arXiv subjects

Boris O. Volkov

Publications and source records attributed to Boris O. Volkov.

8 recordsLinked to original sources

arXiv2024

Control landscapes for high-fidelity generation of C-NOT and C-PHASE gates with coherent and environmental driving

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 qubits are manipulated by coherent and incoherent controls; the latter is used to induce generally time-dependent decoherence rates. For estimating efficiency of optimization methods for high fidelity generation of these gates, we study quantum control landscapes which describe the behaviour of the fidelity as a function of the controls. For this, we generate and analyze the statistical distributions of best objective values obtained by incoherent GRadient Ascent Pulse Engineering (inGRAPE) approach. We also apply inGRAPE and stochastic zero-order method to numerically estimate minimal infidelity values. The results are different from the case of single-qubit gates and indicate a smooth trap-free behaviour of the fidelity.

quant-ph↗

arXiv2022

Modified Lévy Laplacian on manifold and Yang-Mills instantons

An infinite dimensional Laplacian defined as the Cesáro 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 manifold.

math-ph↗

arXiv2021

Lévy Laplacians, holonomy group and instantons on 4-manifolds

The connection between 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 bundle of self-dual 2-forms, the following is proved. There is a modified Lévy Laplacian such that a parallel transport in some vector bundle over the 4-manifold is a solution of the Laplace equation for this modified Lévy Laplacian if and only if the connection corresponding to the parallel transport satisfies the Yang--Mills self-duality (anti-self-duality) equations. An analogous connection between the Laplace equation for the Lévy Laplacian and the Yang--Mills equations was previously known.

math-ph↗

arXiv2021

Quantum control landscape for ultrafast generation of single-qubit phase shift quantum gates

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 arbitrary single-qubit quantum gates for sufficiently long times, as well as for fast control of quantum gates other than phase shift gates. Ultrafast generation of phase-shift gates was missed in the previous analysis. In this work we show, combining analytical and numerical optimization methods such as Gradient Ascent Pulse Engineering (GRAPE), differential evolution, and dual annealing, that control landscape for ultrafast generation of phase shift gates is also free of traps. 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 (GRAPE), differential evolution, and dual annealing, that control landscape for ultrafast generation of phase shift gates is free of traps.

quant-ph↗

arXiv2019

Levy Laplacian on manifold and Yang-Mills heat flow

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 Laplacian on the infinite dimensional manifold.

math-ph↗

arXiv2019

Levy Laplacians and instantons on manifolds

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 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.

math-ph↗

arXiv2017

Levy Laplacians in Hida calculus and Malliavin calculus

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 parametrized by a real parameter with the help of this approach. One of the elements of this chain is the classical Levy Laplacian. Another approach to define the Levy Laplacian is based on the application of the theory of Sobolev spaces over the Wiener measure (the Malliavin calculus). It is proved that the Levy Laplacian defined with the help of the second approach coincides with one of the elements of the chain of Levy Laplacians, which is not the classical Levy 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 Levy Laplacian in the stochastic analysis is connected to the gauge fields.

math-ph↗

arXiv2016

Stochastic Levy Differential Operators and Yang-Mills Equations

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 Laplacian are not equivalent as in the deterministic case. An equation equivalent to the Yang-Mills equations is obtained. The equation contains the stochastic Levy 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 Cesaro averaging.

math-ph↗