Mathematical analysis of quantum control landscapes, which aims to prove either absence or existence of traps for quantum control objective functionals, is an important topic in quantum control. In this work, we provide a rigorous analysis of quantum control landscapes for ultrafast generation of single-qubit quantum gates and show, combining analytical methods based on a sophisticated analysis of spectrum of the Hessian, and numerical optimization methods such as gradient ascent pulse engineering, differential evolution, and dual annealing, that control landscape for ultrafast generation of phase shift gates is free of traps.
Dedicated to the memory of Alexander A. Belyaev Abstract: The equivalence of the anti-selfduality Yang-Mills equations on the 4dimensional orientable Riemannian manifold and Laplace equations for some infinite dimensional Laplacians is proved. A class of modificated Lévy Laplacians parameterized by the choice of a curve in the group SO(4) is introduced. It is shown that a connection is an instanton (a solution of the anti-selfduality Yang-Mills equations) if and only if the parallel transport generalized by this connection is a solution of the Laplace equations for some three modificated Levy Laplacians from this class.
Communicated by O. SmolyanovWe consider a family of infinite dimensional Laplace operators which contains the classical Lévy-Laplacian. We prove a representation of these operators as a quadratic functions of quantum stochastic processes. Particularly, for the classical Lévy-Laplacian, the following formula is proved: ∆ L = lim ε→0 R s−t <ε bsbtdsdt, where bt is the annihilation process.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.