Related papers: Explicit Inverse Confluent Vandermonde Matrices wi…

In this work, we present a method to exponentiate non-sparse indefinite low-rank matrices on a quantum computer. Given an operation for accessing the elements of the matrix, our method allows singular values and associated singular vectors…

In present article the self-contained derivation of eigenvalue inverse problem results is given by using a discrete approximation of the Schroedinger operator on a bounded interval as a finite three-diagonal symmetric Jacobi matrix. This…

In this note, we extend the Vandermonde with Arnoldi method recently advocated by P. D. Brubeck, Y. Nakatsukasa and L. N. Trefethen to dealing with the confluent Vandermonde matrix. To apply the Arnoldi process, it is critical to find a…

An exact invariant is derived for $n$-degree-of-freedom Hamiltonian systems with general time-dependent potentials. The invariant is worked out in two equivalent ways. In the first approach, we define a special {\it Ansatz\/} for the…

Time dependent quantum systems have become indispensable in science and its applications, particularly at the atomic and molecular levels. Here, we discuss the approximation of closed time dependent quantum systems on bounded domains, via…

The quantum singular value transformation is a powerful quantum algorithm that allows one to apply a polynomial transformation to the singular values of a matrix that is embedded as a block of a unitary transformation. This paper shows how…

In this work, new closed-form formulas for the matrix exponential are provided. Our method is direct and elementary, it gives tractable and manageable formulas not current in the extensive literature on this essential subject. Moreover,…

We consider the dynamic problems for the discrete systems with discrete time associated with finite and semi-infinite Jacobi matrices. The result of the paper is a procedure of association of special Hilbert spaces of functions, namely de…

For a quantum many-body problem, effective Hamiltonians that give exact eigenvalues in reduced model space usually have different expressions, diagrams and evaluation rules from effective transition operators that give exact transition…

Convex optimization problems arise naturally in quantum information theory, often in terms of minimizing a convex function over a convex subset of the space of hermitian matrices. In most cases, finding exact solutions to these problems is…

The results on the inversion of convolution operators and Toeplitz matrices in the 1-D (one dimensional) case are classical and have numerous applications. We consider a 2-D case of Toeplitz-block Toeplitz matrices, describe a minimal…

We consider Calder\'{o}n's inverse boundary value problems for a class of nonlinear Helmholtz Schr\"{o}dinger equations and Maxwell's equations in a bounded domain in $\R^n$. The main method is the higher-order linearization of the…

We develop a general approach for monitoring and controlling evolution of open quantum systems. In contrast to the master equations describing time evolution of density operators, here, we formulate a dynamical equation for the evolution of…

The computation of the matrix exponential is a ubiquitous operation in numerical mathematics, and for a general, unstructured $n\times n$ matrix it can be computed in $\mathcal{O}(n^3)$ operations. An interesting problem arises if the input…

We introduce a general method for transforming the equations of motion following from a Das-Jevicki-Sakita Hamiltonian, with boundary conditions, into a boundary value problem in one-dimensional quantum mechanics. For the particular case of…

In this paper we focus on high order finite element approximations of the electric field combined with suitable preconditioners, to solve the time-harmonic Maxwell's equations in waveguide configurations.The implementation of high order…

This paper tackles efficient methods for Bayesian inverse problems with priors based on Whittle--Mat\'ern Gaussian random fields. The Whittle--Mat\'ern prior is characterized by a mean function and a covariance operator that is taken as a…

This is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators. Among the advantages of this approach is that the Nevanlinna functions appear as elements of a transfer matrix…

We present a simple yet powerful framework for solving inverse problems by leveraging automatic differentiation. Our method is broadly applicable whenever a smooth cost function can be defined near the true solution, and a numerical…

The results on the inversion of convolution operators as well as Toeplitz (and block Toeplitz) matrices in the $1$-D (one-dimensional) case are classical and have numerous applications. Last year, we considered the $2$-D case of…