Efficient implementation of iterative polynomial matrix EVD algorithms exploiting structural redundancy and parallelisation

A number of algorithms are capable of iteratively calculating a polynomial matrix eigenvalue decomposition (PEVD), which is a generalisation of the EVD and will diagonalise a parahermitian polynomial matrix via paraunitary operations. While offering promising results in various broadband array processing applications, the PEVD has seen limited deployment in hardware due to the high computational complexity of these algorithms. Akin to low complexity divide-and-conquer (DaC) solutions to eigenproblems, this paper addresses a partially parallelisable DaC approach to the PEVD. A novel algorithm titled parallel-sequential matrix diagonalisation exhibits significantly reduced algorithmic complexity and run-time when compared with existing iterative PEVD methods. The DaC approach, which is shown to be suitable for multi-core implementation, can improve eigenvalue resolution at the expense of decomposition mean squared error, and offers a trade-off between the approximation order and accuracy of the resulting paraunitary matrices

Polynomial matrix eigenvalue decomposition techniques for multichannel signal processing

Polynomial eigenvalue decomposition (PEVD) is an extension of the eigenvalue decomposition (EVD) for para-Hermitian polynomial matrices, and it has been shown to be a powerful tool for broadband extensions of narrowband signal processing problems. In the context of broadband sensor arrays, the PEVD allows the para-Hermitian matrix that results from the calculation of a space-time covariance matrix of the convolutively mixed signals to be diagonalised. Once the matrix is diagonalised, not only can the correlation between different sensor signals be removed but the signal and noise subspaces can also be identified. This process is referred to as broadband subspace decomposition, and it plays a very important role in many areas that require signal separation techniques for multichannel convolutive mixtures, such as speech recognition, radar clutter suppression, underwater acoustics, etc. The multiple shift second order sequential best rotation (MS-SBR2) algorithm, built on the most established SBR2 algorithm, is proposed to compute the PEVD of para-Hermitian matrices. By annihilating multiple off-diagonal elements per iteration, the MS-SBR2 algorithm shows a potential advantage over its predecessor (SBR2) in terms of the computational speed. Furthermore, the MS-SBR2 algorithm permits us to minimise the order growth of polynomial matrices by shifting rows (or columns) in the same direction across iterations, which can potentially reduce the computational load of the algorithm. The effectiveness of the proposed MS-SBR2 algorithm is demonstrated by various para-Hermitian matrix examples, including randomly generated matrices with different sizes and matrices generated from source models with different dynamic ranges and relations between the sources’ power spectral densities. A worked example is presented to demonstrate how the MS-SBR2 algorithm can be used to strongly decorrelate a set of convolutively mixed signals. Furthermore, the performance metrics and computational complexity of MS-SBR2 are analysed and compared to other existing PEVD algorithms by means of numerical examples. Finally, two potential applications of theMS-SBR2 algorithm, includingmultichannel spectral factorisation and decoupling of broadband multiple-input multiple-output (MIMO) systems, are demonstrated in this dissertation

Polynomial matrix algebra with applications

[Abstract unavailable

Polynomial eigenvalue decomposition for multichannel broadband signal processing

This article is devoted to the polynomial eigenvalue decomposition (PEVD) and its applications in broadband multichannel signal processing, motivated by the optimum solutions provided by the eigenvalue decomposition (EVD) for the narrow-band case [1], [2]. In general, the successful techniques from narrowband problems can also be applied to broadband ones, leading to improved solutions. Multichannel broadband signals arise at the core of many essential commercial applications such as telecommunications, speech processing, healthcare monitoring, astronomy and seismic surveillance, and military technologies like radar, sonar and communications [3]. The success of these applications often depends on the performance of signal processing tasks, including data compression [4], source localization [5], channel coding [6], signal enhancement [7], beamforming [8], and source separation [9]. In most cases and for narrowband signals, performing an EVD is the key to the signal processing algorithm. Therefore, this paper aims to introduce PEVD as a novel mathematical technique suitable for many broadband signal processing applications

Source separation and beamforming

As part of the last day of the UDRC 2021 summer school, this presentation provides an overview over polynomial matrix methods. The use of polynomial matrices is motivated through a number of broadband multichannel problems, involving space-time covariance matrices, filter banks, or wideband MIMO systems. We extend the utility of EVD from narrowband to broadband solutions via a number of factorisation algorithms belonging to the second order sequential rotation or sequential matrix diagonalisation families of algorithms. In a second part of this presentation, a number of application areas are explored, ranging from precoder and equaliser design for broadband MIMO communications systems, to broadband angle of arrival estimation, broadband beamforming, and the problem of identifying source-sensor transfer paths from the second order statistics of the sensor signals

Quantum computation: theory and implementation at IBM Q

This end of degree project constitutes an introduction to Quantum Computation. It presents a combination of theoretical concepts, mainly based in the guidelines of “Quantum Computation and Quantum Information” of Michael A. Nielsen & Isaac L. Chuang [NC02], and the implementation of some of them at IBM’s online quantum computers [ibm18]. The aim is therefore to realize a first approach to some basic concepts of Quantum Computation and Quantum Information and put them in practice. Particularly, after the introduction of qubits and essential ideas about entanglement and multiple qubit states, the 14-qubit quantum computer IBM Q Melbourne was employed to generate both Bell and GHZ states. After that, a quantum/classical hybrid algorithm known as Variational Quantum Eigensolver (VQE) [Com16] was introduced as a crucial tool for the next two targets of the project. The first of them consists on exhaustively analyzing and solving an optimization problem named Exact cover problem [Gal17]. The second one relates to find the ground state of a bidimensional Ising model and study the evolution of bipartite entanglement, as measured by the von Neumann entropy, in the approach of the system to its ground state.El presente trabajo de fin de grado constituye una introduccion a la Com- ´ putacion Cu ´ antica. En ´ el se presenta una combinaci ´ on de conceptos te ´ oricos, ´ basados principalmente en las directrices del libro “Quantum Computation and Quantum Information” of Michael A. Nielsen & Isaac L. Chuang [NC02], y la implemantacion de algunos de ellos en los ordenadores cu ´ anticos de ´ IBM disponibles en la nube [ibm18]. El objetivo del proyecto es, por tanto, realizar una primera aproximacion a algunos de los conceptos b ´ asicos de la ´ Computacion Cu ´ antica e Informaci ´ on Cu ´ antica y ponerlos en pr ´ actica. En ´ particular, tras introducir el concepto de cubit y las ideas esenciales sobre en- ´ trelazamiento y estados formados por multiples cubits, el ordenador cu ´ antico ´ de 14 cubits ´ IBM Q Melbourne fue empleado para generar estados de Bell y GHZ. Tras esto, se introdujo un algoritmo h´ıbrido clasico/cu ´ antico cono- ´ cido como Variational Quantum Eigensolver (VQE) [Com16], el cual es una herramienta fundamental a la hora de cumplimentar los siguientes dos objetivos del proyecto. El primero de ellos consiste en el analisis exhaustivo y la ´ resolucion de un problema de optimizaci ´ on conocido como ´ Exact cover problem [Gal17]. El segundo se fundamenta en encontrar el estado fundamental de un modelo de Ising bidimensional y estudiar la evolucion del entrelazamiento ´ bipartito, medido mediante la entrop´ıa de von Neumann, en la evolucion del ´ sistema hacia el estado fundamental

Variational quantum architectures. Applications for noisy intermediate-scale quantum computers

[eng] Quantum algorithms showing promising speedups with respect to their classical counterparts already exist. However, noise limits the quantum circuit depth, making the practical implementation of many such quantum algorithms impossible nowadays. In this sense, variational quantum algorithms offer a new approach, reducing the requisites of quantum computational resources at the expense of classical optimization. Disciplines in which variational quantum algorithms may have practical applications include simulation of quantum systems, solving large systems of linear equations, combinatorial optimization, data compression, quantum state diagonalization, among others. This thesis studies different variational quantum algorithm applications. In Chapter 1, we introduce the main building blocks of variational quantum algorithms. In Chapter 2, we benchmark the seminal variational quantum eigensolver algorithm for condensed matter systems. In Chapter 3, we explore how the task of compressing quantum information is affected by data encoding in variational quantum circuits. In Chapter 4, we propose a novel variational quantum algorithm to compute the singular values of pure bipartite states. In Chapter 5, we develop a new variational quantum algorithm to solve linear systems of equations. Finally, in Chapter 6, we implement quantum generative adversarial networks for generative modeling tasks. The conclusions of this thesis are exposed in Chapter 7. Furthermore, supplementary material can be found in the appendices. Appendix A provides an introduction to Qibo, a framework for quantum simulation. Appendix B presents some results related to the Solovay-Kitaev theorem. Extra results from Chapter 5 and Chapter 6 can be found in Appendix C and Appendix D, respectively.[spa] Algoritmos cuánticos mostrando prometedoras ventajas respecto sus contrapartes clásicas ya existen. Sin embargo, el ruido limita la profundidad de los circuitos cuánticos, lo que hace imposible la aplicación práctica de muchos de estos algoritmos cuánticos en la actualidad. En este sentido, los algoritmos cuánticos variacionales ofrecen un nuevo enfoque, reduciendo los requisitos de recursos computacionales cuánticos a expensas de optimización clásica. Disciplinas en las que los algoritmos cuánticos variacionales pueden tener aplicaciones prácticas incluyen la simulación de sistemas cuánticos, la resolución de grandes sistemas de ecuaciones lineales, la optimización combinatoria, la compresión de datos y la diagonalización de estados cuánticos, entre otras. Esta tesis estudia diferentes aplicaciones de los algoritmos cuánticos variacionales. En el Capítulo 1, presentamos los principales bloques de construcción de los algoritmos cuánticos variacionales. En el Capítulo 2, evaluamos el algoritmo “variational quantum eigensolver” para sistemas de materia condensada. En el capítulo 3, exploramos cómo la tarea de comprimir la información cuántica se ve afectada por la codificación de datos en los circuitos cuánticos variacionales. En el Capítulo 4, proponemos un novedoso algoritmo cuántico variacional para calcular los valores singulares de los estados bipartitos puros. En el Capítulo 5, desarrollamos un nuevo algoritmo cuántico variacional para resolver sistemas lineales de ecuaciones. Finalmente, en el Capítulo 6, implementamos redes generativas adversarias cuánticas para tareas de modelado generativo. Las conclusiones de esta tesis se exponen en el Capítulo 7. Además, se puede encontrar material complementario en los apéndices. El Apéndice A ofrece una introducción a Qibo, un software para la simulación cuántica. El Apéndice B presenta algunos resultados relacionados con el teorema de Solovay-Kitaev. En el Apéndice C y en el Apéndice D se pueden encontrar resultados adicionales del Capítulo 5 y del Capítulo 6, respectivamente

The University Defence Research Collaboration In Signal Processing

This chapter describes the development of algorithms for automatic detection of anomalies from multi-dimensional, undersampled and incomplete datasets. The challenge in this work is to identify and classify behaviours as normal or abnormal, safe or threatening, from an irregular and often heterogeneous sensor network. Many defence and civilian applications can be modelled as complex networks of interconnected nodes with unknown or uncertain spatio-temporal relations. The behavior of such heterogeneous networks can exhibit dynamic properties, reflecting evolution in both network structure (new nodes appearing and existing nodes disappearing), as well as inter-node relations. The UDRC work has addressed not only the detection of anomalies, but also the identification of their nature and their statistical characteristics. Normal patterns and changes in behavior have been incorporated to provide an acceptable balance between true positive rate, false positive rate, performance and computational cost. Data quality measures have been used to ensure the models of normality are not corrupted by unreliable and ambiguous data. The context for the activity of each node in complex networks offers an even more efficient anomaly detection mechanism. This has allowed the development of efficient approaches which not only detect anomalies but which also go on to classify their behaviour