Potential Energy Operator Research Articles

Efficient simulation of potential energy operators on quantum hardware: a study on sodium iodide (NaI)

This study introduces a conceptually novel polynomial encoding algorithm for simulating potential energy operators encoded in diagonal unitary forms in a quantum computing machine. The current trend in quantum computational chemistry is effective experimentation to achieve high-precision quantum computational advantage. However, high computational gate complexity and fidelity loss are some of the impediments to the realization of this advantage in a real quantum hardware. In this study, we address the challenges of building a diagonal Hamiltonian operator having exponential functional form, and its implementation in the context of the time evolution problem (Hamiltonian simulation and encoding). Potential energy operators when represented in the first quantization form is an example of such types of operators. Through systematic decomposition and construction, we demonstrate the efficacy of the proposed polynomial encoding method in reducing gate complexity from O(2n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(2^n)$$\\end{document} to O∑i=1rnCr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}\\left( \\sum _{i=1}^{r} {} ^nC_r \\right)$$\\end{document} (for some r≪n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\ll n$$\\end{document}). This offers a solution with lower complexity in comparison to the conventional Hadamard basis encoding approach. The effectiveness of the proposed algorithm was validated with its implementation in the IBM quantum simulator and IBM quantum hardware. This study demonstrates the proposed approach by taking the example of the potential energy operator of the sodium iodide molecule (NaI) in the first quantization form. The numerical results demonstrate the potential applicability of the proposed method in quantum chemistry problems, while the analytical bound for error analysis and computational gate complexity discussed, throw light on issues regarding its implementation.

Exploring the electronic potential of effective tight-binding hamiltonians

The linear combination of atomic orbitals (LCAO) is a standard method for studying solids and molecules, it is also known as the tight-binding (TB) method. In most of the implementations only the basis set and the coupling constants are provided, without the explicit definition of kinetic and potential energy operators. The tight-binding scheme is, nonetheless, capable of providing an accurate description of properties such as the electronic bands and elastic constants for many materials. However, for some applications, the knowledge of the underlying electronic potential associated with the tight-binding hamiltonian might be important to guarantee that the actual physics is preserved by the semiempirical scheme. In this work the electronic potentials that arise from the use of tight-binding effective hamiltonians it is explored. The formalism is applied to the extended Hückel tight-binding (EHTB) hamiltonian, which is a two-center Slater−Koster approach that makes explicit use of the overlap matrix.

Variable mechanism of electrical impedance for MCT high voltage switch under synergetic action of the mechanical and electric loads

To study the electromechanical coupling effect of Metal Oxide Semiconductor Controlled Thyristor (MCT) high voltage switch under the synergistic action of mechanical load and strong voltage in the actual use of penetrating hard targets and launching, the universal testing machine was used to simulate the overload environment. The rising edge time, falling time and impedance changes in the conduction path of MCT under different stress–strain were tested, respectively. Experimental results showed that the rise time of MCT decreased from 0.08 ms to 0.03 ms when the uniaxial compressive stress increased from 1.20 MPa to 7.3 MPa, and the rising edge time after unloading was 0.04 ms. The falling time had no obvious change with the increase of stress. The total resistance in the path decreased from 833.30 Ω to 564.22 Ω with the increase of strain. The stress–strain of each part of MCT under different uniaxial compressive stress was collected by COMSOL Multiphysics finite element software. Based on the Schrödinger equation, the potential energy operator was established. By introducing the strain Hamiltonian H ε,ν , the E(k)-k model near the minimum value of the conduction band of Si was established by using the k·p perturbation method. Combined with the physical field interface of Schrödinger equation in COMSOL Multiphysics, the valence band structure of Si material under uniaxial strain was studied on the basis of strain Hamiltonian perturbation. The results showed that the 6-degree degenerate valley (Δ6) in the conduction band was split into a 2-degree degenerate valley (Δ2) and a 4-degree degenerate valley (Δ4) due to the stress effect. The strain caused more electrons to occupy the lower Δ4 energy valley, resulting in a decrease in the effective mass of the total conductivity. Therefore, the electron mobility of Si was increased under uniaxial strain. The uniaxial compressive stress could more effectively reduce the curvature radius of valence band top, the effective mass of carriers and the interband scattering between light and heavy hole bands, which was beneficial to improve the hole mobility.

Covariant dynamics on the momentum space

A geometrical interpretation of Schrödinger's kinetic and potential energy operators is proposed, allowing for a covariant momentum space formulation of the dynamics that is relevant for the theories with the deformation of the momentum space structure. Some specific examples are discussed in the context of flat space deformations and the Euclidean Snyder (spherical momentum space) model. In this formulation the dynamics for the deformations of the flat momentum space becomes trivial, while different versions of the Snyder model turn out to be dynamically equivalent.

Regularization of δ′ potential in general case of deformed space with minimal length

In the general case of deformed Heisenberg algebra leading to the minimal length, we present a definition of the δ′(x) potential as a linear kernel of potential energy operator in momentum representation. We find exactly the energy level and corresponding eigenfunction for δ′(x) and δ(x) − δ′(x) potentials in deformed space with arbitrary function of deformation. The energy spectrum for different partial cases of deformation function is analysed.

Quasienergy operators and generalized squeezed states for systems of trapped ions

Collective many-body dynamics for time-dependent quantum Hamiltonians is investigated for a dynamical system that exhibits multiple degrees of freedom, in this case a combined (Paul and Penning) trap. Quantum stability is characterized by a discrete quasienergy spectrum, while the quasienergy states are symplectic coherent states. We introduce the generators of the Lie algebra of the symplectic group SL(2,R), which we use to build the coherent states (CS) associated to the system under investigation. The trapped ion is treated as a harmonic oscillator (HO) to which we associate the quantum Hamilton function. We obtain the kinetic and potential energy operators as functions of the Lie algebra generators and supply the expressions for the classical coordinate, momentum, kinetic and potential energy, along with the total energy. Moreover, we also infer the dispersions for the coordinate and momentum, together with the asymmetry and the flatness parameter for the distribution. The system interaction with laser radiation is also examined for a system of identical two-level atoms. The Hamilton function for the Dicke model is derived. The optical system is modelled as a HO (trapped ion) that undergoes interaction with an external laser field and we use it to engineer a squeezed state of the electromagnetic (EM) field. We consider coherent and squeezed states associated to both ion dynamics and to the EM field. The approach used enables one to build CS in a compact and smart manner by use of the group theory.

Mapping Quantum Chemical Dynamics Problems to Spin-Lattice Simulators.

The accurate computational determination of chemical, materials, biological, and atmospheric properties has a critical impact on a wide range of health and environmental problems, but is deeply limited by the computational scaling of quantum mechanical methods. The complexity of quantum chemical studies arises from the steep algebraic scaling of electron correlation methods and the exponential scaling in studying nuclear dynamics and molecular flexibility. To date, efforts to apply quantum hardware to such quantum chemistry problems have focused primarily on electron correlation. Here, we provide a framework that allows for the solution of quantum chemical nuclear dynamics by mapping these to quantum spin-lattice simulators. Using the example case of a short-strong hydrogen-bonded system, we construct the Hamiltonian for the nuclear degrees of freedom on a single Born-Oppenheimer surface and show how it can be transformed to a generalized Ising model Hamiltonian. We then demonstrate a method to determine the local fields and spin-spin couplings needed to identically match the molecular and spin-lattice Hamiltonians. We describe a protocol to determine the on-site and intersite coupling parameters of this Ising Hamiltonian from the Born-Oppenheimer potential and nuclear kinetic energy operator. Our approach represents a paradigm shift in the methods used to study quantum nuclear dynamics, opening the possibility to solve both electronic structure and nuclear dynamics problems using quantum computing systems.

On the inclusion of cusp effects in expectation values with explicitly correlated Gaussians

AbstractThis paper elaborates the integral transformation technique and uses it for the case of the non‐relativistic kinetic and Coulomb potential energy operators, as well as for the relativistic mass‐velocity and Darwin terms. The techniques are tested for the ground electronic state of the helium atom and perturbative relativistic energies are reported for the ground electronic state of the molecular ion near its equilibrium structure.

Encapsulation of a Water Molecule inside C60 Fullerene: The Impact of Confinement on Quantum Features.

We introduce an efficient quantum fully coupled computational scheme within the multiconfiguration time-dependent Hartree (MCTDH) approach to handle the otherwise extremely costly computations of translational–rotational–vibrational states and energies of light-molecule endofullenes. Quantum calculations on energy levels are reported for a water molecule inside C60 fullerene by means of such a systematic approach that includes all nine degrees of freedom of H2O@C60 and does not consider restrictions above them. The potential energy operator is represented as a sum of natural potentials employing the n-mode expansion, along with the exact kinetic energy operator, by introducing a set of Radau internal coordinates for the H2O molecule. On the basis of the present rigorous computations, various aspects of the quantized intermolecular dynamics upon confinement of H2O@C60 are discussed, such as the rotational energy level splitting and the significant frequency shifts of the encapsulated water molecule vibrations. The impact of water encapsulation on quantum features is explored, and insights into the nature of the underlying forces are provided, highlighting the importance of a reliable first-principles description of the guest–host interactions.

Wave Packet Dynamical Simulation of Quasiparticle Interferences in 2D Materials

Materials consisting of single- or a few atomic layers have extraordinary physical properties, which are influenced by the structural defects. We present two calculation methods based on wave packet (WP) dynamics, where we compute the scattering of quasiparticle WPs on localized defects. The methods are tested on a graphene sheet: (1) We describe the perfect crystal lattice and the electronic structure by a local atomic pseudopotential, then calculate the Bloch eigenstates and build a localized WP from these states. The defect is represented by a local potential, then we compute the scattering by the time development of the WP. (2) We describe the perfect crystal entirely by the kinetic energy operator, then we calculate the scattering on the local defect described by the potential energy operator. The kinetic energy operator is derived from the dispersion relation, which can be obtained from any electronic structure calculation. We also verify the method by calculating Fourier transform images and comparing them with experimental FFT-LDOS images from STM measurements. These calculation methods make it possible to study the quasiparticle interferences, inter- and intra-valley scattering, anisotropic scattering, etc., caused by defect sites for any 2D material.

Collinear configuration of the helium atom and two-electron ions

Collinear configurations of the helium-like atomic systems, relevant, e.g., for the quasifree mechanism of the double photoionization of helium, are studied, parameterized by the single scalar parameter −1≤λ≤1 (“collinear parameter”) where λ=0 corresponds to the electron–nucleus (e −−n) coalescence and λ=1 corresponds to the electron–electron (e −−e) coalescence. In general, λ>0 corresponds to the n–e–e configuration, and λ<0 to the e–n–e configuration. Simple mathematical representations of the expectation values of the Dirac delta function relevant for the collinear configurations are derived and calculated from fully three-body dynamics without approximation for the two-electron atomic wave functions with nuclear charge 1≤Z≤5. Simple formulas for calculating the expectation values of the kinetic and potential energy operators in collinear configurations are derived. Unusual physical properties of the n-e-e collinear configurations found for certain ranges of λ are presented. The first few angular Fock coefficients for collinear configurations are derived as functions of λ. Highly accurate model wave functions describing the ground states of the two-electron atoms with a collinear arrangement of the particles are constructed. All results are illustrated in tables and figures.

Influence of the Pauli principle on two-cluster potential energy

We study effects of the Pauli principle on the potential energy of two-cluster systems. The object of the investigation is the lightest nuclei of p-shell with a dominant \boldsymbol{\alpha}𝛂-cluster channel. For this aim we construct matrix elements of two-cluster potential energy between cluster oscillator functions with and without full antisymmetrization. Eigenvalues and eigenfunctions of the potential energy matrix are studied in detail. Eigenfunctions of the potential energy operator are presented in oscillator, coordinate and momentum spaces. We demonstrate that the Pauli principle affects more strongly the eigenfunctions than the eigenvalues of the matrix and leads to the formation of resonance and trapped states.

How the antisymmetrization affects a cluster–cluster interaction: Two-cluster systems

We study effects of the antisymmetrization on the potential energy of two-cluster systems. The object of the investigation is the lightest nuclei of p-shell with a dominant alpha-cluster channel. For this aim we construct matrix elements of two-cluster potential energy between cluster oscillator functions with and without full antisymmetrization. Eigenvalues and eigenfunctions of the potential energy matrix are studied in detail. Eigenfunctions of the potential energy operator are presented in oscillator, coordinate and momentum spaces. We demonstrate that the Pauli principle affects more strongly the eigenfunctions than the eigenvalues of the matrix and leads to the formation of resonance and trapped states.

Relativistic Expressions for the r.m.s. Radii of the λ-Particle Orbits in Hypernuclei and of the Corresponding Potential Energies

The root mean square radii of the Λ-particle orbits in hypemuclei are calculated semi-analytically for every bound state, using the Dirac equation with a scalar potential Us{r) and the fourth component of a vector potential Uv{f) in the case of rectangular shapes of these potentials with the same radius R.ln addition an analytic expression of the expectation value of the corresponding potential energy operator is derived. For the above quantities, expressions of the energy eigenvalues in terms of the potential parameters are needed and approximate formulae may be used, in certain cases. The variation of these quantities with the mass number is also investigated and numerical calculations are performed.

Quantum mechanical virial-like theorem for confined quantum systems

Confinement of atoms inside impenetrable (hard) and penetrable (soft) cavities has been studied for nearly eight decades. However, a unified virial theorem for such systems has not yet been found. Here we provide a general virial-like equation in terms of mean square and expectation values of potential and kinetic energy operators. It appears to be applicable in both free and confined situations. Apart from that, we have derived an equation using the time-independent Schr\"odinger equation, which can be treated as a sufficient condition for a given stationary quantum state. A change of boundary condition does not affect these virial equations. In the hard confining condition, the perturbing (confining potential) does not affect the expression; it merely shifts the boundary from infinity to a finite region. In the soft case, on the contrary, the final expression includes contributions from the perturbing term. These are demonstrated numerically for several representative enclosed systems like harmonic oscillators (one-dimensional and three-dimensional) and hydrogen atoms. Its applicability in various other confinements (including angular) has been discussed. In essence, a virial equation has been proposed for free and confined quantum systems, from simple arguments.

MCTDH on-the-fly: Efficient grid-based quantum dynamics without pre-computed potential energy surfaces.

We present significant algorithmic improvements to a recently proposed direct quantum dynamics method, based upon combining well established grid-based quantum dynamics approaches and expansions of the potential energy operator in terms of a weighted sum of Gaussian functions. Specifically, using a sum of low-dimensional Gaussian functions to represent the potential energy surface (PES), combined with a secondary fitting of the PES using singular value decomposition, we show how standard grid-based quantum dynamics methods can be dramatically accelerated without loss of accuracy. This is demonstrated by on-the-fly simulations (using both standard grid-based methods and multi-configuration time-dependent Hartree) of both proton transfer on the electronic ground state of salicylaldimine and the non-adiabatic dynamics of pyrazine.

Generalized Potential Operator and the Comments on Its Time Evolution

Potential energy inherited by a system plays a significant role in the system analysis. The potential energy operators are not generalized till date. Each system has its unique basis to explain the same. This article provides a novel approach to determine a generalized potential operator for a particle using Poisson's equation in a set basis in accordance with the corresponding characteristics manifested by the unit elements of basis. The mathematics includes the fundamental definitions of quantum physics and geometrical inference of manifolds. We shall see that the operator itself is time variant and medium dependent, which in turn depicts the evolution of the system in different mediums.

Design of specially adapted reactive coordinates to economically compute potential and kinetic energy operators including geometry relaxation.

Quantum dynamics simulations require prior knowledge of the potential energy surface as well as the kinetic energy operator. Typically, they are evaluated in a low-dimensional subspace of the full configuration space of the molecule as its dimensionality increases proportional to the number of atoms. This entails the challenge to find the most suitable subspace. We present an approach to design specially adapted reactive coordinates spanning this subspace. In addition to the essential geometric changes, these coordinates take into account the relaxation of the non-reactive coordinates without the necessity of performing geometry optimizations at each grid point. The method is demonstrated for an ultrafast photoinduced bond cleavage in a commonly used organic precursor for the generation of electrophiles. The potential energy surfaces for the reaction as well as the Wilson G-matrix as part of the kinetic energy operator are shown for a complex chemical reaction, both including the relaxation of the non-reactive coordinates on equal footing. A microscopic interpretation of the shape of the G-matrix elements allows to analyze the impact of the non-reactive coordinates on the kinetic energy operator. Additionally, we compare quantum dynamics simulations with and without the relaxation of the non-reactive coordinates included in the kinetic energy operator to demonstrate its influence.

Exactly solvable problems in the momentum space with a minimum uncertainty in position

A new approach in solution of simple quantum mechanical problems in deformed space with minimal length is presented. We propose the generalization of Schrödinger equation in momentum representation on the case of deformed Heisenberg algebra with minimal length. Assuming that the kernel of potential energy operator does not change in the case of deformation, we obtain exact solution of eigenproblem of a particle in delta potential as well as double delta potential. Particle in Coulomb like potential is revisited and the problem of inversibility and hermicity of inverse coordinate operator is solved.

Vibrational Calculations of Higher-Order Weakly Bound Complexes: The He3,4I2 Cases

The structure and relative stability of higher-order He3,4I2 clusters are investigated by carrying out full-dimensional quantum calculations within the multiconfiguration time-dependent Hartree framework. The full interaction between the I2 molecule and the He atoms is based on analytical three-body ab initio He-I2 potentials obtained from high level ab initio calculations plus the He-He interaction. The low-lying minima on the potential surfaces are found to be very close in energy with the He atoms in a ring encircling the dopant for the global minimum structure, while for the local minima one or two of the He atoms prefer the linear arrangements along the I2-axis. Such classical description on the basis of the potential energy landscape is corrected by including anharmonic quantum effects, present in highly floppy systems, by carrying out full dimensional quantum calculations. The potential energy operator was constructed by natural potential fits, while a mode combination scheme was employed to optimize the computational cost of the improved relaxation calculations. The obtained results predict the relative stability of the He3,4I2 isomers at zero temperature and provide benchmark data on binding energies and structural properties of these van der Waals systems. The (2,1) and (2,2), involving two He atoms in the T-shape and one or two He atoms in the linear configurations, respectively, are found to be the most stable isomers, although extremely close in energy with the (3,0) and (4,0) ones as predicted by classical optimizations. Comparison with experimental data on similar systems at low temperatures is also discussed. This analysis indicates once more the importance of quantum delocalization and the need of accurate quantum-mechanical treatments to characterize such doped helium nanosystems.