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.

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Abstract:The relationship between the Yang-Mills equations and the stochastic analogue of Lévy differential operators is studied. The value of the stochastic Lévy Laplacian is found by means of Cèsaro averaging of directional derivatives on the stochastic parallel transport. It is shown that the Yang-Mills equations and the Lévy-Laplace equation for such Laplacian are not equivalent as in the deterministic case. An equation equivalent to the Yang-Mills equations is obtained. The equation contains the stochastic Lévy divergence. It is proved that the Yang-Mills action functional can be represented as an infinite-dimensional analogue of the Direchlet functional of chiral field. This analogue is also derived using Cèsaro averaging.

The equivalence of the anti-selfduality Yang–Mills equations on the four-dimensional orientable Riemannian manifold and the Laplace equations for some infinite-dimensional Laplacians is proved. A class of modified Lévy Laplacians parameterized by the choice of a curve in the group [Formula: see text] 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 modified Lévy Laplacians from this class.

Some connections between different definitions of Lévy Laplacians in the stochastic analysis are considered. Two approaches are used to define these operators. The standard one is based on the application of the theory of Sobolev-Schwartz distributions over the Wiener measure (the Hida calculus). One can consider the chain of Lévy Laplacians parametrized by a real parameter with the help of this approach. One of the elements of this chain is the classical Lévy Laplacian. Another approach to define the Lévy Laplacian is based on the application of the theory of Sobolev spaces over the Wiener measure (the Malliavin calculus). It is proved that the Lévy Laplacian defined with the help of the second approach coincides with one of the elements of the chain of Lévy Laplacians, which is not the classical Lévy Laplacian, under the imbedding of the Sobolev space over the Wiener measure into the space of generalized functionals over this measure. It is shown which Lévy Laplacian in the stochastic analysis is connected to the gauge fields.

We study the Lévy infinite-dimensional differential operators (differential operators defined by the analogy with the Lévy Laplacian) and their relationship to the Yang-Mills equations. We consider the parallel transport on the space of curves as an infinite-dimensional analogue of chiral fields and show that it is a solution to the system of differential equations if and only if the associated connection is a solution to the Yang-Mills equations. This system is an analogue of the equation of motion of chiral fields and contains the Lévy divergence. The systems of infinite-dimensional equations containing Lévy differential operators, that are equivalent to the Yang-Mills-Higgs equations and the Yang-Mills-Dirac equations (the equations of quantum chromodynamics), are obtained. The equivalence of two ways to define Lévy differential operators is shown.