Spectral Theory of Sparse Non-Hermitian Random Matrices

Sparse non-Hermitian random matrices arise in the study of disordered physical systems with asymmetric local interactions, and have applications ranging from neural networks to ecosystem dynamics. The spectral characteristics of these matrices provide crucial information on system stability and susceptibility, however, their study is greatly complicated by the twin challenges of a lack of symmetry and a sparse interaction structure. In this review we provide a concise and systematic introduction to the main tools and results in this field. We show how the spectra of sparse non-Hermitian matrices can be computed via an analogy with infinite dimensional operators obeying certain recursion relations. With reference to three illustrative examples --- adjacency matrices of regular oriented graphs, adjacency matrices of oriented Erd\H{o}s-R\'{e}nyi graphs, and adjacency matrices of weighted oriented Erd\H{o}s-R\'{e}nyi graphs --- we demonstrate the use of these methods to obtain both analytic and numerical results for the spectrum, the spectral distribution, the location of outlier eigenvalues, and the statistical properties of eigenvectors.Comment: 60 pages, 10 figure

From Quantum Entanglement to Interactions of Elementary Excitations in Coupled Spin Chains : An Introduction to Numerical Many-Body Physics with Matrix Product States and Tensor Networks

Matrix product states provide an efficient parametrisation of low-entanglement many-body quantum states. In this thesis, the underlying theory is developed from scratch, requiring only basic notions of quantum mechanics and quantum information theory. A full introduction to matrix product state algebra and matrix product operators is given, culminating in the derivation of the density matrix renormalisation group algorithm. The latter provides a simple variational scheme to determine the ground state of arbitrary one-dimensional many-body quantum systems with supreme precision. As an application of matrix-product state technology, the kernel polynomial method is introduced in detail as a state-of-the art numerical tool to find the spectral function or the dynamical correlator of a given quantum system. This in turn gives access to the elementary excitations of the system, such that the locations of the low-energy eigenstates can be studied directly in real space. To illustrate those theoretical tools concretely, the ground state energy, the entanglement entropy and the elementary excitations of a simple interface model of a Heisenberg ferromagnet and a Heisenberg antiferromagnet are studied. By changing the location of the model in parameter space, the dependence of the above-mentioned quantities on the transverse field and the coupling strength is investigated. Most notably, we find that the entanglement entropy characteristic to the antiferromagnetic ground state stretches across the interface into the ferromagnetic half-chain. The dependence of the physics on the value of the coupling strength is, overall, small, with exception of the appearance of a boundary mode whose eigenenergy grows with the coupling. A comparison with a localised edge field shows however that the boundary mode is a true interaction effect of the two half-chains. Various algorithmic and physics extensions of the present project are discussed, such that the code written as part of this thesis could be turned into a state-of-the-art MPS library with managable effort. In particular, an application of the kernel polynomial method to calculate finite-temperature correlators is derived in detail

Polynomial matrix algebra with applications

[Abstract unavailable

The Application of Spectral Clustering in Drug Discovery

The application of clustering algorithms to chemical datasets is well established and has been reviewed extensively. Recently, a number of ‘modern’ clustering algorithms have been reported in other fields. One example is spectral clustering, which has yielded promising results in areas such as protein library analysis. The term spectral clustering is used to describe any clustering algorithm that utilises the eigenpairs of a matrix as the basis for partitioning a dataset. This thesis describes the development and optimisation of a non-overlapping spectral clustering method that is based upon a study by Brewer. The initial version of the spectral clustering algorithm was closely related to Brewer’s method and used a full matrix diagonalisation procedure to identify the eigenpairs of an input matrix. This spectral clustering method was compared to the k-means and Ward’s algorithms, producing encouraging results, for example, when coupled with extended connectivity fingerprints, this method outperformed the other clustering algorithms according to the QCI measure. Although the spectral clustering algorithm showed promising results, its operational costs restricted its application to small datasets. Hence, the method was optimised in successive studies. Firstly, the effect of matrix sparsity on the spectral clustering was examined and showed that spectral clustering with sparse input matrices can lead to an improvement in the results. Despite this improvement, the costs of spectral clustering remained prohibitive, so the full matrix diagonalisation procedure was replaced with the Lanczos algorithm that has lower associated costs, as suggested by Brewer. This method led to a significant decrease in the computational costs when identifying a small number of clusters, however a number of issues remained; leading to the adoption of a SVD-based eigendecomposition method. The SVD-based algorithm was shown to be highly efficient, accurate and scalable through a number of studies

Induced Ginibre ensemble of random matrices and quantum operations

A generalisation of the Ginibre ensemble of non-Hermitian random square matrices is introduced. The corresponding probability measure is induced by the ensemble of rectangular Gaussian matrices via a quadratisation procedure. We derive the joint probability density of eigenvalues for such induced Ginibre ensemble and study various spectral correlation functions for complex and real matrices, and analyse universal behaviour in the limit of large dimensions. In this limit the eigenvalues of the induced Ginibre ensemble cover uniformly a ring in the complex plane. The real induced Ginibre ensemble is shown to be useful to describe statistical properties of evolution operators associated with random quantum operations, for which the dimensions of the input state and the output state do differ.Comment: 2nd version, 34 pages, 5 figure