A generalized finite-difference method based on minimizing global residual

An improved generalized finite-difference method is proposed in this paper, as an alternative meshless method to solve differential equations. The method establishes discrete equations by minimizing a global residual. A general frame for constructing difference schemes is first described. As one choice the moving least square method is used in this paper. Compared with other generalized finite-difference methods, the improved method yields a set of discrete equations having the favorable properties such as symmetric, positive definite and well conditioned. Compared with meshless methods based on a variational principle or a weak form, the method described in this paper does not need a numerical integration and thus provides an alternative way to avoid the difficulties in implementing a numerical integration. In the proposed method there is no such inconvenience in applying essential boundary conditions as commonly encountered in other meshless methods. Numerical examples show that the improved method has a high convergence rate and can produce accurate results even with a coarse mesh.     

Pentadiagonal alternating-direction-implicit finite-difference time-domain method for two-dimensional Schrödinger equation

In this paper, we have proposed a pentadiagonal alternating-direction-implicit (Penta-ADI) finite-difference time-domain (FDTD) method for the two-dimensional Schrödinger equation. Through the separation of complex wave function into real and imaginary parts, a pentadiagonal system of equations for the ADI method is obtained, which results in our Penta-ADI method. The Penta-ADI method is further simplified into pentadiagonal fundamental ADI (Penta-FADI) method, which has matrix-operator-free right-hand-sides (RHS), leading to the simplest and most concise update equations. As the Penta-FADI method involves five stencils in the left-hand-sides (LHS) of the pentadiagonal update equations, special treatments that are required for the implementation of the Dirichlet’s boundary conditions will be discussed. Using the Penta-FADI method, a significantly higher efficiency gain can be achieved over the conventional Tri-ADI method, which involves a tridiagonal system of equations.     

A finite-difference method for parameter-plane analysis

Factorial series are used for deriving a method of successive solutions of parameter-plane equations for real, uniform and positive increments in the complex variable. A difference table of trigonometric sine and cosine functions is formulated for evaluating the real and imaginary parts of complex products occurring in the factorial series expansion. Coefficients, nonlinear in the parameters, are approximated by Taylor's series, and an estimate is given for the total remainder neglected in one iteration.     

Rigorous interference and diffraction analysis of diffractive optic elements using the finite-difference time-domain method

The Finite-Difference Time-Domain (FDTD) method has proven to be a useful tool to analyze electromagnetic scattering phenomena. In this work, the FDTD method is applied at optical wavelengths. More precisely, we present the results obtained using the FDTD algorithm to simulate the performance of optical devices such as volume diffraction gratings. The Perfectly Matched Layers (PML), Total-Field Scattered-Field formulation (TF/SF) and Near-Field to Far-Field transformation (NF/FF) are some add-ons included in order to correctly calculate the far field distribution obtained from the numerical near-field values computed in the simulation region. These values in the near-field region are computed by illuminating the grating with of a plane wave at the Bragg angle of incidence. In addition, we compare the results obtained by the FDTD method to those obtained using the Rigorous Coupled Wave Theory (RCWT) applied to diffraction gratings. As will be seen in this paper there is good agreement between the two approaches, thus validating our FDTD implementation.     

给定环境下稳定开放量子系统的哈密顿量方法

针对耗散已知情况下Lindblad主方程描述的开放量子系统,本文通过哈密顿量的设计实现了系统对于目标平衡态的稳定化.借助相干矢量体系,将矩阵形式下的原始系统模型转换为了一个矢量形式的线性系统,并证明了变换前后系统稳定属性的等价性.通过保证矢量化线性系统模型的稳定性,并使系统的唯一平衡态等于期望的目标态,得到了系统哈密顿量的设计框架.特别地,本文讨论了这两类条件下系统哈密顿量各元素的范围,并指出根据它们的交集即可构成所设计的系统哈密顿量.最后,在一个两能级系统上进行了数值仿真实验,验证了本文哈密顿量稳定化方案的有效性.     

A generalized circuit for the Hamiltonian dynamics through the truncated series

In this paper, we present a method for Hamiltonian simulation in the context of eigenvalue estimation problems, which improves earlier results dealing with Hamiltonian simulation through the truncated Taylor series. In particular, we present a fixed-quantum circuit design for the simulation of the Hamiltonian dynamics, \({\mathcal {H}}(t)\), through the truncated Taylor series method described by Berry et al. (Phys Rev Lett 114:090502, 2015). The circuit is general and can be used to simulate any given matrix in the phase estimation algorithm by only changing the angle values of the quantum gates implementing the time variable t in the series. The circuit complexity depends on the number of summation terms composing the Hamiltonian and requires O(Ln) number of quantum gates for the simulation of a molecular Hamiltonian. Here, n is the number of states of a spin orbital, and L is the number of terms in the molecular Hamiltonian and generally is bounded by \(O(n^4)\). We also discuss how to use the circuit in adaptive processes and eigenvalue-related problems along with a slightly modified version of the iterative phase estimation algorithm. In addition, a simple divide-and-conquer method is presented for mapping a matrix which are not given as sums of unitary matrices into the circuit. The complexity of the circuit is directly related to the structure of the matrix and can be bounded by \(O(\mathrm{poly}(n))\) for a matrix with \(\mathrm{poly}(n)\)-sparsity.     

Observer-based Hamiltonian identification for quantum systems

A symmetry-preserving observer-based parameter identification algorithm for quantum systems is proposed. Starting with a 2-level quantum system (qubit), where the unknown parameters consist of the atom-laser frequency detuning and coupling constant, we prove an exponential convergence result. The analysis is inspired by Lyapunov and adaptive control techniques and is based on averaging theory. The observer is then extended to the multi-level case where all the atom-laser coupling constants are unknown. The extension of the convergence analysis is discussed through some heuristic arguments. The relevance and the robustness with respect to various noises are tested through numerical simulations.     

基于量子系综分类的量子系统哈密顿量辨识

针对量子系统中的哈密顿量辨识问题,提出了一种基于量子系综分类的量子系统参数辨识方法.首先,所采用的量子系综分类方法结合了基于采样的学习控制方法和梯度流算法,可利用所设计的控制场有效区分具有不同哈密顿量参数的量子系统;其次,以交叉验证的方式对于所需估计的哈密顿量参数值进行区间判定,提高估计可靠性;再次,采用逐次细化判定区间的方法,辨识出最终的哈密顿量参数;最后,通过数值仿真验证了所提出的量子系统哈密顿量辨识方法的有效性和实用性.     

Implementation of the iterative finite-difference time-domain technique for simulation of periodic structures at oblique incidence

In this paper we review a recently developed finite-difference time-domain (FDTD) iterative technique for the analysis of periodic structures at oblique incidence. We show how it can be implemented in FDTD code and estimate required computer memory and time resources. To illustrate performance of our technique we demonstrate the plasmon formation in a thin gold film placed at air/glass interface and calculate reflectance from silicon textured coating at oblique incidence.     

A linearized implicit finite-difference method for solving the equal width wave equation

A linearized implicit finite-difference method is presented to find numerical solutions of the equal width wave equation. The method has been used successfully to investigate the motion of a single solitary wave, the development of the interaction of two solitary waves and an undular bore. The obtained results are compared with other numerical results in the literature. A stability analysis of the scheme is also investigated.     

On the efficiency of quantum algorithms for Hamiltonian simulation

We study algorithms simulating a system evolving with Hamiltonian H = ?_j=1^m H_j{H = \sum_{j=1}^m H_j} , where each of the H _j , j = 1, . . . ,m, can be simulated efficiently. We are interested in the cost for approximating e^-iHt, t ? \mathbbR{e^{-iHt}, t \in \mathbb{R}} , with error e{\varepsilon} . We consider algorithms based on high order splitting formulas that play an important role in quantum Hamiltonian simulation. These formulas approximate e ^−iHt by a product of exponentials involving the H _j , j = 1, . . . ,m. We obtain an upper bound for the number of required exponentials. Moreover, we derive the order of the optimal splitting method that minimizes our upper bound. We show significant speedups relative to previously known results.     

Effective Hamiltonian for the hybrid double quantum dot qubit

Quantum dot hybrid qubits formed from three electrons in double quantum dots represent a promising compromise between high speed and simple fabrication for solid state implementations of single-qubit and two-qubits quantum logic ports. We derive the Schrieffer–Wolff effective Hamiltonian that describes in a simple and intuitive way the qubit by combining a Hubbard-like model with a projector operator method. As a result, the Hubbard-like Hamiltonian is transformed in an equivalent expression in terms of the exchange coupling interactions between pairs of electrons. The effective Hamiltonian is exploited to derive the dynamical behavior of the system and its eigenstates on the Bloch sphere to generate qubits operation for quantum logic ports. A realistic implementation in silicon and the coupling of the qubit with a detector are discussed.     

基于时域有限差分和非参数核回归的室内移动通信干扰信号预测模型研究

针对网络优化和干扰实施中电磁辐射预测准确率不高的问题,本文利用多制式移动通信干扰器对室内的复杂场景进行了干扰防护的信号强度预测,对实际场景建立几何模型后采用时域有限差分算法获得信号强度预测值,并结合实际测量改进非参数核回归预测算法获得统计分析预测值,最后提出混合加权预测模型,对电磁辐射进行定量预测分析。预测结果表明,本文提出的基于时域有限差分算法和非参数核回归算法的混合加权预测模型能对复杂室内场景下信号强度进行预测,准确率较高。     

基于广义Hamilton系统的机械人双目视觉伺服控制器

对于手眼双目视觉伺服机器人,考虑一类新的双目视觉模型,研究了新模型视觉伺服控制器的设计问题。首先,基于新视觉模型和机器人动力学方程,构建了广义Hamilton框架下的视觉反馈系统(GHVFS)。针对GHVFS提出了基于广义Hamilton框架下的视觉伺服控制器设计方法。此外,研究了GHVFS下的L2增益干扰抑制问题,并证明了该闭环系统的渐近稳定性。最后给出的仿真结果验证了该方法的有效性。     

A parallel Self Mesh-Adaptive N-body method based on approximate inverses

A new parallel Self Mesh-Adaptive N-body method based on approximate inverses is proposed. The scheme is a three-dimensional Cartesian-based method that solves the Poisson equation directly in physical space, using modified multipole expansion formulas for the boundary conditions. Moreover, adaptive-mesh techniques are utilized to form a class of separate smaller n-body problems that can be solved in parallel and increase the total resolution of the system. The solution method is based on multigrid method in conjunction with the symmetric factored approximate sparse inverse matrix as smoother. The design of the parallel Self Mesh-Adaptive method along with discussion on implementation issues for shared memory computer systems is presented. The new parallel method is evaluated through a series of benchmark simulations using N-body models of isolated galaxies or galaxies interacting with dwarf companions. Furthermore, numerical results on the performance and the speedups of the scheme are presented.     

A linearization method for generalized complementarity problems

This correspondence presents a Newton-type iterative method for the generalized complementarity problems (i.e., nonlinear complementarity problems over a closed convex cone). The solution is obtained by a sequence of linear approximations. A sufficient condition for the global convergence and the rate of convergence of the algorithm are studied.     

A generalized fractional KN equation hierarchy and its fractional Hamiltonian structure

A generalized Hamiltonian structure of the fractional soliton equation hierarchy is presented by using differential forms and exterior derivatives of fractional orders. We construct the generalized fractional trace identity through the Riemann-Liouville fractional derivative. An example of the fractional KN soliton equation hierarchy and Hamiltonian structure is presented, which is a new integrable hierarchy and possesses Hamiltonian structure.