Quantum chaos as delocalization in Krylov space

Quantum chaos, delocalization, and entanglement in disordered Heisenberg models.

We investigate disordered one- and two-dimensional Heisenberg spin lattices across the transition from integrability to quantum chaos from both statistical many-body and quantum-information perspectives. Special emphasis is devoted to quantitatively exploring the interplay between eigenvector statistics, delocalization, and entanglement in the presence of nontrivial symmetries. The implication ...

Space-Time Approach to Quantum Chaos

Delocalization border and onset of chaos in a model of quantum computation.

We study the properties of spectra and eigenfunctions for a chain of 1/2 spins (qubits) in an external time-dependent magnetic field and under the conditions of nonselective excitation (when the amplitude of the magnetic field is large). This model is known as a possible candidate for experimental realization of quantum computation. We present the theory for finding delocalization transitions a...

Fractional Inversion in Krylov Space

The fractional inverse M (real γ > 0) of a matrix M is expanded in a series of Gegenbauer polynomials. If the spectrum of M is confined to an ellipse not including the origin, convergence is exponential, with the same rate as for Chebyshev inversion. The approximants can be improved recursively and lead to an iterative solver for Mx = b in Krylov space. In case of γ = 1/2, the expansion is in t...

Krylov Space Methods on State - Space Control

We give an overview of various Lanczos/Krylov space methods and how they are being used for solving certain problems in Control Systems Theory based on state-space models. The matrix methods used are based on Krylov sequences and are closely related to modern iterative methods for standard matrix problems such as sets of linear equations and eigenvalue calculations. We show how these methods ca...