computers as well as simulations of the proposed quantum toolchain

Alternative Methods for Non-Linearity Estimation in High-Resolution Analog-to-Digital Converters

The evaluation of the linearity performance of a high resolution Analog-to- Digital Converter (ADC) by the Standard Histogram method is an outstanding challenge due to the requirement of high purity of the input signal and the high number of output data that must be acquired to obtain an acceptable accuracy on the estimation. These requirements become major application drawbacks when the measures have to be performed multiple times within long test flows and for many parts, and under an industrial environment that seeks to reduce costs and lead times as is the case in the New Space sector. This thesis introduces two alternative methods that succeed in relaxing the two previous requirements for the estimation of the Integral Nonlinearity (INL) parameter in ADCs. The methods have been evaluated by estimating the Integral Non-Linearity pattern by simulation using realistic high-resolution ADC models and experimentally by applying them to real high performance ADCs. First, the challenge of applying the Standard Histogram method for the evaluation of static parameters in high resolution ADCs and how the drawbacks are accentuated in the New Space industry is analysed, being a highly expensive method for an industrial environment where cost and lead time reduction is demanded. Several alternative methods to the Standard Histogram for estimating Integral Nonlinearity in high resolution ADCs are reviewed and studied. As the number of existing works in the literature is very large and addressing all of them is a challenge in itself, only those most relevant to the development of this thesis have been included. Methods based on spectral processing to reduce the number of data acquired for the linearity test and methods based on a double histogram to be able to use generators that do not meet the the purity requirement against the ADC to be tested are further analysed. Two novel contributions are presented in this work for the estimation of the Integral Nonlinearity in ADCs, as possible alternatives to the Standard Histogram method. The first method, referred to as SSA (Simple Spectral Approach), seeks to reduce the number of output data that need to be acquired and focuses on INL estimation using an algorithm based on processing the spectrum of the output signal when a sinusoidal input stimulus is used. This type of approach requires a much smaller number of samples than the Standard Histogram method, although the estimation accuracy will depend on how smooth or abrupt the ADC nonlinearity pattern is. In general, this algorithm cannot be used to perform a calibration of the ADC nonlinearity error, but it can be applied to find out between which limits it lies and what its approximate shape is. The second method, named SDH (Simplified Double Histogram)aims to estimate the Non-Linearity of the ADC using a poor linearity generator. The approach uses two histograms constructed from the two set of output data in response to two identical input signals except for a dc offset between them. Using a simple adder model, an extended approach named ESDH (Extended Simplified Double Histogram) addresses and corrects for possible time drifts during the two data acquisitions, so that it can be successfully applied in a non-stationary test environment. According to the experimental results obtained, the proposed algorithm achieves high estimation accuracy. Both contributions have been successfully tested in high-resolution ADCs with both simulated and real laboratory experiments, the latter using a commercial ADC with 14-bit resolution and 65Msps sampling rate (AD6644 from Analog Devices).La medida de la característica de linealidad de un convertidor analógicodigital (ADC) de alta resolución mediante el método estándar del Histograma constituye un gran desafío debido los requisitos de alta pureza de la señal de entrada y del elevado número de datos de salida que deben adquirirse para obtener una precisión aceptable en la estimación. Estos requisitos encuentran importantes inconvenientes para su aplicación cuando las medidas deben realizarse dentro de largos flujos de pruebas, múltiples veces y en un gran número de piezas, y todo bajo un entorno industrial que busca reducir costes y plazos de entrega como es el caso del sector del Nuevo Espacio. Esta tesis introduce dos métodos alternativos que consiguen relajar los dos requisitos anteriores para la estimación de los parámetros de no linealidad en los ADCs. Los métodos se han evaluado estimando el patrón de No Linealidad Integral (INL) mediante simulación utilizando modelos realistas de ADC de alta resolución y experimentalmente aplicándolos en ADCs reales. Inicialmente se analiza el reto que supone la aplicación del método estándar del Histograma para la evaluación de los parámetros estáticos en ADCs de alta resolución y cómo sus inconvenientes se acentúan en la industria del Nuevo Espacio, siendo un método altamente costoso para un entorno industrial donde se exige la reducción de costes y plazos de entrega. Se estudian métodos alternativos al Histograma estándar para la estimación de la No Linealidad Integral en ADCs de alta resolución. Como el número de trabajos es muy amplio y abordarlos todos es ya en sí un desafío, se han incluido aquellos más relevantes para el desarrollo de esta tesis. Se analizan especialmente los métodos basados en el procesamiento espectral para reducir el número de datos que necesitan ser adquiridos y los métodos basados en un doble histograma para poder utilizar generadores que no cumplen el requisito de precisión frente al ADC a medir. En este trabajo se presentan dos novedosas aportaciones para la estimación de la No Linealidad Integral en ADCs, como posibles alternativas al método estándar del Histograma. El primer método, denominado SSA (Simple Spectral Approach), busca reducir el número de datos de salida que es necesario adquirir y se centra en la estimación de la INL mediante un algoritmo basado en el procesamiento del espectro de la señal de salida cuando se utiliza un estímulo de entrada sinusoidal. Este tipo de enfoque requiere un número mucho menor de muestras que el método estándar del Histograma, aunque la precisión de la estimación dependerá de lo suave o abrupto que sea el patrón de no-linealidad del ADC a medir. En general, este algoritmo no puede utilizarse para realizar una calibración del error de no linealidad del ADC, pero puede aplicarse para averiguar entre qué límites se encuentra y cuál es su forma aproximada. El segundo método, denominado SDH (Simplified Double Histogram) tiene como objetivo estimar la no linealidad del ADC utilizando un generador de baja pureza. El algoritmo utiliza dos histogramas, construidos a partir de dos conjuntos de datos de salida en respuesta a dos señales de entrada idénticas, excepto por un desplazamiento constante entre ellas. Utilizando un modelo simple de sumador, un enfoque ampliado denominado ESDH (Extended Simplified Double Histogram) aborda y corrige las posibles derivas temporales durante las dos adquisiciones de datos, de modo que puede aplicarse con éxito en un entorno de prueba no estacionario. De acuerdo con los resultados experimentales obtenidos, el algoritmo propuesto alcanza una alta precisión de estimación. Ambas contribuciones han sido probadas en ADCs de alta resolución con experimentos tanto simulados como reales en laboratorio, estos últimos utilizando un ADC comercial con una resolución de 14 bits y una tasa de muestreo de 65Msps (AD6644 de Analog Devices)

Workshop on Fuzzy Control Systems and Space Station Applications

The Workshop on Fuzzy Control Systems and Space Station Applications was held on 14-15 Nov. 1990. The workshop was co-sponsored by McDonnell Douglas Space Systems Company and NASA Ames Research Center. Proceedings of the workshop are presented

Nonlinear Systems

Open Mathematics is a challenging notion for theoretical modeling, technical analysis, and numerical simulation in physics and mathematics, as well as in many other fields, as highly correlated nonlinear phenomena, evolving over a large range of time scales and length scales, control the underlying systems and processes in their spatiotemporal evolution. Indeed, available data, be they physical, biological, or financial, and technologically complex systems and stochastic systems, such as mechanical or electronic devices, can be managed from the same conceptual approach, both analytically and through computer simulation, using effective nonlinear dynamics methods. The aim of this Special Issue is to highlight papers that show the dynamics, control, optimization and applications of nonlinear systems. This has recently become an increasingly popular subject, with impressive growth concerning applications in engineering, economics, biology, and medicine, and can be considered a veritable contribution to the literature. Original papers relating to the objective presented above are especially welcome subjects. Potential topics include, but are not limited to: Stability analysis of discrete and continuous dynamical systems; Nonlinear dynamics in biological complex systems; Stability and stabilization of stochastic systems; Mathematical models in statistics and probability; Synchronization of oscillators and chaotic systems; Optimization methods of complex systems; Reliability modeling and system optimization; Computation and control over networked systems

Dynamics of Macrosystems; Proceedings of a Workshop, September 3-7, 1984

There is an increasing awareness of the important and persuasive role that instability and random, chaotic motion play in the dynamics of macrosystems. Further research in the field should aim at providing useful tools, and therefore the motivation should come from important questions arising in specific macrosystems. Such systems include biochemical networks, genetic mechanisms, biological communities, neutral networks, cognitive processes and economic structures. This list may seem heterogeneous, but there are similarities between evolution in the different fields. It is not surprising that mathematical methods devised in one field can also be used to describe the dynamics of another. IIASA is attempting to make progress in this direction. With this aim in view this workshop was held at Laxenburg over the period 3-7 September 1984. These Proceedings cover a broad canvas, ranging from specific biological and economic problems to general aspects of dynamical systems and evolutionary theory