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DOE OSTI2023

A solvable model of a nonlinear extension of quantum mechanics

We introduce a particular nonlinear generalization of quantum mechanics which has the property that it is exactly solvable in terms of the eigenvalues and eigenfunctions of the Hamiltonian of the usual linear quantum mechanics problem. Here, we hope that this simple example will elucidate some of the issues of interpreting nonlinear generalization of quantum mechanics that have been put forth to resolve questions about quantum measurement theory.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Geometric Interpretation of a Non-Linear Extension of Quantum Mechanics

We recently introduced a particular non-linear generalization of quantum mechanics that has the property that it is exactly solvable in terms of the eigenvalues and eigenfunctions of the Hamiltonian of the usual linear quantum mechanics problem. In this paper, we suggest that the two components of the wave function represent the system described by the Hamiltonian H in two different asymptotic regions of spacetime and we show that the non-linear terms can be viewed as giving rise to gravitational effects.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

Stochastic Schrödinger evolution over piecewise enlarged filtrations

This paper constructs a nonlinear filtering framework that admits appearances of new information processes at random times by introducing piecewise enlargements of filtrations and proposes a new energy-based Schrodinger evolution expressed as a stochastic differential equation on a complex Hilbert space. Each information process is modeled as the sum of a random variable taking the eigenvalues of a Hamiltonian and an independent Brownian bridge noise. It is shown that under a piecewise enlarged filtration, the wave function is a jump-diffusion process until it collapses at some terminal time. In between discontinuities, the dynamics of the state vector are governed by different Wiener processes and diffusion coefficients. This motivates the introduction of an inclusive chain of Kolmogorov probability spaces or a *-isomorphic chain of commutative von Neumann probability spaces, on which the quantum system evolves differently based on the number of active information processes. The expectation of the Hamiltonian at a given state is the solution of a second-order nonlinear differential equation determined by one of the possible regimes that the quantum system belongs to. It is shown that the collapse rate is a submartingale with positive jumps and the Shannon entropy process is a supermartingale with expected negative jumps when passing to higher-order probability spaces. The framework is extended to the case when the Hamiltonian is modeled as a function of a set of commutative operators, where each operator is associated with a different piecewise enlarged filtration.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2017

Mechanisms of interfacial reactivity in near surface and extreme environments

The local water structure surrounding ions in aqueous solutions greatly affects their chemical properties such as reaction rates, ion association, and proton and electron transport. These properties result in the behavior of ions in natural aqueous environments. For example ore transport is facilitated by chloride ion pair formation and the reaction of ions in an interface is strongly dependent on the dehydration of the ion hydration shell. We are developing the use of high-­resolution XAFS observations and 1st principles based MD-­XAFS analysis (spectra simulated using 1st principle methods with no adjustable parameters, AIMD) to interpret the solution properties of strongly interacting aqueous solutes under arbitrary pressure and temperature conditions. In the 1st principle MD-­XAFS method density functional theory (DFT) based MD simulations(Car and Parrinello, 1985) are used to generate a large ensemble of structural snap shots of the hydration region. These are then used to generate scattering intensities. I emphasize three points about this novel approach to analyzing XAFS data. 1st: As illustrated in Figure 1, the level of agreement between the calculated and observed intensities is considerably higher than has been obtained by any XAFS analysis to date (note 2nd shell region, R> 2 Å). 2nd: This result was obtained from a parameter free simulation with no fitting of the interaction potentials to any data. This supports the use of these methods for more difficult environments and more complex solutes (polyions). 3rd: New information about the shell structure (Figure 1) is now available because of this more detailed agreement. We note also that both multiple scattering and second shell features are well represented in the analysis. As far as we know this is the 1st analysis of second shell structure and multiple scattering. Excellent agreement has been obtained for most of the third row metal ions: Ca 2+ , Zn 2+ , Cu 2+ , Ni 2+ , Co 2+ , Mn 2+ , Fe 3+ , Cr 3+ . Calculations on these systems are demanding because of their open electronic shells, and high ionic charge. Principal Investigator: Professor John Weare (University of California, San Diego) The prediction of the interactions of geochemical fluids with minerals, nanoparticles, and colloids under extreme near surface conditions of temperature (T) and pressure (P) is a grand challenge research need in geosciences (U.S. DOE 2007, Basic Research Needs for Geosciences: Facilitating the 21st Energy Systems.). To evaluate the impact of these processes on energy production and management strategies it is necessary to have a high level of understanding of the interaction between complex natural fluids and mineral formations. This program emphasizes 1st principle parameter free simulations of complex chemical processes in solutions, in the mineral phase, and in the interfaces between these phases The development of new computational tools (with emphasis on oxide materials and reaction dynamics) tailored to treat wide range of conditions and time scales experienced in such geochemical applications is has been developed. Because of the sensitivity of the interaction in these systems to electronic structure and local bonding environments, and of the need to describe bond breaking/formation, our simulations are based on interactions calculated at the electronic structure level (ab-initio molecular dynamics, AIMD). The progress in the computational aspects of program may be summarized in terms of the following themes (objectives); Development of efficient parameter free dynamical simulation technology based on 1st principles force and energy calculations especially adapted for geochemical applications (e.g., mineral, interfaces and aqueous solutions) (continuing program); Calculation of the dynamics of water structure of in the surface-water interface of transition metal oxides and oxihydroxides; and Development of improved (beyond DFT+GGA) electronic structure calculations for minerals and the interface region that more accurately calculate electron correlation, spin density, and localization. The focus of the program is also on the iron oxide and oxihydroxide minerals and Fe 2+ (aq)/Fe 3+ (cr) oxidation in the mineral solution interface region. These methods included the development of model Hamilitonian methods that can be solved to near convergence for single site models (DMFT) and many-body perturbation methods (MP2, GW); Development of time decomposition methods to extend time scales of molecular dynamics (MD) simulations and support the use of high complexity electronic structure calculations (MP2, CCSD(T)) of forces for use in dynamical simulations where very high chemical accuracy is required (microsolvated reactions in absorbed surface layers); and The development of a new linear scaling finite element solver for eigenvalue problem that supports solution of quantum problems with unusual potential and boundary values. Application progress of the above new simulation technology to problems of geochemical interests includes; The prediction of metal oxide surface structure and the reduction/oxidation of Fe 3+ (cr)/Fe 2+ (aq) in metal oxide (hematite, goethite)/solution interfaces. Result: water interacts strongly with the 001 Hematite surface; interaction of water with the 100 goethite is weak; The study of ion solvation and the composition of ion hydration shells under extreme conditions (focus on Fe 3+/2+ , Al 3+ and Mg 2+ and their hydroxide speciation). Result: Ion association in water solutions can be calculated from 1st principle methods. Efficient sampling of the free energy requires more development; The continued development of new high resolution analysis of XAFS scattering of disordered systems (particularly Al, Mg) and of XANES calculations for aqueous ions. Result: EXAFS spectra can be calculated to high accuracy with DFT level dynamic simulations; The exploration of electron localization and electron transport in metal oxides (highly correlated materials). Result: Proper description of electron localization requires levels of calculation beyond DFT; and Localization of electrons in DFT type Hamiltonians was studied. Result: For very Dirac high exchange new solutions (New unphysical bifrucations) to the eigenvalue problem are found. The program was highly collaborative involving faculty and students in mathematics, physics and computer science departments as well as coworkers at the Pacific Northwest National Laboratories (PNNL). The students in this program had the opportunity to develop skills in the development of methods, the implementation of method on high performance parallel computers and the application of these methods to problem in geochemical science. Much of the software that was developed was incorporated in the NWchem software package maintained by PNNL.

58 GEOSCIENCES↗
DOE OSTI2021

Adiabatic quantum decoherence in many non-interacting subsystems induced by the coupling with a common boson bath

Highlights: • System–environment quantum correlation: a main solid state NMR decoherence channel. • Non-separable system–environment model yields realistic spin decoherence rates. • New open quantum system approach explains irreversible decay of refocused NMR echoes. • Adiabatic quantum decoherence is inherently irreversible and eigen-selective. This work addresses adiabatic quantum decoherence of many-body spin systems coupled with a boson field in the framework of open quantum systems theory. We generalize the traditional spin-boson model by considering a system–environment interaction Hamiltonian that represents a partition of non-interacting subsystems and highlights the collective correlation that appears exclusively due to the coupling with a common environment. Remarkably, this simple, exactly solvable model encompasses relevant aspects of a many-body open quantum system and features the subtle quantum effects that arise when the size scales up to a macroscopic level. We derive an analytical expression for the time dependence of the density matrix elements (in the preferred basis) without assuming coarse-graining. The resulting decoherence function is eigen-selective and is a complex exponential whose exponent has a real part that introduces a decay similar to that in the spin-boson model. On the contrary, the imaginary part depends on the quantum numbers and geometry of the whole partition and does not reflect the system temperature. Motivated by decoherence in solid-state NMR, and in search of realistic numerical estimations, we apply the theoretical results to a partition of dipole-coupled spin pairs in contact with a common phonon bath, using typical parameters of hydrated salts. The proposal allows estimating the decoherence time scale in terms of the system physical constants: sound velocity and eigenvalue distribution width. As a significant novelty, the decoherence function phase depends on the eigenvalue distribution throughout the sample. It plays the leading role, overshadowing the mechanism associated with the bath thermal state. Finally, we apply the formalism to describe decoherence in the “magic echo” NMR reversal experiment. We find that the system–environment correlation explains the origin of irreversibility, and both the decoherence rate value and its dependence on the dipolar frequency, are remarkably similar to the experiment.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Kohn-Sham Solver (KSSOLV) v2.0

KSSOLV is a MATLAB toolbox for solving Kohn-Sham density functional theory based electronic structure eigenvalue problems. It uses an object oriented features of MATLAB to represent atom, molecules, wavefunctions and Hamiltonians and their operations. It is designed to make it easier for users to prototype and test new algorithms for solving the Kohn-Sham problem. KSSOLV2.0 contains significant improvement over the original KSSOLV described in a paper published in ACM Transaction on Mathematical Software (attached). In addition to performing ground state calculation for small molecules, it can also perform geometry optimization for both molecules and solids. It uses standard pseudopotentials and implements local density approximation, generalized gradient approximation and hybrid functionals. Future releases will also include time-dependent DFT and post DFT calculations such as the GW quasi-particle energy calculation and Bethe-Salpeter equation solver for optical absorption.

Yang, Chao↗
DOE OSTI2021

Reduction of the molecular hamiltonian matrix using quantum community detection

Abstract Quantum chemistry is interested in calculating ground and excited states of molecular systems by solving the electronic Schrödinger equation. The exact numerical solution of this equation, frequently represented as an eigenvalue problem, remains unfeasible for most molecules and requires approximate methods. In this paper we introduce the use of Quantum Community Detection performed using the D-Wave quantum annealer to reduce the molecular Hamiltonian matrix in Slater determinant basis without chemical knowledge. Given a molecule represented by a matrix of Slater determinants, the connectivity between Slater determinants (as off-diagonal elements) is viewed as a graph adjacency matrix for determining multiple communities based on modularity maximization. A gauge metric based on perturbation theory is used to determine the lowest energy cluster. This cluster or sub-matrix of Slater determinants is used to calculate approximate ground state and excited state energies within chemical accuracy. The details of this method are described along with demonstrating its performance across multiple molecules of interest and bond dissociation cases. These examples provide proof-of-principle results for approximate solution of the electronic structure problem using quantum computing. This approach is general and shows potential to reduce the computational complexity of post-Hartree–Fock methods as future advances in quantum hardware become available.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2015

Using an iterative eigensolver to compute vibrational energies with phase-spaced localized basis functions

Although phase-space localized Gaussians are themselves poor basis functions, they can be used to effectively contract a discrete variable representation basis [A. Shimshovitz and D. J. Tannor, Phys. Rev. Lett. 109, 070402 (2012)]. This works despite the fact that elements of the Hamiltonian and overlap matrices labelled by discarded Gaussians are not small. By formulating the matrix problem as a regular (i.e., not a generalized) matrix eigenvalue problem, we show that it is possible to use an iterative eigensolver to compute vibrational energy levels in the Gaussian basis.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

Chebyshev-filtered subspace iteration method free of sparse diagonalization for solving the Kohn–Sham equation

First-principles density functional theory (DFT) calculations for the electronic structure problem require a solution of the Kohn–Sham equation, which requires one to solve a nonlinear eigenvalue problem. Solving the eigenvalue problem is usually the most expensive part in DFT calculations. Sparse iterative diagonalization methods that compute explicit eigenvectors can quickly become prohibitive for large scale problems. The Chebyshev-filtered subspace iteration (CheFSI) method avoids most of the explicit computation of eigenvectors and results in a significant speedup over iterative diagonalization methods for the DFT self-consistent field (SCF) calculations. However, the original formulation of the CheFSI method utilizes a sparse iterative diagonalization at the first SCF step to provide initial vectors for subspace filtering at latter SCF steps. This diagonalization is expensive for large scale problems. We develop a new initial filtering step to avoid completely this diagonalization, thus making the CheFSI method free of sparse iterative diagonalizations at all SCF steps. Our new approach saves memory usage and can be two to three times faster than the original CheFSI method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2017

Paradeisos: A perfect hashing algorithm for many-body eigenvalue problems

Here, we describe an essentially perfect hashing algorithm for calculating the position of an element in an ordered list, appropriate for the construction and manipulation of many-body Hamiltonian, sparse matrices. Each element of the list corresponds to an integer value whose binary representation reflects the occupation of single-particle basis states for each element in the many-body Hilbert space. The algorithm replaces conventional methods, such as binary search, for locating the elements of the ordered list, eliminating the need to store the integer representation for each element, without increasing the computational complexity. Combined with the “checkerboard” decomposition of the Hamiltonian matrix for distribution over parallel computing environments, this leads to a substantial savings in aggregate memory. While the algorithm can be applied broadly to many-body, correlated problems, we demonstrate its utility in reducing total memory consumption for a series of fermionic single-band Hubbard model calculations on small clusters with progressively larger Hilbert space dimension.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

Shift-and-invert parallel spectral transformation eigensolver: Massively parallel performance for density-functional based tight-binding

The Shift-and-invert parallel spectral transformations (SIPs), a computational approach to solve sparse eigenvalue problems, is developed for massively parallel architectures with exceptional parallel scalability and robustness. The capabilities of SIPs are demonstrated by diagonalization of density-functional based tight-binding (DFTB) Hamiltonian and overlap matrices for single-wall metallic carbon nanotubes, diamond nanowires, and bulk diamond crystals. The largest (smallest) example studied is a 128,000 (2000) atom nanotube for which ~330,000 (~5600) eigenvalues and eigenfunctions are obtained in ~190 (~5) seconds when parallelized over 266,144 (16,384) Blue Gene/Q cores. Weak scaling and strong scaling of SIPs are analyzed and the performance of SIPs is compared with other novel methods. Different matrix ordering methods are investigated to reduce the cost of the factorization step, which dominates the time-to-solution at the strong scaling limit. As a result, a parallel implementation of assembling the density matrix from the distributed eigenvectors is demonstrated.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Enhancing scalability of a matrix-free eigensolver for studying many-body localization

We propose several techniques to enhance the parallel scalability of a matrix-free eigensolver designed for studying many-body localization (MBL) of quantum spin chain models with nearest-neighbor interactions and on-site disorder. This type of problem is computationally challenging because the dimension of the associated Hamiltonian matrix grows exponentially with respect to the number of spins L, and we need to average over different realizations of the random disorder to obtain relevant statistical behavior. For each disorder realization, we need to compute eigenvalues from different regions of the spectrum and their corresponding eigenvectors. In previous work, the interior eigenstates for a single eigenvalue problem are computed via the shift-and-invert Lanczos algorithm. Due to the extremely high memory footprint of the LU factorizations, this technique is not well suited for large L’s. For example, we need thousands of compute nodes on modern high performance computing infrastructures to go beyond L = 24. The matrix-free approach does not suffer from this memory bottleneck, however, its scalability is limited by a computation and communication load imbalance. To reduce this imbalance and to significantly enhance the scalability of the matrix-free eigensolver, we reorder the matrix and leverage the consistent space runtime, CSPACER. We also show its efficiency in managing irregular communication patterns at scale compared to optimized MPI non-blocking two-sided and one-sided RMA implementation variants. This effort enables us to study MBL for spin chains with a larger number of spins. The efficiency and effectiveness of the proposed algorithm is demonstrated by computing eigenstates on a massively parallel many-core high performance computer.

METIS↗
DOE OSTI2018

MiniApp for Density Matrix Renormalization Group Hamiltonian Application Kernel

We present two miniapps that implement the core computational kernel of the DMRG++ application, a generic C++ code that implements the Density Matrix Renormalization Group (DMRG) algorithm. The DMRG++ core Kronecker multiplication kernel is formulated using a batched BLAS approach, with implementation that targets both multi-core CPUs using OpenMP and GPGPU using the MAGMA library. The kernel evaluates the matrix-vector multiplication of the target Hamiltonian matrix used in Lanczos algorithm for computing the lowest eigenvalue and eigenvector. The Hamiltonian matrix is expressed compactly as sums of Kronecker products of small dense matrices. We demonstrate improved performance of the miniapp on synthetic problem, and show the performance of the DMRG++ application using a plugin based on the miniapp. We also present an OpenMP miniapp that explores the use of nested parallel constructs to implement the Kronecker multiplication kernel, exploring the use of nested OpenMP worksharing and tasking abstractions to implement the multi-level parallel multiplication algorithm. The miniapp has been used as a co-design vehicle for evaluating features in the OpenMP-4.5 and upcoming OpenMP-5.0 standards.

Elwasif, Wael↗
DOE OSTI2018

Witnessing eigenstates for quantum simulation of Hamiltonian spectra

The efficient calculation of Hamiltonian spectra, a problem often intractable on classical machines, can find application in many fields, from physics to chemistry. We introduce the concept of an “eigenstate witness” and, through it, provide a new quantum approach that combines variational methods and phase estimation to approximate eigenvalues for both ground and excited states. This protocol is experimentally verified on a programmable silicon quantum photonic chip, a mass-manufacturable platform, which embeds entangled state generation, arbitrary controlled unitary operations, and projective measurements. Both ground and excited states are experimentally found with fidelities >99%, and their eigenvalues are estimated with 32 bits of precision. We also investigate and discuss the scalability of the approach and study its performance through numerical simulations of more complex Hamiltonians. This result shows promising progress toward quantum chemistry on quantum computers.

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

On quark-antiquark approximation in light front QCD with zero gluon modes

We consider a transition to the light front Hamiltonian from theories quantized on spacelike planes approaching to the light front. In this approach we preserve the dynamics of zero mode present in the theories near the light front. We make the limit transition differently for zero and nonzero modes. This leads to the appearance of some phenomenological parameter which can be used to describe vacuum effects. Also we use a lattice gauge invariant regularization in transverse coordinate space. As an illustration of our scheme we consider the quark-antiquark bound states problem in 2+1 dimensions. We construct basis states in the light front Fock space and provide detailed computations of the Hamiltonian matrix in this basis. These steps allow us to construct the matrix eigenvalue equation. In conclusion we discuss the nuances of obtained results.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Effect of emitters on quantum state transfer in coupled cavity arrays

Over the last decade, conditions for perfect state transfer in quantum spin chains have been discovered and their experimental realizations addressed, as have their extensions to more complex geometries of coupled cavity-emitter arrays. In this paper we further consider such studies and situations in which quantum state transfer can occur with high fidelity, even when the cavity-cavity coupling rates and cavity-emitter interaction rates are comparable. This is accomplished through the development and use of a Monte Carlo approach to the inverse eigenvalue problem, which allows the determination of coupling rates which optimize quantum state transfer fidelity and subsequent time evolution of the polariton wave function through exact diagonalization of the resulting Jaynes-Cummings-Hubbard Hamiltonian. The effect of inhomogeneous emitter locations is also evaluated. Furthermore, our key results include the demonstration that our methodology can be used successfully to establish Hamiltonian parameters for high-fidelity state transfer in more general lattice geometries and excitation number sectors, and also a determination of the effects of fluctuations in those parameters about their optimal values.

77 NANOSCIENCE AND NANOTECHNOLOGY↗
DOE OSTI2024

Unleashed from constrained optimization: quantum computing for quantum chemistry employing generator coordinate inspired method

Hybrid quantum-classical approaches offer potential solutions to quantum chemistry problems, yet they often manifest as constrained optimization problems. Here, we explore the interconnection between constrained optimization and generalized eigenvalue problems through the Unitary Coupled Cluster (UCC) excitation generators. Inspired by the generator coordinate method, we employ these UCC excitation generators to construct non-orthogonal, overcomplete many-body bases, projecting the system Hamiltonian into an effective Hamiltonian, which bypasses issues such as barren plateaus that heuristic numerical minimizers often encountered in standard variational quantum eigensolver (VQE). Diverging from conventional quantum subspace expansion methods, we introduce an adaptive scheme that robustly constructs the many-body basis sets from a pool of the UCC excitation generators. This scheme supports the development of a hierarchical ADAPT quantum-classical strategy, enabling a balanced interplay between subspace expansion and ansatz optimization to address complex, strongly correlated quantum chemical systems cost-effectively, setting the stage for more advanced quantum simulations in chemistry.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2019

Quantum computation of nanosheets in a background magnetic field for external control of nanosystems

We ran quantum simulations on lattices of interacting fermions in a background magnetic field [1]. We created discretized Hamiltonian matrices for each system and pre-processed them so that they could be read by the quantum simulator implemented in IBM's Qiskit package [2]. The simulator uses the variational quantum eigensolver (VQE) algorithm [3] to approximate the ground state energy of the input Hamiltonian, which is then compared to the classically-calculated exact value. This classical evaluation is possible because the systems studied are still within the grasp of classical computers. We also investigated ways to simplify the system before the pre-processing step. We found that in almost all cases these simplifications made our quantum calculations converge to better answers using simpler starting conditions. We are exploring other ways to interface with IBM's quantum computing software so that we no longer need to explicitly define the Hamiltonian matrices for our systems, a problem which will itself soon become intractable.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗