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Paola Boito

Publications and source records attributed to Paola Boito.

14 recordsLinked to original sources
arXiv2026

Quantum block encoding for one-pair semiseparable matrices

Quantum block encoding (QBE) is a crucial step in the development of most quantum algorithms, as it provides an embedding of a given matrix into a suitable larger unitary matrix. Historically, the development of efficient techniques for QBE has mostly focused on sparse matrices; less effort has been devoted to data-sparse (e.g., rank-structured) matrices. In this work we examine a particular case of rank structure, namely, one-pair semiseparable matrices. We present a new block encoding approach that relies on a suitable factorization of the given matrix as the product of triangular and diagonal factors. To encode the matrix, the algorithm needs $2\log(N)+7$ ancillary qubits. Assuming that the data input oracles can be implemented with polylogarithmic depth, or that a QRAM input model is available, our proposed method requires $\mathcal{O}({\rm polylog} (N))$ time and has an error of $\mathcal{O}(N^2)$, where $N$ is the matrix size.

quant-ph↗
arXiv2026

Advances on the recovery of (perturbed) Cauchy matrices

Given a (possibly approximate) Cauchy matrix, how can we efficiently compute its generators? Expanding on previous work by Liesen and Luce [Linear Algebra Appl. 493 (2016) 261--280], we present a general family of algorithms for Cauchy parameter recovery, together with new error estimates. We also introduce a displacement-based approximation, which leads to a new algorithm for Cauchy parameter recovery. Numerical experiments show that the algorithm based on the displacement approximation is generally more accurate than the other algorithms.

math.NA↗
arXiv2025

Quasiseparable LU decay bounds for inverses of banded matrices

We develop new, easily computable exponential decay bounds for inverses of banded matrices, based on the quasiseparable representation of Green matrices. The bounds rely on a diagonal dominance hypothesis and do not require explicit spectral information. Numerical experiments and comparisons show that these new bounds can be advantageous especially for nonsymmetric or symmetric indefinite matrices.

math.NA↗
arXiv2024

Numerical Solution of Nonclassical Boundary Value Problems

We provide a new approach to obtain solutions of certain evolution equations set in a Banach space and equipped with nonlocal boundary conditions. From this approach we derive a family of numerical schemes for the approximation of the solutions. We show by numerical tests that these schemes are numerically robust and computationally efficient.

math.NA↗
arXiv2023

Computation of quasiseparable representations of Green matrices

The well-known Asplund theorem states that the inverse of a (possibly one-sided) band matrix $A$ is a Green matrix. In accordance with quasiseparable theory, such a matrix admits a quasiseparable representation in its rank-structured part. Based on this idea, we derive algorithms that compute a quasiseparable representation of $A^{-1}$ with linear complexity. Many inversion algorithms for band matrices exist in the literature. However, algorithms based on a computation of the rank structure performed theoretically via the Asplund theorem appear for the first time in this paper. Numerical experiments confirm complexity estimates and offer insight into stability properties.

math.NA↗
arXiv2022

Ranking nodes in directed networks via continuous-time quantum walks

Four new centrality measures for directed networks based on unitary, continuous-time quantum walks (CTQW) in $n$ dimensions -- where $n$ is the number of nodes -- are presented, tested and discussed. The main idea behind these methods consists in re-casting the classical HITS and PageRank algorithms as eigenvector problems for symmetric matrices, and using these symmetric matrices as Hamiltonians for CTQWs, in order to obtain a unitary evolution operator. The choice of the initial state is also crucial. Two options were tested: a vector with uniform occupation and a vector weighted w.r.t.~in- or out-degrees (for authority and hub centrality, respectively). Two methods are based on a HITS-derived Hamiltonian, and two use a PageRank-derived Hamiltonian. Centrality scores for the nodes are defined as the average occupation values. All the methods have been tested on a set of small, simple graphs in order to spot possible evident drawbacks, and then on a larger number of artificially generated larger-sized graphs, in order to draw a comparison with classical HITS and PageRank. Numerical results show that, despite some pathologies found in three of the methods when analyzing small graphs, all the methods are effective in finding the first and top ten nodes in larger graphs. We comment on the results and offer some insight into the good accordance between classical and quantum approaches.

quant-ph↗
arXiv2021

Computing the Reciprocal of a $ϕ$-function by Rational Approximation

In this paper we introduce a family of rational approximations of the reciprocal of a $ϕ$-function involved in the explicit solutions of certain linear differential equations, as well as in integration schemes evolving on manifolds. The derivation and properties of this family of approximations applied to scalar and matrix arguments are presented. Moreover, we show that the matrix functions computed by these approximations exhibit decaying properties comparable to the best existing theoretical bounds. Numerical examples highlight the benefits of the proposed rational approximations w.r.t.~the classical Taylor polynomials and other rational functions.

math.NA↗
arXiv2021

Quantum hub and authority centrality measures for directed networks based on continuous-time quantum walks

In this work we introduce, test and discuss three quantum methods for computing hub and authority centrality scores in directed networks. The methods are based on unitary, continuous-time quantum walks; the construction of a suitable Hermitian Hamiltonian is achieved by performing a quantum walk on the associated bipartite graph. Two methods, called CQAu and CQAw, use the same evolution operator, inspired by the classical HITS algorithm, but with different initial states; the computation of hub and authority scores is performed simultaneously. The third method, called CQG and inspired by classical PageRank, requires instead two separate runs with different evolution operators, one for hub and one for authority scores. The methods are tested on several directed graphs with different sizes and properties; a comparison with other well-established ranking algorithms is provided. CQAw emerges as the most reliable of the three methods and yields rankings that are largely compatible with results from HITS, although CQAu and CQG also present interesting features and potential for applications.

math.NA↗
arXiv2017

Efficient Solution of Parameter Dependent Quasiseparable Systems and Computation of Meromorphic Matrix Functions

In this paper we focus on the solution of shifted quasiseparable systems and of more general parameter dependent matrix equations with quasiseparable representations. We propose an efficient algorithm exploiting the invariance of the quasiseparable structure under diagonal shifting and inversion. This algorithm is applied to compute various functions of matrices. Numerical experiments show that this approach is fast and numerically robust.

math.NA↗
arXiv2017

A Real QZ Algorithm for Structured Companion Pencils

We design a fast implicit real QZ algorithm for eigenvalue computation of structured companion pencils arising from linearizations of polynomial rootfinding problems. The modified QZ algorithm computes the generalized eigenvalues of an $N\times N$ structured matrix pencil using $O(N)$ flops per iteration and $O(N)$ memory storage. Numerical experiments and comparisons confirm the effectiveness and the stability of the proposed method.

math.NA↗
arXiv2014

Implicit QR for Companion-like Pencils

A fast implicit QR algorithm for eigenvalue computation of low rank corrections of unitary matrices is adjusted to work with matrix pencils arising from polynomial zerofinding problems . The modified QZ algorithm computes the generalized eigenvalues of certain NxN rank structured matrix pencils using O(N^2) ops and O(N) memory storage. Numerical experiments and comparisons confirm the effectiveness and the stability of the proposed method.

math.NA↗
arXiv2012

Decay properties of spectral projectors with applications to electronic structure

Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our theory leads to a rigorous proof of the exponential off-diagonal decay ("nearsightedness") for the density matrix of gapped systems at zero electronic temperature in both orthogonal and non-orthogonal representations, thus providing a firm theoretical basis for the possibility of linear scaling methods in electronic structure calculations for non-metallic systems. We further discuss the case of density matrices for metallic systems at positive electronic temperature. A few other possible applications are also discussed.

math.NA↗
arXiv2011

Generalized companion matrix for approximate GCD

We study a variant of the univariate approximate GCD problem, where the coefficients of one polynomial f(x)are known exactly, whereas the coefficients of the second polynomial g(x)may be perturbed. Our approach relies on the properties of the matrix which describes the operator of multiplication by gin the quotient ring C[x]=(f). In particular, the structure of the null space of the multiplication matrix contains all the essential information about GCD(f; g). Moreover, the multiplication matrix exhibits a displacement structure that allows us to design a fast algorithm for approximate GCD computation with quadratic complexity w.r.t. polynomial degrees.

cs.SC↗