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All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Quantum Davidson algorithm for excited states

Abstract Excited state properties play a pivotal role in various chemical and physical phenomena, such as charge separation and light emission. However, the primary focus of most existing quantum algorithms has been the ground state, as seen in quantum phase estimation and the variational quantum eigensolver (VQE). Although VQE-type methods have been extended to explore excited states, these methods grapple with optimization challenges. In contrast, the quantum Krylov subspace (QKS) method has been introduced to address both ground and excited states, positioning itself as a cost-effective alternative to quantum phase estimation. However, conventional QKS methodologies depend on a pre-generated subspace through real or imaginary-time evolutions. This subspace is inherently expansive and can be plagued with issues like slow convergence or numerical instabilities, often leading to relatively deep circuits. Our research presents an economic QKS algorithm, which we term the quantum Davidson (QDavidson) algorithm. This innovation hinges on the iterative expansion of the Krylov subspace and the incorporation of a pre-conditioner within the Davidson framework. By using the residues of eigenstates to expand the Krylov subspace, we manage to formulate a compact subspace that aligns closely with the exact solutions. This iterative subspace expansion paves the way for a more rapid convergence in comparison to other QKS techniques, such as the quantum Lanczos. Using quantum simulators, we employ the novel QDavidson algorithm to delve into the excited state properties of various systems, spanning from the Heisenberg spin model to real molecules. Compared to the existing QKS methods, the QDavidson algorithm not only converges swiftly but also demands a significantly shallower circuit. This efficiency establishes the QDavidson method as a pragmatic tool for elucidating both ground and excited state properties on quantum computing platforms.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Implementation of McMurchie–Davidson Algorithm for Gaussian AO Integrals Suited for SIMD Processors

We report an implementation of the McMurchie− Davidson evaluation scheme for 1- and 2-particle Gaussian AO integrals designed for processors with Single Instruction Multiple Data (SIMD) instruction sets. Like in our recent MD implementation for graphical processing units (GPUs) [Asadchev, A.; Valeev, E. F.. J. Chem. Phys. 2024, 160, 244109.], variable-sized batches of shellsets of integrals are evaluated at a time. By optimizing for the floating point instruction throughput rather than minimizing the number of operations, this approach achieves up to 50% of the theoretical hardware peak FP64 performance for many common SIMD-equipped platforms (AVX2, AVX512, NEON), which translates to speedups of up to 30 over the state-of-the-art one-shellset-at-a-time implementation of Obara−Saika-type schemes in Libint for a variety of primitive and contracted integrals. As with our previous work, we rely on the standard C++ programming language such as the std::simd standard library feature to be included in the 2026 ISO C++ standard without any explicit code generation to keep the code base small and portable. The implementation is part of the open source LibintX library freely available at https://github.com/ValeevGroup/libintx.

Basis sets↗
DOE OSTI2015

A projected preconditioned conjugate gradient algorithm for computing many extreme eigenpairs of a Hermitian matrix

We present an iterative algorithm for computing an invariant subspace associated with the algebraically smallest eigenvalues of a large sparse or structured Hermitian matrix A. We are interested in the case in which the dimension of the invariant subspace is large (e.g., over several hundreds or thousands) even though it may still be small relative to the dimension of A. These problems arise from, for example, density functional theory (DFT) based electronic structure calculations for complex materials. The key feature of our algorithm is that it performs fewer Rayleigh–Ritz calculations compared to existing algorithms such as the locally optimal block preconditioned conjugate gradient or the Davidson algorithm. It is a block algorithm, and hence can take advantage of efficient BLAS3 operations and be implemented with multiple levels of concurrency. We discuss a number of practical issues that must be addressed in order to implement the algorithm efficiently on a high performance computer.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2017

Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

We present two efficient iterative algorithms for solving the linear response eigen- value problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into a product eigenvalue problem that is self-adjoint with respect to a K-inner product. This product eigenvalue problem can be solved efficiently by a modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. The solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. However, the other component of the eigenvector can be easily recovered in a postprocessing procedure. Therefore, the algorithms we present here are more efficient than existing algorithms that try to approximate both components of the eigenvectors simultaneously. The efficiency of the new algorithms is demonstrated by numerical examples.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

New algorithms for iterative matrix‐free eigensolvers in quantum chemistry

New algorithms for iterative diagonalization procedures that solve for a small set of eigen‐states of a large matrix are described. The performance of the algorithms is illustrated by calculations of low and high‐lying ionized and electronically excited states using equation‐of‐motion coupled‐cluster methods with single and double substitutions (EOM‐IP‐CCSD and EOM‐EE‐CCSD). We present two algorithms suitable for calculating excited states that are close to a specified energy shift (interior eigenvalues). One solver is based on the Davidson algorithm, a diagonalization procedure commonly used in quantum‐chemical calculations. The second is a recently developed solver, called the “Generalized Preconditioned Locally Harmonic Residual (GPLHR) method.” We also present a modification of the Davidson procedure that allows one to solve for a specific transition. The details of the algorithms, their computational scaling, and memory requirements are described. The new algorithms are implemented within the EOM‐CC suite of methods in the Q‐Chem electronic structure program. © 2014 Wiley Periodicals, Inc.

Zuev, Dmitry↗
DOE OSTI2014

Tunneling splitting in double-proton transfer: Direct diagonalization results for porphycene

Zero-point and excited level splittings due to double-proton tunneling are calculated for porphycene and the results are compared with experiment. The calculation makes use of a multidimensional imaginary-mode Hamiltonian, diagonalized directly by an effective reduction of its dimensionality. Porphycene has a complex potential energy surface with nine stationary configurations that allow a variety of tunneling paths, many of which include classically accessible regions. A symmetry-based approach is used to show that the zero-point level, although located above the cis minimum, corresponds to concerted tunneling along a direct trans − trans path; a corresponding cis − cis path is predicted at higher energy. This supports the conclusion of a previous paper [Z. Smedarchina, W. Siebrand, and A. Fernández-Ramos, J. Chem. Phys. 127, 174513 (2007)] based on the instanton approach to a model Hamiltonian of correlated double-proton transfer. A multidimensional tunneling Hamiltonian is then generated, based on a double-minimum potential along the coordinate of concerted proton motion, which is newly evaluated at the RI-CC2/cc-pVTZ level of theory. To make it suitable for diagonalization, its dimensionality is reduced by treating fast weakly coupled modes in the adiabatic approximation. This results in a coordinate-dependent mass of tunneling, which is included in a unique Hermitian form into the kinetic energy operator. The reduced Hamiltonian contains three symmetric and one antisymmetric mode coupled to the tunneling mode and is diagonalized by a modified Jacobi-Davidson algorithm implemented in the Jadamilu software for sparse matrices. The results are in satisfactory agreement with the observed splitting of the zero-point level and several vibrational fundamentals after a partial reassignment, imposed by recently derived selection rules. They also agree well with instanton calculations based on the same Hamiltonian.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2016

Iterative diagonalization of the non-Hermitian transcorrelated Hamiltonian using a plane-wave basis set: Application to sp-electron systems with deep core states

We develop an iterative diagonalization scheme in solving a one-body self-consistent-field equation in the transcorrelated (TC) method using a plane-wave basis set. Non-Hermiticity in the TC method is well handled with a block-Davidson algorithm. We verify that the required computational cost is efficiently reduced by our algorithm. In addition, we apply our plane-wave-basis TC calculation to some simple sp-electron systems with deep core states to elucidate an impact of the pseudopotential approximation to the calculated band structures. We find that a position of the deep valence bands is improved by an explicit inclusion of core states, but an overall band structure is consistent with a regular setup that includes core states into the pseudopotentials. This study offers an important understanding for the future application of the TC method to strongly correlated solids.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

Accurate and efficient calculation of excitation energies with the active-space particle-particle random phase approximation

An efficient method for calculating excitation energies based on the particle-particle random phase approximation (ppRPA) is presented. Neglecting the contributions from the high-lying virtual states and the low-lying core states leads to the significantly smaller active-space ppRPA matrix while keeping the error to within 0.05 eV from the corresponding full ppRPA excitation energies. The resulting computational cost is significantly reduced and becomes less than the construction of the non-local Fock exchange potential matrix in the self-consistent-field (SCF) procedure. With only a modest number of active orbitals, the original ppRPA singlet-triplet (ST) gaps as well as the low-lying single and double excitation energies can be accurately reproduced at much reduced computational costs, up to 100 times faster than the iterative Davidson diagonalization of the original full ppRPA matrix. For high-lying Rydberg excitations where the Davidson algorithm fails, the computational savings of active-space ppRPA with respect to the direct diagonalization is even more dramatic. The virtues of the underlying full ppRPA combined with the significantly lower computational cost of the active-space approach will significantly expand the applicability of the ppRPA method to calculate excitation energies at a cost of O(K^{4}), with a prefactor much smaller than a single SCF Hartree-Fock (HF)/hybrid functional calculation, thus opening up new possibilities for the quantum mechanical study of excited state electronic structure of large systems.

74 ATOMIC AND MOLECULAR PHYSICS↗
NASA NTRS2006

Measuring Constraint-Set Utility for Partitional Clustering Algorithms

Clustering with constraints is an active area of machine learning and data mining research. Previous empirical work has convincingly shown that adding constraints to clustering improves the performance of a variety of algorithms. However, in most of these experiments, results are averaged over different randomly chosen constraint sets from a given set of labels, thereby masking interesting properties of individual sets. We demonstrate that constraint sets vary significantly in how useful they are for constrained clustering; some constraint sets can actually decrease algorithm performance. We create two quantitative measures, informativeness and coherence, that can be used to identify useful constraint sets. We show that these measures can also help explain differences in performance for four particular constrained clustering algorithms.

constraints↗
DOE OSTI2024

3-center and 4-center 2-particle Gaussian AO integrals on modern accelerated processors

We report an implementation of the McMurchie–Davidson (MD) algorithm for 3-center and 4-center 2-particle integrals over Gaussian atomic orbitals (AOs) with low and high angular momenta l and varying degrees of contraction for graphical processing units (GPUs). This work builds upon our recent implementation of a matrix form of the MD algorithm that is efficient for GPU evaluation of 4-center 2-particle integrals over Gaussian AOs of high angular momenta (l ≥ 4) [A. Asadchev and E. F. Valeev, J. Phys. Chem. A 127, 10889–10895 (2023)]. The use of unconventional data layouts and three variants of the MD algorithm allow for the evaluation of integrals with double precision and sustained performance between 25% and 70% of the theoretical hardware peak. Performance assessment includes integrals over AOs with l ≤ 6 (a higher l is supported). Preliminary implementation of the Hartree–Fock exchange operator is presented and assessed for computations with up to a quadruple-zeta basis and more than 20 000 AOs. The corresponding C++ code is part of the experimental open-source LibintX library available at https://github.com/ValeevGroup/libintx.

Chemistry↗
DOE OSTI2014

Erratum: “A finite difference Davidson procedure to sidestep full ab initio hessian calculation: Application to characterization of stationary points and transition state searches”

Certain key contributions to the field were not presented in the Introduction section of our recent paper. In addition to the variational method described in Ref. 14 and iterative diagonalization methods in Refs. 15–17, we wish to direct the attention of the reader to other significant efforts that have utilized similar principles for various applications. Reiher and Neugebauer developed a mode-tracking algorithm using the Davidson method for calculating only the modes relevant for vibrational analysis. Similarly, a block-Davidson approach via finite differentiation of the gradient was implemented for calculating hessian-vector products for normal mode analysis. Deglmann and Furche developed a method for stationary point characterization with density functional theory where full solution of all coupled-perturbed Kohn-Sham equations, the bottleneck in hessian calculation, could be avoided through iterative calculation of hessian-vector products to determine only the lowest eigenvalues.

99 GENERAL AND MISCELLANEOUS↗
NASA NTRS1975

A variable-metric algorithm employing linear and quadratic penalties

A variable-metric algorithm is described that uses both linear and quadratic penalty terms for handling nonlinear constraints. Quadratic penalty coefficients are adjusted in a process which maintains a positive-definite matrix of second partial derivatives of the function without generating the large positive eigenvalues which cause zigzagging and slow convergence. The schemes suggested use inferred second-order properties not only in terms of the variable metric of the Davidson-Fletcher-Powell algorithm (or its relatives) but by estimating of second directional derivatives by fitting cubics to various functions along search directions.

Kelley, H. J.↗
NASA NTRS2006

When is Constrained Clustering Beneficial, and Why?

Several researchers have shown that constraints can improve the results of a variety of clustering algorithms. However, there can be a large variation in this improvement, even for a fixed number of constraints for a given data set. We present the first attempt to provide insight into this phenomenon by characterizing two constraint set properties: informativeness and coherence. We show that these measures can help explain why some constraint sets are more beneficial to clustering algorithms than others. Since they can be computed prior to clustering, these measures can aid in deciding which constraints to use in practice.

clustering algorithms↗
NASA NTRS2000

Ordering Unstructured Meshes for Sparse Matrix Computations on Leading Parallel Systems

The ability of computers to solve hitherto intractable problems and simulate complex processes using mathematical models makes them an indispensable part of modern science and engineering. Computer simulations of large-scale realistic applications usually require solving a set of non-linear partial differential equations (PDES) over a finite region. For example, one thrust area in the DOE Grand Challenge projects is to design future accelerators such as the SpaHation Neutron Source (SNS). Our colleagues at SLAC need to model complex RFQ cavities with large aspect ratios. Unstructured grids are currently used to resolve the small features in a large computational domain; dynamic mesh adaptation will be added in the future for additional efficiency. The PDEs for electromagnetics are discretized by the FEM method, which leads to a generalized eigenvalue problem Kx = AMx, where K and M are the stiffness and mass matrices, and are very sparse. In a typical cavity model, the number of degrees of freedom is about one million. For such large eigenproblems, direct solution techniques quickly reach the memory limits. Instead, the most widely-used methods are Krylov subspace methods, such as Lanczos or Jacobi-Davidson. In all the Krylov-based algorithms, sparse matrix-vector multiplication (SPMV) must be performed repeatedly. Therefore, the efficiency of SPMV usually determines the eigensolver speed. SPMV is also one of the most heavily used kernels in large-scale numerical simulations.

Oliker, Leonid↗
NASA NTRS1994

The relationship between the microwave radar cross section and both wind speed and stress: Model function studies using Frontal Air-Sea Interaction Experiment data

The Frontal Air-Sea Interaction Experiment (FASINEX) provided a unique data set with coincident airborne scatterometer measurements of the ocean surface radar cross section (RCS)(at Ku band) and near-surface wind and wind stress. These data have been analyzed to study new model functions which relate wind speed and surface friction velocity (square root of the kinematic wind stress) to the radar cross section and to better understand the processes in the boundary layer that have a strong influence on the radar backscatter. Studies of data from FASINEX indicate that the RCS has a different relation to the friction velocity than to the wind speed. The difference between the RCS models using these two variables depends on the polarization and the incidence angle. The radar data have been acquired from the Jet Propulsion Laboratory airborne scatterometer. These data span 10 different flight days. Stress measurements were inferred from shipboard instruments and from aircraft flying at low altitudes, closely following the scatterometer. Wide ranges of radar incidence angles and environmental conditions needed to fully develop algorithms are available from this experiment.

Weissman, David E.↗
DOE OSTI2016

SU-F-J-161: Prostate Contouring in Patients with Bilateral Hip Prostheses: Impact of Using Artifact-Reduced CT Images and MRI

Purpose: To investigate the usefulness of metal artifact reduction in CT images of patients with bilateral hip prostheses (BHP) for contouring the prostate and determine if the inclusion of MR images provides additional benefits. Methods: Five patients with BHP were CT scanned using our clinical protocol (140kV, 300mAs, 3mm slices, 1.5mm increment, Philips Medical Systems, OH). Images were reconstructed with the orthopaedic metal artifact reduction (O-MAR) algorithm. MRI scanning was then performed (1.5T, GE Healthcare, WI) with a flat table-top (T{sub 2}-weighted, inherent body coil, FRFSE, 3mm slices with 0mm gap). All images were transferred to Pinnacle (Version 9.2, Philips Medical Systems). For each patient, two data sets were produced: one containing the O-MAR-corrected CT images and another containing fused MRI and O-MAR-corrected CT images. Four genito-urinary radiation oncologists contoured the prostate of each patient on the O-MAR-corrected CT data. Two weeks later, they contoured the prostate on the fused data set, blinded to all other contours. During each contouring session, the oncologists reported their confidence in the contours (1=very confident, 3=not confident) and the contouring difficulty that they experienced (1=really easy, 4=very challenging). Prostate volumes were computed from the contours and the conformity index was used to evaluate inter-observer variability. Results: Larger prostate volumes were found on the O-MAR-corrected CT set than on the fused set (p< 0.05, median=36.9cm{sup 3} vs. 26.63 cm{sup 3}). No significant differences were noted in the inter-observer variability between the two data sets (p=0.3). Contouring difficulty decreased with the addition of MRI (p<0.05) while the radiation oncologists reported more confidence in their contours when MRI was fused with the O-MAR-corrected CT data (p<0.05). Conclusion: This preliminary work demonstrated that, while O-MAR correction to CT images improves visualization of anatomy, the addition of MRI enhanced the oncologists’ confidence in contouring the prostate in patients with BHP.

60 APPLIED LIFE SCIENCES↗