Related papers: Loschmidt echo approach to Krylov-subspace approxi…
We obtain an expression for the error in the approximation of $f(A) \boldsymbol{b}$ and $\boldsymbol{b}^T f(A) \boldsymbol{b}$ with rational Krylov methods, where $A$ is a symmetric matrix, $\boldsymbol{b}$ is a vector and the function $f$…
The notion of Loschmidt echo (also called "quantum fidelity") has been introduced in order to study the (in)-stability of the quantum dynamics under perturbations of the Hamiltonian. It has been extensively studied in the past few years in…
Model reduction is a powerful tool in dealing with numerical simulation of large scale dynamic systems for studying complex physical systems. Two major types of model reduction methods for linear time-invariant dynamic systems are Krylov…
We consider iterative (`turbo') algorithms for compressed sensing. First, a unified exposition of the different approaches available in the literature is given, thereby enlightening the general principles and main differences. In particular…
The Loschmidt echo is a measure of the stability and reversibility of quantum evolution under perturbations of the Hamiltonian. One of the expected and most relevant characteristics of this quantity for chaotic systems is an exponential…
Quantum Krylov subspace methods can extract ground and excited states by diagonalizing the Hamiltonian in a compact variational space. In practice, these spaces are almost always generated by real or imaginary time evolution, forcing a…
The Cahn-Hilliard equation has been widely employed within various mathematical models in physics, chemistry and engineering. Explicit stabilized time stepping methods can be attractive for time integration of the Cahn-Hilliard equation,…
Quantum Krylov subspace diagonalization (QKSD) algorithms provide a low-cost alternative to the conventional quantum phase estimation algorithm for estimating the ground and excited-state energies of a quantum many-body system. While QKSD…
Despite the rapid development of quantum science and technology, errors are inevitable and play a crucial role in quantum simulation and quantum computation. In quantum chaotic systems, coherent errors arising from imperfect Hamiltonian…
We provide convergence rates for Krylov subspace solutions to the trust-region and cubic-regularized (nonconvex) quadratic problems. Such solutions may be efficiently computed by the Lanczos method and have long been used in practice. We…
Quantum Krylov algorithms have emerged as a promising approach for ground-state energy estimation in the near-term quantum computing era. A major challenge, however, lies in their inherently substantial sampling cost, primarily due to the…
Randomized block Krylov subspace methods form a powerful class of algorithms for computing the extreme eigenvalues of a symmetric matrix or the extreme singular values of a general matrix. The purpose of this paper is to develop new…
For several classes of mathematical models that yield linear systems, the splitting of the matrix into its Hermitian and skew Hermitian parts is naturally related to properties of the underlying model. This is particularly so for…
We are interested in the simulation of open quantum systems governed by the Lindblad master equation in an infinite-dimensional Hilbert space. To simulate the solution of this equation, the standard approach involves two sequential…
We propose a class of randomized quantum Krylov diagonalization (rQKD) algorithms capable of solving the eigenstate estimation problem with modest quantum resource requirements. Compared to previous real-time evolution quantum Krylov…
As computational machines become larger and more complex, the probability of hardware failure rises. ``Silent errors'', or bit flips, may not be immediately apparent but can cause detrimental effects to algorithm behavior. In this work, we…
The quantum dynamics of a complex system can be efficiently described in Krylov space, the minimal subspace in which the dynamics unfolds. We apply the Krylov subspace method for Hamiltonian deformations, which provides a systematic way of…
Krylov quantum diagonalization methods have emerged as a promising use case for quantum computers. However, many existing implementations rely on controlled operations, which pose challenges to near-term quantum hardware. We introduce a…
Several problems in machine learning, statistics, and other fields rely on computing eigenvectors. For large scale problems, the computation of these eigenvectors is typically performed via iterative schemes such as subspace iteration or…
When a solution to an abstract inverse linear problem on Hilbert space is approximable by finite linear combinations of vectors from the cyclic subspace associated with the datum and with the linear operator of the problem, the solution is…