Hamiltonian For Use Research Articles

Digital-Analog Quantum Simulations Using the Cross-Resonance Effect

Digital-analog quantum computation aims to reduce the currently infeasible resource requirements needed for near-term quantum information processing by replacing sequences of one- and two-qubit gates with a unitary transformation generated by the systems' underlying Hamiltonian. Inspired by this paradigm, we consider superconducting architectures and extend the cross-resonance effect, up to first order in perturbation theory, from a two-qubit interaction to an analog Hamiltonian acting on 1D chains and 2D square lattices which, in an appropriate reference frame, results in a purely two-local Hamiltonian. By augmenting the analog Hamiltonian dynamics with single-qubit gates we show how one may generate a larger variety of distinct analog Hamiltonians. We then synthesize unitary sequences, in which we toggle between the various analog Hamiltonians as needed, simulating the dynamics of Ising, $XY$, and Heisenberg spin models. Our dynamics simulations are Trotter error-free for the Ising and $XY$ models in 1D. We also show that the Trotter errors for 2D $XY$ and 1D Heisenberg chains are reduced, with respect to a digital decomposition, by a constant factor. In order to realize these important near-term speedups, we discuss the practical considerations needed to accurately characterize and calibrate our analog Hamiltonians for use in quantum simulations. We conclude with a discussion of how the Hamiltonian toggling techniques could be extended to derive new analog Hamiltonians which may be of use in more complex digital-analog quantum simulations for various models of interacting spins.

Additive atomic approximation for relativistic effects: A two-component Hamiltonian for molecular electronic structure calculations.

An approximate relativistic two-component Hamiltonian for use in molecular electronic structure calculations is derived in the form of a sum of fixed atom-centered kinetic and spin-orbit operators added to the non-relativistic Hamiltonian. Starting from the well-known zeroth-order regular approximation, further steps are taken to get rid of its nonlinearity in the potential, ending up with a simple formulation with easily computable integrals that can seamlessly work with any traditional electronic structure method. Molecular tests show a good accuracy of this approximation.

Shell-model interactions from chiral effective field theory

We construct valence-space Hamiltonians for use in shell-model calculations, where the residual two-body interaction is based on symmetry principles and the low-momentum expansion from chiral effective field theory. In addition to the usual free-space contact interactions, we also include novel center-of-mass--dependent operators that arise due to the Galilean invariance breaking by in-medium effects. We fitted the low-energy constants to 441 ground- and excited-state energies in the sd shell and obtained a root-mean-square derivation of 1.8 MeV at leading order and of 0.5 MeV at next-to-leading order, with natural low-energy constants in all cases. The developed chiral shell-model interactions enable order-by-order uncertainty estimates and show promising predictions for neutron-rich isotopes beyond the fitted data set.

Preface: Trends in Computational Catalysis

Computation has continued to grow in significance in the field of catalysis. One routinely sees a wide range of relevant properties computed ranging from adsorption energies and reaction barriers, to full reaction mechanisms, trends across series of catalysts, as well as in developing models of what surfaces might look like under reaction conditions. Computations are performed using density functional theory (DFT), molecular dynamics, kinetic Monte Carlo, and microkinetic models among other approaches. This growth has been spurred by continued development of faster computer hardware, the ready availability of high quality software, and the development of new algorithms and methodologies. In this special issue, we present several current trends in computational catalysis, broadly organized in the areas of (1) adsorption and molecular interactions, (2) kinetics and mechanistic studies, and (3) new methodologies. Complex adsorbates and surfaces are increasingly examined using computational methods. For example, in this issue Sitamraju et al., examined the adsorption of aromatic sulfur-containing molecules on oxide surfaces, and Han examined the subtle interactions of chiral molecules with chiral bimetallic surfaces. In work by Curet-Arana et al., trends in the reaction of CO2 with organometallic molecules were studied. In these three papers, computation provided insight into the nature of the adsorbate–surface or adsorbate-molecule interactions, and provided some insight into how to choose or design materials with desired properties. A growing trend has been the use of DFT calculations to model coverage dependent adsorption properties and to parameterize coarse-grained models such as lattice gas Hamiltonians for use in Monte Carlo simulations. Alfonso illustrates this approach in studying the effects of sulfur poisoning on CO adsorption and shows that in addition to reducing the coverage of CO due to sulfur poisoning, the mobility of CO is dramatically reduced by sulfur poisoning. Kinetic and mechanistic computations have traditionally been fairly expensive compared to computation of adsorption properties. The calculation of reaction barriers and pre-exponential factors remains more expensive than simple adsorption calculations. The continued development of faster and highly parallel computing hardware as well as software enhancement has made many aspects of these calculations routine. It is becoming increasingly common to see studies in reactivity trends of molecules on different surfaces. In this issue, the selectivity in various decomposition pathways of glycerol on Pt(111) are considered by Liu and Greeley, and the decomposition of furan on Pd(111) is considered by Xu. These molecules are significant in biomass-conversion strategies. In a combined experimental and computational study, McCalman et al. consider the reduction of N2O by H2 on Pd catalysts in water, including the effects of adsorbed water molecules in the simulations. The catalyst structures being used in computation continue to grow in complexity. Wei et al. examine the mechanism of the water gas shift reaction over two alloy surfaces using DFT. Balbuena et al. examine effects of segregation and dissolution in fuel cell alloy electrocatalysts using computational methods ranging from DFT to molecular dynamics. Direct computation of properties on nanoparticles is becoming increasingly common. Grabow et al. investigate the synthesis of hydrogen peroxide from hydrogen and oxygen over gold nanoparticles, while Cheng J. Kitchin (&) Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA, USA e-mail: jkitchin@andrew.cmu.edu

Relativistic Hamiltonians for Chemistry: A Primer

Entering Dirac territory: This Minireview provides a guide to two- and four-component relativistic Hamiltonians for use in quantum chemistry with particular emphasis on the recently developed eXact two-component (X2C) Hamiltonian.

Hamiltonian mechanics and relative equilibria of orbiting gyrostats

We use a noncanonical Hamiltonian approach to study the equilibrium attitudes of a rigid satellite with N rotors in a central gravitational field. The relative equilibria of this system of equations represent steady motions of the body as seen in the body frame, and correspond to stationary points of the Hamiltonian constrained by Casimir functions. This approach leads to an algorithm for computing the equilibria, and simultaneously providing direct stability information, since the calculations required to solve the constrained minimization problem are also involved in computing the positive definiteness of the constrained Hamiltonian for use as a Lyapunov function.

Systematic treatment of relativistic effects accurate through arbitrarily high order in α2

A systematic method for the generation of two-component relativistic Hamiltonians for use in relativistic quantum chemistry is presented and discussed. The free particle Foldy–Wouthuysen transformation of the Dirac Hamiltonian is performed prior to the determination of the block-diagonalizing unitary transformation. The latter can be determined iteratively through arbitrarily high leading order with respect to α (fine structure constant). Certain freedom in the initialization of the iterative solution leads to the whole class of two-component Hamiltonians h2k which are exact through the order of α2k and differ in contributions of all higher orders in α2. The efficiency of different iterative schemes is analyzed. Also the relation between the present method and the Douglas–Kroll approximation is investigated. The performance of two-component Hamiltonians for k=2, 3, and 4 is studied numerically in calculations of energies of the 1s1/2 level in heavy hydrogen-like ions. Their performance in calculations of the valence-determined atomic and molecular properties is investigated by computing the ionization potential of Au and spectroscopic constants of the AuH molecule. The total energy of these systems strongly depends on the level of exactness with respect to α2. However, for moderately relativistic systems the α4-class Hamiltonian derived in this paper is found to be sufficient for accurate calculations of valence-determined properties.

A quasiparticle derivation of a zeroth-order Hamiltonian for use in multiconfigurational perturbation theories

A derivation of a zeroth-order Hamiltonian for use in multiconfigurational perturbation theories is presented. This derivation is based on a quasiparticle Green function formalism. It is shown that this formalism can naturally describe the physical picture of the zeroth-order Hamiltonian as an approximate one-particle potential set up by the multiconfigurational reference wave function. The derivation also provides a physically meaningful definition of single-particle eigenvalues which arise from the multiconfigurational perturbation theory.

Iteratively determined effective Hamiltonians for the adiabatically reduced coupled equations approach to intramolecular dynamics calculations

An iterative procedure is proposed for determining increasingly accurate effective Hamiltonians for use in the adiabatically reduced coupled equations approach to intramolecular dynamics calculations [J. Chem. Phys. 84, 2254 (1986)]. The relationships between this iterative determination of the effective Hamiltonian, which is based on an adiabatic approximation, and some other partitioning methods for determining an effective Hamiltonian are discussed. The present iterative procedure provides accurate agreement with the exact dynamics for the two specific model systems studied.

Electronic and nuclear order in non-magnetic Γ 3 (E) doublet systems

Effective many-body Hamiltonians for use in non-magnetic Γ 3 (E) doublet systems are presented and discussed. It is argued, for example, that such compounds may be characterized by two transition temperatures: the first to an electronic quadrupolar state ( T q ≲ 1 K), and the second to a nuclear ordered state ( T n ≈ 1 mK). It is suggested that the driving force behind the electronic-quadrupolar state in the elpasolite Γ 3 compounds is the cooperative Jahn-Teller mechanism, whereas the nuclear ordered state is driven by dipolar interactions between enhanced nuclear spins.

Orientation of rigid linear molecules: A Hamiltonian for use in lattice dynamics

A previous treatment for non-linear molecules is adapted to the case of linear molecules. The quaternions of the former case are replaced by direction cosines of the linear axis. The resulting Hamiltonian is developed for application to lattice dynamics and other problems involving orientational displacements.

Screened polarization waves and the energies of simple metals: Formulation

We investigate the effect that core polarization has on the energy of a simple metal. To do this we establish an approximate Hamiltonian based on the standard result for nearly-free-electron metals but generalized to include interactions between localized core electrons. An application of perturbation theory then leads to expressions for various contributions to the total energy and a reformulation of the one-electron Hamiltonian for use in structural applications. The fluctuating dipole interactions between the ions are dynamically screened by the valence electrons. In the crystalline context the corresponding excitations are screened polarization waves and are manifested as screened van der Waals forces between ions. The core electrons also participate in the screening of other interactions in the metal.