Related papers: Rayleigh Quotient Iteration, cubic convergence, an…
The absolute change in the Rayleigh quotient (RQ) for a Hermitian matrix with respect to vectors is bounded in terms of the norms of the residual vectors and the angle between vectors in [\doi{10.1137/120884468}]. We substitute…
In this study, a recursive solution technique in conjunction with generalized integrating factors is presented and applied to address first and second order linear differential equations. This approach demonstrates practical utility in…
Many theoretical problems in quantum technology can be formulated and addressed as constrained optimization problems. The most common quantum mechanical constraints such as, e.g., orthogonality of isometric and unitary matrices, CPTP…
We present the first application of quantics tensor trains (QTTs) and tensor cross interpolation (TCI) to the solution of a full set of self-consistent equations for multivariate functions, the so-called parquet equations. We show that the…
Using the Riesz-Feller fractional derivative, we apply the factorization algorithm to the fractional quantum harmonic oscillator along the lines previously proposed by Olivar-Romero and Rosas-Ortiz, extending their results. We solve the…
We introduce a new iterative method for computing solutions of elliptic equations with random rapidly oscillating coefficients. Similarly to a multigrid method, each step of the iteration involves different computations meant to address…
This work is on constrained large-scale non-convex optimization where the constraint set implies a manifold structure. Solving such problems is important in a multitude of fundamental machine learning tasks. Recent advances on Riemannian…
High-order tensor methods for solving both convex and nonconvex optimization problems have generated significant research interest, leading to algorithms with optimal global rates of convergence and local rates that are faster than Newton's…
This paper builds and extends on the authors' previous work related to the algorithmic tool, Cylindrical Algebraic Decomposition (CAD), and one of its core applications, Real Quantifier Elimination (QE). These topics are at the heart of…
We develop a new method for equality constrained optimization problems based on a sequential cubic programming framework. Each iteration utilizes a step decomposition based on the Jacobian of the constraints into a normal and a tangential…
While quantum computing provides an exponential advantage in solving system of linear equations, there is little work to solve system of nonlinear equations with quantum computing. We propose quantum Newton's method (QNM) for solving…
In this paper we consider the cubic regularization (CR) method for minimizing a twice continuously differentiable function. While the CR method is widely recognized as a globally convergent variant of Newton's method with superior iteration…
Quantum optimization algorithms hold the promise of solving classically hard, discrete optimization problems in practice. The requirement of encoding such problems in a Hamiltonian realized with a finite -- and currently small -- number of…
In this paper, we consider the nonconvex quadratically constrained quadratic programming (QCQP) with one quadratic constraint. By employing the conjugate gradient method, an efficient algorithm is proposed to solve QCQP that exploits the…
The computation of generalized inverses of quaternion matrices is a fundamental problem in quaternion linear algebra, with wide-ranging applications in signal processing, image restoration, and multidimensional data analysis. This paper…
Designing quantum circuits for ground state preparation is a fundamental task in quantum information science. However, standard Variational Quantum Algorithms (VQAs) are often constrained by limited ansatz expressivity and difficult…
A new algorithm, denoted by RSRR, is presented for solving large-scale nonlinear eigenvalue problems (NEPs) with a focus on improving the robustness and reliability of the solution, which is a challenging task in computational science and…
We analyze the problem of a quantum computer in a correlated environment protected from decoherence by QEC using a perturbative renormalization group approach. The scaling equation obtained reflects the competition between the dimension of…
Many applications of generalised linear models (GLMs) can be improved by applying constraints that impose assumptions on the associations or improve consistency of the estimators. Yet, there are still barriers to the implementation and…
We propose Quantum Riemannian Hamiltonian Descent (QRHD), a quantum algorithm for continuous optimization on Riemannian manifolds that extends Quantum Hamiltonian Descent (QHD) by incorporating geometric structure of the parameter space via…