Boris Volkov

Moscow Institute of Physics and Technology | MIPT · Department of Higher Mathematics

We describe dual and antidual solutions of the Yang–Mills equations by means of L´evy Laplacians. To this end, we introduce a class of L´evy Laplacians parameterized by the choice of a curve in the group SO(4). Two approaches are used to define such Laplacians: (i) the Lévy Laplacian can be defined as an integral functional defined by a curve in SO...

A covariant definition of the Levy Laplacian on an infinite dimensional manifold is introduced. It is shown that a time-depended connection in a finite dimensional vector bundle is a solution of the Yang—Mills heat equations if and only if the associated flow of the parallel transports is a solution of the heat equation for the covariant Levy Lapla...

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...

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 analy...

The relationship between the Yang-Mills equations and the stochastic analogue of Levy differential operators is studied. The value of the stochastic Levy Laplacian is found by means of Cesaro averaging of directional derivatives on the stochastic parallel transport. It is shown that the Yang-Mills equations and the Levy-Laplace equation for such La...

Most general dynamics of an open quantum system is commonly represented by a quantum channel, which is a completely positive trace-preserving map (CPTP or Kraus map). Well-known are the representations of quantum channels by Choi matrices and by Kraus operator-sum representation (OSR). As was shown before, one can use Kraus OSR to parameterize quan...

High fidelity generation of two-qubit gates is important for quantum computation, since such gates are components of popular universal sets of gates. Here we consider the problem of high fidelity generation of two-qubit C-NOT and C-PHASE (with a detailed study of C-Z) gates in presence of the environment. We consider the general situation when qubi...

High-fidelity generation of two-qubit gates is important for quantum computation, since such gates are components of popular universal sets of gates. Here, we consider the problem of high-fidelity generation of two-qubit C-NOT and C-PHASE (with a detailed study of C–Z) gates in presence of the environment. We consider the general situation when qub...

In this work, we study the detailed structure of quantum control landscape for the problem of single-qubit phase shift gate generation on the fast time scale. In previous works, the absence of traps for this problem was proved on various time scales. A special critical point which was known to exist in quantum control landscapes was shown to be eit...

In this work, we study the detailed structure of quantum control landscape for the problem of single-qubit phase shift gate generation on the fast time scale. In previous works, the absence of traps for this problem was proven on various time scales. A special critical point which was known to exist in quantum control landscapes was shown to be eit...

Quantum control is necessary for a variety of modern quantum technologies as it allows to optimally manipulate quantum systems. An important problem in quantum control is to establish whether the control objective functional has trapping behaviour or no, namely if it has or no traps -- controls from which it is difficult to escape by local search o...

In this paper, an infinite-dimensional Laplacian defined as the Cesáro mean of the second-order directional derivatives on manifold is considered. This Laplacian is parametrized by the choice of a curve in the group of orthogonal rotations. It is shown that under certain conditions on the curve, this operator is related to instantons on a four-dime...

A connection between the Yang–Mills gauge fields on 4-dimensional orientable compact Riemannian manifolds and modified Lévy Laplacians is studied. A modified Lévy Laplacian is obtained from the Lévy Laplacian by the action of an infinite dimensional rotation. Under the assumption that the 4-manifold has a nontrivial restricted holonomy group of the...

An infinite dimensional Laplacian defined as the Ces\'aro mean of the second order directional derivatives on manifold is considered. This Laplacian is parameterized by the choice of a curve in the group of orthogonal rotations. It is shown that, under certain conditions on the curve, this operator is related to instantons on a 4-dimensional manifo...

In this work, we study the detailed structure of quantum control landscape for the problem of single-qubit phase shift gate generation on the fast time scale. In previous works, the absence of traps for this problem was proven on various time scales. A special critical point which was known to exist in quantum control landscapes was shown to be eit...

The connection between Yang--Mills gauge fields on $4$-dimensional orientable compact Riemannian manifolds and modified L\'evy Laplacians is studied. A modified L\'evy Laplacian is obtained from the L\'evy Laplacian by the action of an infinite dimensional rotation. Under the assumption that the 4-manifold has a nontrivial restricted holonomy group...

In this work, we consider the problem of ultrafast controlled generation of single-qubit phase shift quantum gates. Globally optimal control is a control which realizes the gate with maximal possible fidelity. Trap is a control which is optimal only locally but not globally. It was shown before that traps do not exist for controlled generation of a...

This paper is a survey of results on the relationship between gauge fields and infinitedimensional equations for parallel transport that contain the Lévy Laplacian or the divergence associated with this Laplacian. Also we analyze the deterministic case where parallel transports are operator-valued functionals on the space of curves and the case of...

We use the Chern–Simons action for a [Formula: see text]-connection for the description of point disclinations in the geometric theory of defects. The most general spherically symmetric [Formula: see text]-connection with zero curvature is found. The corresponding orthogonal spherically symmetric [Formula: see text] matrix and [Formula: see text]-f...

The equivalence of the anti-selfduality Yang-Mills equations on the 4-dimensional orientable Riemannian manifold and Laplace equations for some infinite dimensional Laplacians is proved. A class of modificated Levy 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...

We use the Chern-Simons action for a SO(3)-connection for the description of point disclinations in the geometric theory of defects. The most general spherically symmetric SO(3)-connection with zero curvature is found. The corresponding orthogonal spherically symmetric SO(3) matrix and n-field are computed. Two examples of point disclinations are d...

A covariant definition of the Levy Laplacian on an infinite dimensional manifold is introduced. It is shown that a time-depended connection in a finite dimensional vector bundle is a solution of the Yang-Mills heat equations if and only if the associated flow of the parallel transports is a solution of the heat equation for the covariant Levy Lapla...

The Levy Laplacians are infinite-dimensional Laplace operators defined as the Cesaro meanof the second-order directional derivatives. In the theory of Sobolev–Schwarz distributions overa Gaussian measure on an infinite-dimensional space (the Hida calculus), we can consider two canonical Levy Laplacians. The first Laplacian, the so-called classical...

Some connections between different definitions of Levy 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 Levy Laplacians parame...

We study the Levy infinite-dimensional differential operators (differential operators defined by the analogy with the Levy 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 diffe...

One of the main reasons for interest in the Levy Laplacian and its analogues such as Levy d'Alembertian is a connection of these operators with gauge fields. The theorem proved by Accardi, Gibillisco and Volovich stated that a connection in a bundle over a Euclidean space or over a Minkowski space is a solution of the Yang-Mills equations if and on...

One of the main reasons for interest in the Levy Laplacian and its analogues such as Levy d'Alembertian is a connection of these operators with gauge fields. The theorem proved by Accardi, Gibillisco and Volovich stated that a connection in a bundle over a Euclidean space or over a Minkowski space is a solution of the Yang-Mills equations if and on...

The Lévy d'Alambertian is the natural analogue of the well-known Lévy-Laplacian. The aim of the paper is the following. We study the relationship between different definitions of the Lévy d'Alambertian and the relationship between the Lévy d'Alambertian and the QCD equations (the Yang–Mills–Dirac equations). There are two different definitions of t...

We 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: ΔL = limε→0 ∫‖s-t‖<ε bsbtdsdt, where bt is the annihilatio...

The formula ΔL = limɛ→0 ∫∥s–t∥<ɛb s b t dsdt for the Levy Laplacian is obtained, where b t stands for an annihilation process. The formula is extended to some generalizations of the Levy Laplacian.

The following statement is proved for the Lévy–Laplacian defined as the Cesàro mean of second-order directional derivatives: a connection form on a base Riemannian C3-smooth manifold satisfies the Yang–Mills equations if and only if the parallel transport associated with the connection is Lévy harmonic. This statement is an improvement of the well-...