scholarly journals A fixed point algorithm for improving fidelity of quantum gates

2020 ◽  
Author(s):  
Paulo Sergio Pereira da Silva ◽  
Pierre Rouchon ◽  
Hector Bessa Silveira
Keyword(s):  
Fixed Point ◽  
Approximate Solution ◽  
Computational Effort ◽  
Banach Fixed Point Theorem ◽  
Fixed Point Algorithm ◽  
Piecewise Constant ◽  
System A ◽  
Piecewise Constant Control ◽  
Qubit System ◽  
The One

This work considers the problem of quantum gate generation for controllable quantum systems with drift.  It is assumed that an approximate solution called seed is pre-computed  by some known algorithm. This work presents a method, called   Fixed-Point Algorithm (FPA)  that is able to improve arbitrarily the fidelity of the given seed. When  the infidelity of the seed is small enough and the approximate solution is attractive in  the context of a tracking control problem (that is verified with probability one, in some sense), the Banach Fixed Point Theorem allows to prove the exponential convergence of the FPA. Even when the FPA  does not converge, several iterated applications of the FPA  may produce the desired fidelity. The FPA produces only small corrections in the control pulses and preserves the original bandwidth  of the seed. The computational effort of each step of the FPA corresponds to the one of the numerical integration of a stabilized closed loop system. A piecewise-constant and a smooth numerical implementations are developed. Several numerical experiments with a N-qubit system  illustrates the effectiveness of the method in several different applications including the conversion of piecewise-constant control pulses into smooth ones and the reduction of their bandwidth.

Solving of nonlinear Fredholm integro-differential equation in a complex plane with rationalized Haar wavelet bases

2019 ◽  
Vol 12 (04) ◽  
pp. 1950055 ◽  
Author(s):  
Majid Erfanian ◽  
Hamed Zeidabadi
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Approximate Solution ◽  
Integral Operator ◽  
Complex Plane ◽  
Haar Wavelet ◽  
Banach Fixed Point Theorem ◽  
Numerical Examples ◽  
Integro Differential Equation ◽  
Rationalized Haar Wavelet

Everyone knows about the complicated solution of the nonlinear Fredholm integro-differential equation in general. Hence, often, authors attempt to obtain the approximate solution. In this paper, a numerical method for the solutions of the nonlinear Fredholm integro-differential equation (NFIDE) of the second kind in the complex plane is presented. In fact, by using the properties of Rationalized Haar (RH) wavelet, we try to give the solution of the problem. So far, as we know, no study has yet been attempted for solving the NFIDE in the complex plane. For this purpose, we introduce the continuous integral operator and real valued function. The Banach fixed point theorem guarantees that, under certain assumptions, the integral operator has a unique solution. Furthermore, we give an upper bound for the error analysis. An algorithm is presented to compute and illustrate the solutions for some numerical examples.

scholarly journals Some qualitative properties of nonlinear fractional integro-differential equations of variable order

2021 ◽  
Vol 11 (3) ◽  
pp. 68-78
Author(s):  
Ahmed Refice ◽  
Mohammed Said Souid ◽  
Ali Yakar
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Existence And Uniqueness ◽  
Banach Fixed Point Theorem ◽  
Variable Order ◽  
Qualitative Properties ◽  
Piecewise Constant ◽  
Fractional Differential ◽  
The Stability ◽  
Uniqueness Criteria

The existence-uniqueness criteria of nonlinear fractional integro-differential equations of variable order with multiterm boundary value conditions are considered in this work. By utilizing the concepts of generalized intervals combined with the piecewise constant functions, we transform our problem into usual Caputo’s fractional differential equations of constant order. We develop the necessary criteria for assuring the solution's existence and uniqueness by applying Schauder and Banach fixed point theorem. We also examine the stability of the derived solution in the Ulam-Hyers-Rassias (UHR) sense and provide an example to demonstrate the credibility of the results.

Structure Information From Electron Diffraction Intensities

1990 ◽  
Vol 48 (2) ◽  
pp. 516-517
Author(s):  
J. Gjønnes ◽  
N. Bøe ◽  
K. Gjønnes
Keyword(s):  
Computational Effort ◽  
Unit Cell ◽  
Small Unit ◽  
Structure Information ◽  
Dispersion Surface ◽  
Integrated Intensity ◽  
The One ◽  
Image Position ◽  
Convergent Beam ◽  
Beam Patterns

Structure information of high precision can be extracted from intentsity details in convergent beam patterns like the one reproduced in Fig 1. From low order reflections for small unit cell crystals,bonding charges, ionicities and atomic parameters can be derived, (Zuo, Spence and O’Keefe, 1988; Zuo, Spence and Høier 1989; Gjønnes, Matsuhata and Taftø, 1989) , but extension to larger unit cell ma seem difficult. The disks must then be reduced in order to avoid overlap calculations will become more complex and intensity features often less distinct Several avenues may be then explored: increased computational effort in order to handle the necessary many-parameter dynamical calculations; use of zone axis intensities at symmetry positions within the CBED disks, as in Figure 2 measurement of integrated intensity across K-line segments. In the last case measurable quantities which are well defined also from a theoretical viewpoint can be related to a two-beam like expression for the intensity profile:With as an effective Fourier potential equated to a gap at the dispersion surface, this intensity can be integrated across the line, with kinematical and dynamical limits proportional to and at low and high thickness respctively (Blackman, 1939).

Purification of a Specific Placental Plasminogen Activator Inhibitor by Monoclonal Antibody and Its Complex Formation with Plasminogen Activator

1985 ◽  
Vol 53 (01) ◽  
pp. 122-125 ◽  
Author(s):  
B Åstedt ◽  
Ingegerd Lecander ◽  
T Brodin ◽  
A Lundblad ◽  
Karin Löw
Keyword(s):  
Monoclonal Antibody ◽  
Complex Formation ◽  
Plasminogen Activator ◽  
Plasminogen Activator Inhibitor ◽  
Tissue Type ◽  
System A ◽  
The One ◽  
Fast Acting ◽  
Chain Form ◽  
Activator Inhibitor

SummaryA monoclonal antibody of IgG2a-type was obtained against a specific fast acting plasminogen activator inhibitor found in placenta. The placental inhibitor was purified by affinity chromatography using the monoclonal antibody and additionally in a FPLC-system. A strong complex formation was found between the inhibitor and urokinase and also with the two-chain form of plasminogen activator of the tissue-type. A weaker complex was found between the placental inhibitor and the one- chain form of the tissue-type activator.

scholarly journals Gauge theories on compact toric manifolds

2021 ◽  
Vol 111 (3) ◽  
Author(s):  
Giulio Bonelli ◽  
Francesco Fucito ◽  
Jose Francisco Morales ◽  
Massimiliano Ronzani ◽  
Ekaterina Sysoeva ◽  
...  
Keyword(s):  
Partition Function ◽  
Gauge Theories ◽  
Blow Up ◽  
Gauge Coupling ◽  
Piecewise Constant Function ◽  
Partition Functions ◽  
Piecewise Constant ◽  
Toric Manifolds ◽  
Holomorphic Anomaly ◽  
The One

AbstractWe compute the $$\mathcal{N}=2$$ N = 2 supersymmetric partition function of a gauge theory on a four-dimensional compact toric manifold via equivariant localization. The result is given by a piecewise constant function of the Kähler form with jumps along the walls where the gauge symmetry gets enhanced. The partition function on such manifolds is written as a sum over the residues of a product of partition functions on $$\mathbb {C}^2$$ C 2 . The evaluation of these residues is greatly simplified by using an “abstruse duality” that relates the residues at the poles of the one-loop and instanton parts of the $$\mathbb {C}^2$$ C 2 partition function. As particular cases, our formulae compute the SU(2) and SU(3) equivariant Donaldson invariants of $$\mathbb {P}^2$$ P 2 and $$\mathbb {F}_n$$ F n and in the non-equivariant limit reproduce the results obtained via wall-crossing and blow up methods in the SU(2) case. Finally, we show that the U(1) self-dual connections induce an anomalous dependence on the gauge coupling, which turns out to satisfy a $$\mathcal {N}=2$$ N = 2 analog of the $$\mathcal {N}=4$$ N = 4 holomorphic anomaly equations.

scholarly journals A study of Babylonian planetary theory III. The planet Mercury

2021 ◽  
Author(s):  
Teije de Jong
Keyword(s):  
Direct Method ◽  
Step Function ◽  
Computational Effort ◽  
Stationary Points ◽  
Planetary Motion ◽  
Time Intervals ◽  
Outer Planets ◽  
Step Functions ◽  
System A ◽  
Computational Systems

AbstractIn this series of papers I attempt to provide an answer to the question how the Babylonian scholars arrived at their mathematical theory of planetary motion. Papers I and II were devoted to system A theory of the outer planets and of the planet Venus. In this third and last paper I will study system A theory of the planet Mercury. Our knowledge of the Babylonian theory of Mercury is at present based on twelve Ephemerides and seven Procedure Texts. Three computational systems of Mercury are known, all of system A. System A1 is represented by nine Ephemerides covering the years 190 BC to 100 BC and system A2 by two Ephemerides covering the years 310 to 290 BC. System A3 is known from a Procedure Text and from Text M, an Ephemeris of the last evening visibility of Mercury for the years 424 to 403 BC. From an analysis of the Babylonian observations of Mercury preserved in the Astronomical Diaries and Planetary Texts we find: (1) that dates on which Mercury reaches its stationary points are not recorded, (2) that Normal Star observations on or near dates of first and last appearance of Mercury are rare (about once every twenty observations), and (3) that about one out of every seven pairs of first and last appearances is recorded as “omitted” when Mercury remains invisible due to a combination of the low inclination of its orbit to the horizon and the attenuation by atmospheric extinction. To be able to study the way in which the Babylonian scholars constructed their system A models of Mercury from the available observational material I have created a database of synthetic observations by computing the dates and zodiacal longitudes of all first and last appearances and of all stationary points of Mercury in Babylon between 450 and 50 BC. Of the data required for the construction of an ephemeris synodic time intervals Δt can be directly derived from observed dates but zodiacal longitudes and synodic arcs Δλ must be determined in some other way. Because for Mercury positions with respect to Normal Stars can only rarely be determined at its first or last appearance I propose that the Babylonian scholars used the relation Δλ = Δt −3;39,40, which follows from the period relations, to compute synodic arcs of Mercury from the observed synodic time intervals. An additional difficulty in the construction of System A step functions is that most amplitudes are larger than the associated zone lengths so that in the computation of the longitudes of the synodic phases of Mercury quite often two zone boundaries are crossed. This complication makes it difficult to understand how the Babylonian scholars managed to construct System A models for Mercury that fitted the observations so well because it requires an excessive amount of computational effort to find the best possible step function in a complicated trial and error fitting process with four or five free parameters. To circumvent this difficulty I propose that the Babylonian scholars used an alternative more direct method to fit System A-type models to the observational data of Mercury. This alternative method is based on the fact that after three synodic intervals Mercury returns to a position in the sky which is on average only 17.4° less in longitude. Using reduced amplitudes of about 14°–25° but keeping the same zone boundaries, the computation of what I will call 3-synarc system A models of Mercury is significantly simplified. A full ephemeris of a synodic phase of Mercury can then be composed by combining three columns of longitudes computed with 3-synarc step functions, each column starting with a longitude of Mercury one synodic event apart. Confirmation that this method was indeed used by the Babylonian astronomers comes from Text M (BM 36551+), a very early ephemeris of the last appearances in the evening of Mercury from 424 to 403 BC, computed in three columns according to System A3. Based on an analysis of Text M I suggest that around 400 BC the initial approach in system A modelling of Mercury may have been directed towards choosing “nice” sexagesimal numbers for the amplitudes of the system A step functions while in the later final models, dating from around 300 BC onwards, more emphasis was put on selecting numerical values for the amplitudes such that they were related by simple ratios. The fact that different ephemeris periods were used for each of the four synodic phases of Mercury in the later models may be related to the selection of a best fitting set of System A step function amplitudes for each synodic phase.

scholarly journals Local Information as an Essential Factor for Quantum Entanglement

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 728
Author(s):  
Zhaofeng Su
Keyword(s):  
Quantum Entanglement ◽  
Quantum Information Processing ◽  
Local Information ◽  
Essential Factor ◽  
Qubit System ◽  
Local Vectors ◽  
Correlation Parameters ◽  
The One ◽  
Local Parameters

Quantum entanglement is not only a fundamental concept in quantum mechanics but also a special resource for many important quantum information processing tasks. An intuitive way to understand quantum entanglement is to analyze its geometric parameters which include local parameters and correlation parameters. The correlation parameters have been extensively studied while the role of local parameters have not been drawn attention. In this paper, we investigate the question how local parameters of a two-qubit system affect quantum entanglement in both quantitative and qualitative perspective. Firstly, we find that the concurrence, a measure of quantum entanglement, of a general two-qubit state is bounded by the norms of local vectors and correlations matrix. Then, we derive a sufficient condition for a two-qubit being separable in perspective of local parameters. Finally, we find that different local parameters could make a state with fixed correlation matrix separable, entangled or even more qualitatively entangled than the one with vanished local parameters.

Piecewise constant control of linear mechanical systems in the general case

2018 ◽  
Author(s):  
I. M. Alesova ◽  
L. K. Babadzanjanz ◽  
A. M. Bregman ◽  
K. M. Bregman ◽  
I. Yu. Pototskaya ◽  
...  
Keyword(s):  
Mechanical Systems ◽  
Piecewise Constant ◽  
Constant Control ◽  
Piecewise Constant Control

scholarly journals On Almost Automorphic Mild Solutions for Nonautonomous Stochastic Evolution Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Jing Cui ◽  
Litan Yan
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Banach Fixed Point Theorem ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Almost Automorphic ◽  
Composition Theorem

We consider a class of nonautonomous stochastic evolution equations in real separable Hilbert spaces. We establish a new composition theorem for square-mean almost automorphic functions under non-Lipschitz conditions. We apply this new composition theorem as well as intermediate space techniques, Krasnoselskii fixed point theorem, and Banach fixed point theorem to investigate the existence of square-mean almost automorphic mild solutions. Some known results are generalized and improved.

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