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This paper introduces state polynomials, i.e., polynomials in noncommuting variables and formal states of their products. A state analog of Artin's solution to Hilbert's 17th problem is proved showing that state polynomials, positive over all matrices and matricial states, are sums of squares with denominators. Somewhat surprisingly, it is also established that a Krivine–Stengle Positivstellensatz fails to hold in the state polynomial setting. Further, archimedean Positivstellensätze in the spirit of Putinar and Helton–McCullough are presented leading to a hierarchy of semidefinite relaxations converging monotonically to the optimum of a state polynomial subject to state constraints. This hierarchy can be seen as a state analog of the Lasserre hierarchy for optimization of polynomials, and the Navascués–Pironio–Acín scheme for optimization of noncommutative polynomials. The motivation behind this theory arises from the study of correlations in quantum networks. Determining the maximal quantum violation of a polynomial Bell inequality for an arbitrary network is reformulated as a state polynomial optimization problem. Several examples of quadratic Bell inequalities in the bipartite and the bilocal tripartite scenario are analyzed. To reduce the size of the constructed SDPs, sparsity, sign symmetry and conditional expectation of the observables' group structure are exploited. To obtain the above-mentioned results, techniques from noncommutative algebra, real algebraic geometry, operator theory, and convex optimization are employed. [ABSTRACT FROM AUTHOR]

We consider a Dirac operator with constant magnetic field defined on a half-plane with boundary conditions that interpolate between infinite mass and zigzag. By a detailed study of the energy dispersion curves, we show that the infinite mass case generically captures the profile of these curves, which undergoes a continuous pointwise deformation into the topologically different zigzag profile. Moreover, these results are applied to the bulk-edge correspondence. In particular, by means of a counterexample, we show that this correspondence does not always hold true in the zigzag case. [ABSTRACT FROM AUTHOR]

We establish the improved uniform error bounds on a Lawson-type exponential integrator Fourier pseudospectral (LEI-FP) method for the long-time dynamics of sine-Gordon equation where the amplitude of the initial data is O (ε) with 0 < ε ≪ 1 a dimensionless parameter up to the time at O (1 / ε 2) . The numerical scheme combines a Lawson-type exponential integrator in time with a Fourier pseudospectral method for spatial discretization, which is fully explicit and efficient in practical computation thanks to the fast Fourier transform. By separating the linear part from the sine function and employing the regularity compensation oscillation (RCO) technique which is introduced to deal with the polynomial nonlinearity by phase cancellation, we carry out the improved error bounds for the semi-discretization at O (ε 2 τ) instead of O (τ) according to classical error estimates and at O (h m + ε 2 τ) for the full-discretization up to the time T ε = T / ε 2 with T > 0 fixed. This is the first work to establish the improved uniform error bound for the long-time dynamics of the nonlinear Klein–Gordon equation with non-polynomial nonlinearity. The improved error bound is extended to an oscillatory sine-Gordon equation with O (ε 2) wavelength in time and O (ε - 2) wave speed, which indicates that the temporal error is independent of ε when the time step size is chosen as O (ε 2) . Finally, numerical examples are shown to confirm the improved error bounds and to demonstrate that they are sharp. [ABSTRACT FROM AUTHOR]

Tsirelson and Vershik (In Rev Math Phys 10(1):81–145, 1998) introduced the notion of a mathematical noise, which possesses completely opposite properties to those of a white noise. Afterward, Tsirelson (In Lectures on probability theory and statistics, Lecture Notes in Math., 1840, Springer, Berlin, pp 1–106, 2004) called this noise: 'black noise.' In this paper, we provide a method to simulate black noise using a modified Bayesian convolutional neural network. Then we study the behavior of black noise both numerically and visually. [ABSTRACT FROM AUTHOR]

In this paper we develop a stochastic simulation algorithm for electron transport in a DA-pHEMT heterostructure. Mathematical formulation of the problem of electron gas transport in the heterostructure in the form of a coupled system of Poisson, Schrödinger and kinetic Boltzmann equations is given. A Monte Carlo model of electron transport in DA-pHEMT heterostructures which accounts for multivalley parabolic band structure, as well as relevant formulas for calculating electron scattering rates and scattering phase functions on polar optical, intervalley phonons and on impurities are developed. The results of a computational experiment involving the solution of the system of Poisson–Schrödinger–Boltzmann equations for the AlGaAs/GaAs/InGaAs/GaAs/AlGaAs heterostructure are presented. The distribution of electrons by energy subband in the main and satellite valleys and the field dependences of the electron drift velocity in each valley are calculated. It was discovered that there is no spatial transfer of electrons into wide-gap AlGaAs layers due to high barriers created by modulated-doped impurities. A comparative analysis of the electron drift velocities in the studied DA-pHEMT heterostructures and in the unstrained layer of the InGaAs is given. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of the real-time equilibrium Dyson equation, which is used in calculations of the dynamical properties of quantum many-body systems. We show that this equation can be written as a system of coupled, nonlinear, convolutional Volterra integro-differential equations, for which the kernel depends self-consistently on the solution. As is typical in the numerical solution of Volterra-type equations, the computational bottleneck is the quadratic-scaling cost of history integration. However, the structure of the nonlinear Volterra integral operator precludes the use of standard fast algorithms. We propose a quasilinear-scaling FFT-based algorithm which respects the structure of the nonlinear integral operator. The resulting method can reach large propagation times and is thus well-suited to explore quantum many-body phenomena at low energy scales. We demonstrate the solver with two standard model systems: the Bethe graph and the Sachdev-Ye-Kitaev model. [ABSTRACT FROM AUTHOR]

Response calculations in density functional theory aim at computing the change in ground-state density induced by an external perturbation. At finite temperature, these are usually performed by computing variations of orbitals, which involve the iterative solution of potentially badly conditioned linear systems, the Sternheimer equations. Since many sets of variations of orbitals yield the same variation of density matrix, this involves a choice of gauge. Taking a numerical analysis point of view, we present the various gauge choices proposed in the literature in a common framework and study their stability. Beyond existing methods, we propose a new approach, based on a Schur complement using extra orbitals from the self-consistent field calculations, to improve the stability and efficiency of the iterative solution of Sternheimer equations. We show the success of this strategy on nontrivial examples of practical interest, such as Heusler transition metal alloy compounds, where savings of around 40% in the number of required cost-determining Hamiltonian applications have been achieved. [ABSTRACT FROM AUTHOR]

The bifunctional formalism presents an alternative how to obtain the functional value from its functional derivative by exploiting homogeneous density scaling. In the bifunctional formalism the density dependence of the functional derivative is suppressed. Consequently, those derivatives have to be treated as formal functional derivatives. For a pointwise correspondence between the true and the formal functional derivative, the bifunctional expression yields the same value as the density functional. Within the bifunctional formalism the functional value can directly be obtained from its derivative (while the functional itself remains unknown). Since functional derivatives are up to a constant uniquely defined, this approach allows for a pointwise comparison between approximate potentials and reference potentials. This aspect is especially important in the field of orbital-free density functional theory, where the burden is to approximate the kinetic energy. Since in the bifunctional approach the potential is approximated directly, full control is given over the latter, and consequently over the final electron densities obtained from variational procedure. Besides the bifunctional formalism itself another concept is introduced, dividing the total non-interacting kinetic energy into a known functional part and a remainder, called Pauli kinetic energy. Only the remainder requires further approximations. For practical purposes sufficiently accurate Pauli potentials for application on atoms, molecular and solid-state systems are presented. [ABSTRACT FROM AUTHOR]

We present mixed quantum-classical approaches based on the exact factorization framework. The electron–nuclear correlation term in the exact factorization enables us to deal with quantum coherences by accounting for electronic and nuclear nonadiabatic couplings effectively within classical nuclei approximation. We compare coupled- and independent-trajectory approximations with each other to understand algorithms in description of the bifurcation of nuclear wave packets and the correct spatial distribution of electronic wave functions along with nuclear trajectories. Finally, we show numerical results for comparisons of coupled- and independent-trajectory approaches for the photoisomerization of a protonated Schiff base from excited state molecular dynamics (ESMD) simulations with the recently developed Python-based ESMD code, namely, the PyUNIxMD program. [ABSTRACT FROM AUTHOR]

Motivated by recent progress in quantum information theory, this article aims at optimizing trace polynomials, i.e., polynomials in noncommuting variables and traces of their products. A novel Positivstellensatz certifying positivity of trace polynomials subject to trace constraints is presented, and a hierarchy of semidefinite relaxations converging monotonically to the optimum of a trace polynomial subject to tracial constraints is provided. This hierarchy can be seen as a tracial analog of the Pironio, Navascués and Acín scheme (Pironio et al. in New J. Phys. 10(7):073013, 2008) for optimization of noncommutative polynomials. The Gelfand–Naimark–Segal (GNS) construction is applied to extract optimizers of the trace optimization problem if flatness and extremality conditions are satisfied. These conditions are sufficient to obtain finite convergence of our hierarchy. The results obtained are applied to violations of polynomial Bell inequalities in quantum information theory. The main techniques used in this paper are inspired by real algebraic geometry, operator theory, and noncommutative algebra. [ABSTRACT FROM AUTHOR]

In this work, a high-order numerical method with mass and energy conservation obtained by the operator-compensation (OC) technique is proposed for the coupled Gross–Pitaevskii equations (CGPEs) modelling the spin–orbit-coupled Bose–Einstein condensates. The OC technique is utilized to seek high-order approximations to the first- and the second-order spatial derivatives in the CGPEs, after which a time-dependent semi-discrete system is obtained. Then we obtain a fully discrete scheme by applying the Crank–Nicolson method to discretize this system in time. We theoretically show that the semi-discrete system and the fully discrete scheme conserve both mass and energy on the discrete level. In addition, the stability analysis is discussed for the proposed method. Numerical experiments are given to test accuracy and verify conservative properties. The dynamics of the two-dimensional CGPEs and the effects of the parameters in the CGPEs are simulated. [ABSTRACT FROM AUTHOR]

The dilemma with the deficiencies of the nonlocal kernel functions as the building blocks of the Eringen's nonlocal theory has been of concern. The aim of the current work is to provide a remedy for the calculation of the components of the nonlocal moduli tensor pertinent to face center cubic (fcc) crystals accounting for their true symmetry group. To this end, three new distinct nonlocal kernel functions which are the discrete atomistic Green's functions in the stress space are obtained through the nonlocal dispersion relations associated with the longitudinal and shear waves in fcc crystals combined with the corresponding ones calculated via ab initio based on density functional perturbation theory (DFPT). In other words, based on nonlocal continuum and quantum mechanical considerations different kernel functions pertinent to each component of the elastic moduli tensor are calculated for the first time. Moreover, the nonzero components of the classical elastic moduli as well as the equilibrium lattice parameter associated with Al, Cu, Ni, Pd, and Ag fcc crystals have been computed using ab initio density functional theory (DFT) and compared with the available experimental data. In order to show the importance of the anisotropy of the kernel functions, the nonlocal stress field of a screw dislocation inside an infinitely extended fcc medium has been calculated; the solutions for two cases where the dislocation line is along the [ 0 0 1 ] and [ 1 0 1 ] directions are compared. Within this theory the unrealistic singular behavior of the classical stresses at the dislocation core is eliminated. [ABSTRACT FROM AUTHOR]

In this paper, an effective spectral-Galerkin method was proposed for the dissipative dynamics of a Hamiltonian system. By introducing a suitable Sobolev space, we establish a weak form and corresponding discrete scheme. Compared with the existing methods, our method can achieve higher accuracy and shorter computational time. We also simulate quantum cooling in an optomechanical system as an example to show the advantage of our method. Our method provides a promising platform for studying the dissipative dynamics of nonlinear and complex Hamiltonian systems. [ABSTRACT FROM AUTHOR]

We use projection methods to construct (global) quantum states with prescribed reduced (marginal) states, and possibly with some special properties such as having specific eigenvalues, having specific rank and extreme von Neumann or Rényi entropy. Using convex analysis, optimization techniques on matrix manifolds, we obtain algorithms to solve the problem. MATLAB programs are written based on these algorithms, and numerical examples are illustrated. The numerical results reveal new patterns leading to new insights and research problems on the topic. [ABSTRACT FROM AUTHOR]

The fine physical details for the quantum computation can easily be provided by high-order conservative schemes. By compensating the high-order difference operator (D μ p − 1) δ to the central difference operator δ, a time-dependent semi-discrete system, which conserves both charge and energy, is derived for the two-dimensional nonlinear Dirac equation (NLDE). Two kinds of fully discrete schemes are obtained by discretizing this semi-discrete system in time with the time-midpoint method and the time-splitting method. We prove theoretically that the former one conserves both charge and energy while the latter one only keeps the charge conservation. Some numerical experiments are given to verify the accuracy order, the stability and the conservative properties. In addition, the dynamics of the NLDE in one dimension and two dimensions are simulated. [ABSTRACT FROM AUTHOR]

In this work, a fourth-order in space and second-order in time compact scheme, a sixth-order in space and second-order in time compact scheme and two linearized compact schemes are proposed for the (1+1)-dimensional nonlinear Dirac equation. The iterative algorithm is used to compute the nonlinear algebraic system and the Thomas algorithm in the matrix form is adopted to enhance the computational efficiency. It is proved that all of the schemes are unconditionally stable in the linear sense. Numerical experiments are given to test the accuracy order of the presented schemes, record the error history for all of the schemes with respect to t, discuss the conservation laws of discrete charge and energy from the numerical point of view, study the stability of the solitary waves by adding a small random perturbation to the initial data, and simulate the collision of two and three solitary waves. [ABSTRACT FROM AUTHOR]

We derive new expressions for the Rayleigh-Schrödinger series describing the perturbation of eigenvalues of quantum Hamiltonians. The method, somehow close to the so-called dimensional renormalization in quantum field theory, involves the Birkhoff decomposition of some Laurent series built up out of explicit fully non-resonant terms present in the usual expression of the Rayleigh-Schrödinger series. Our results provide new combinatorial formulae and a new way of deriving perturbation series in quantum mechanics. More generally we prove that such a decomposition provides solutions of general normal form problems in Lie algebras. [ABSTRACT FROM AUTHOR]

Time dependent quantum systems have become indispensable in science and nanotechnology. Disciplines including chemical physics and electrical engineering have used approximate evolution operators to solve these systems for targeted physical quantities. Here, we discuss the approximation of closed time dependent quantum systems on bounded domains via evolution operators. The work builds upon the use of weak solutions, which includes a framework for the evolution operator based upon dual spaces. We are able to derive the corresponding Faedo-Galerkin equation as well as its time discretization, yielding a fully discrete theory. We obtain corresponding approximation estimates. These estimates make no regularity assumptions on the weak solutions, other than their inherent properties. Of necessity, the estimates are in the dual norm, which is natural for weak solutions. This appears to be a novel aspect of this approach. [ABSTRACT FROM AUTHOR]

We address two sets of long-standing open questions in linear algebra and probability theory, from a computational complexity perspective: stochastic matrix divisibility, and divisibility and decomposability of probability distributions. We prove that finite divisibility of stochastic matrices is an NP -complete problem, and extend this result to nonnegative matrices, and completely-positive trace-preserving maps, i.e. the quantum analogue of stochastic matrices. We further prove a complexity hierarchy for the divisibility and decomposability of probability distributions, showing that finite distribution divisibility is in P , but decomposability is NP -hard. For the former, we give an explicit polynomial-time algorithm. All results on distributions extend to weak-membership formulations, proving that the complexity of these problems is robust to perturbations. [ABSTRACT FROM AUTHOR]

We simulate the dynamics of a quantum dot coupled to the single resonating mode of a metal nanoparticle. Systems like this are known as metamolecules. In this study, we consider a time-dependent driving field acting onto the metamolecule. We use the Heisenberg equations of motion for the entire system, while representing the resonating mode in Wigner phase space. A time-dependent basis is adopted for the quantum dot. We integrate the dynamics of the metamolecule for a range of coupling strengths between the quantum dot and the driving field, while restricting the coupling between the quantum dot and the resonant mode to weak values. By monitoring the average of the time variation in the energy of the metamolecule model, as well as the coherence and the population difference in the quantum dot, we observe distinct nonlinear behavior in the case of strong coupling to the driving field. [ABSTRACT FROM AUTHOR]

In this article, we shall present in detail the results announced [5]. We consider a model of a linear time dependent Schrödinger equation with a time dependent potential. This model arises, for example, in underwater acoustics and has been studied by Akrivis and Dougalis [3]. We derive a new finite volume scheme on general nonconforming multidimensional spatial meshes introduced recently by Eymard et al. [18] for stationary anisotropic heterogeneous diffusion problems. The unknowns of this scheme are the values at the centre of the control volumes, at some internal interfaces, and at the mesh points of the time discretization. The discretization of the initial condition is performed using a discrete orthogonal projection. A newa priori estimatefor the discrete problem is proved. Thanks to thisa priori estimatefor the discrete problem, we derive error estimates in discrete norms ofandW1, ∞(L2). Moreover, we establish an error estimate for an approximation for the gradient, in a general framework. We prove that the convergence order ish𝒟 + k, whereh𝒟(resp.k) is the mesh size of the spatial (resp. time) discretization. This estimate is valid under the regularity assumptionfor the exact solutionu. These error estimates are useful because they allow us to get error estimates for the approximations of the exact solution and its first derivatives. Results of the present work have been obtained by a comparison with an appropriately chosen auxiliary finite volume scheme. [ABSTRACT FROM AUTHOR]