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DOE OSTI2023

Mean Flow from Phase Averages in the 2D Boussinesq Equations

The atmosphere and ocean are described by highly oscillatory PDEs that challenge both our understanding of their dynamics and their numerical approximation. This paper presents a preliminary numerical study of one type of phase averaging applied to mean flows in the 2D Boussinesq equations that also has application to numerical methods. The phase averaging technique, well-known in dynamical systems theory, relies on a mapping using the exponential operator, and then an averaging over the phase. The exponential operator has connections to the Craya–Herring basis pioneered by Jack Herring to study the fluid dynamics of oscillatory, nonlinear fluid dynamics. In this paper, we perform numerical experiments to study the effect of this averaging technique on the time evolution of the solution. We explore its potential as a definition for mean flows. We also show that, as expected from theory, the phase-averaging method can reduce the magnitude of the time rate of change in the PDEs, making them potentially suitable for time stepping methods.

Wingate, Beth A.↗

NASA NTRS2015

Theoretical Studies of Spectroscopic Line Mixing in Remote Sensing Applications

The phenomenon of collisional transfer of intensity due to line mixing has an increasing importance for atmospheric monitoring. From a theoretical point of view, all relevant information about the collisional processes is contained in the relaxation matrix where the diagonal elements give half-widths and shifts, and the off-diagonal elements correspond to line interferences. For simple systems such as those consisting of diatom-atom or diatom-diatom, accurate fully quantum calculations based on interaction potentials are feasible. However, fully quantum calculations become unrealistic for more complex systems. On the other hand, the semi-classical Robert-Bonamy (RB) formalism, which has been widely used to calculate half-widths and shifts for decades, fails in calculating the off-diagonal matrix elements. As a result, in order to simulate atmospheric spectra where the effects from line mixing are important, semi-empirical fitting or scaling laws such as the ECS (Energy-Corrected Sudden) and IOS (Infinite-Order Sudden) models are commonly used. Recently, while scrutinizing the development of the RB formalism, we have found that these authors applied the isolated line approximation in their evaluating matrix elements of the Liouville scattering operator given in exponential form. Since the criterion of this assumption is so stringent, it is not valid for many systems of interest in atmospheric applications. Furthermore, it is this assumption that blocks the possibility to calculate the whole relaxation matrix at all. By eliminating this unjustified application, and accurately evaluating matrix elements of the exponential operators, we have developed a more capable formalism. With this new formalism, we are now able not only to reduce uncertainties for calculated half-widths and shifts, but also to remove a once insurmountable obstacle to calculate the whole relaxation matrix. This implies that we can address the line mixing with the semi-classical theory based on interaction potentials between molecular absorber and molecular perturber. We have applied this formalism to address the line mixing for Raman and infrared spectra of molecules such as N2, C2H2, CO2, NH3, and H2O. By carrying out rigorous calculations, our calculated relaxation matrices are in good agreement with both experimental data and results derived from the ECS model.

Formalism↗

DOE OSTI2020

Optimized Lie–Trotter–Suzuki decompositions for two and three non-commuting terms

Lie–Trotter–Suzuki decompositions are an efficient way to approximate operator exponentials exp ( t H ) when H is a sum of n (non-commuting) terms which, individually, can be exponentiated easily. They are employed in time-evolution algorithms for tensor network states, digital quantum simulation protocols, path integral methods like quantum Monte Carlo, and splitting methods for symplectic integrators in classical Hamiltonian systems. Here, we provide optimized decompositions up to order t 6 . The leading error term is expanded in nested commutators (Hall bases) and we minimize the 1-norm of the coefficients. For n = 2 terms, several of the optima we find are close to those in McLachlan (1995). Generally, our results substantially improve over unoptimized decompositions by Forest, Ruth, Yoshida, and Suzuki. We explain why these decompositions are sufficient to efficiently simulate any one- or two-dimensional lattice model with finite-range interactions. This follows by solving a partitioning problem for the interaction graph.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1974

On evaluating compliance with air pollution levels 'not to be exceeded more than once a year'

Under a sampling program operated in conformity with EPA guidelines, calculational procedures are described for estimating the corresponding ambient pollution level. The statistical independence of the data, the commutativity of expectation and exponential operations, and the statistical variability of the calculated results are quantitatively evaluated. It is shown that the recommended sampling frequencies of every third to every sixth day are inadequate in determining conformity when the standards are expressed as levels not to be exceeded more than once a year.

Neustadter, H. E.↗

DOE OSTI2021

Computing partition functions in the one-clean-qubit model

We present a method to approximate partition functions of quantum systems using mixed-state quantum computation. For positive-semidefinite Hamiltonians, our method has an expected running-time that is almost linear in [M/( ε rel Z )] 2 , where M is the dimension of the quantum system, Z is the partition function, and ε rel is the relative precision. It is based on approximations of the exponential operator as linear combinations of certain operators related to block-encoding of Hamiltonians or Hamiltonian evolutions. The trace of each operator is estimated using a standard algorithm in the one-clean-qubit model. For large values of Z , our method may run faster than exact classical methods, whose complexities are polynomial in M . We also prove that a version of the partition function estimation problem within additive error is complete for the so-called DQC1 complexity class, suggesting that our method provides a superpolynomial speedup for certain parameter values. Overall, to attain a desired relative precision, we develop a classical procedure based on a sequence of approximations within predetermined additive errors that may be of independent interest.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1977

Numerical methods for inviscid/viscous flows

A study of numerical schemes for solving viscous fluid flow problems with sizable regions of predominantly inviscid flow is presented. Difficulties associated with the familiar central difference approach for such problems were analyzed and alternative finite difference approaches employing windward concepts are presented. In addition, difference relations based on exponential operators were developed. All such schemes were demonstrated and evaluated through application to the case of Falkner Skan flow with blowing - a problem in which a sizable region of predominantly inviscid flow develops near the injection surface that traditionally causes numerical difficulty.

El-Mistikawy, T. M.↗

DOE OSTI2025

Navigating Integration: Key Challenges for Data Centers, Nuclear Stakeholders, and Utility Operators

he exponential growth of data centers—driven by artificial intelligence and cloud computing—is reshaping the U.S. energy landscape, presenting urgent challenges and transformative opportunities for data center developers, nuclear energy providers, and utility operators. As data centers are projected to consume up to 12% of U.S. electricity by 2028, stakeholders must address rapid deployment needs, grid congestion, and the demand for reliable, high-quality power. This presentation explores the multifaceted barriers to integrating data centers with nuclear and utility infrastructure, including land use constraints, public perception, regulatory complexity, and workforce alignment. It highlights the distinct priorities and operational cultures of each sector, and the friction that arises from misaligned planning horizons and risk tolerances. We examine collaborative strategies such as co-siting, hybrid power-purchase agreements, unified community engagement, and innovative financing models.

22 - GENERAL STUDIES OF NUCLEAR REACTORS↗

NASA NTRS1996

Exponential Formulae and Effective Operations

One of standard methods to predict the phenomena of squeezing consists in splitting the unitary evolution operator into the product of simpler operations. The technique, while mathematically general, is not so simple in applications and leaves some pragmatic problems open. We report an extended class of exponential formulae, which yield a quicker insight into the laboratory details for a class of squeezing operations, and moreover, can be alternatively used to programme different type of operations, as: (1) the free evolution inversion; and (2) the soft simulations of the sharp kicks (so that all abstract results involving the kicks of the oscillator potential, become realistic laboratory prescriptions).

Mielnik, Bogdan↗

DOE OSTI2024

The black hole interior from non-isometric codes and complexity

Quantum error correction has given us a natural language for the emergence of spacetime, but the black hole interior poses a challenge for this framework: at late times the apparent number of interior degrees of freedom in effective field theory can vastly exceed the true number of fundamental degrees of freedom, so there can be no isometric (i.e. inner-product preserving) encoding of the former into the latter. In this paper we explain how quantum error correction nonetheless can be used to explain the emergence of the black hole interior, via the idea of “non-isometric codes protected by computational complexity”. We show that many previous ideas, such as the existence of a large number of “null states”, a breakdown of effective field theory for operations of exponential complexity, the quantum extremal surface calculation of the Page curve, post-selection, “state-dependent/state-specific” operator reconstruction, and the “simple entropy” approach to complexity coarse-graining, all fit naturally into this framework, and we illustrate all of these phenomena simultaneously in a soluble model.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2023

Exploiting Kronecker structure in exponential integrators: Fast approximation of the action of φ $-$functions of matrices via quadrature

Here, in this article, we propose an algorithm for approximating the action of φ $-$ functions of matrices against vectors, which is a key operation in exponential time integrators. In particular, we consider matrices with Kronecker sum structure, which arise from problems admitting a tensor product representation. The method is based on quadrature approximations of the integral form of the φ $-$ functions combined with a scaling and modified squaring method. Owing to the Kronecker sum representation, only actions of 1D matrix exponentials are needed at each quadrature node and assembly of the full matrix can be avoided. Additionally, we derive a priori bounds for the quadrature error, which show that, as expected by classical theory, the rate of convergence of our method is supergeometric. Guided by our analysis, we construct a fast and robust method for estimating the optimal scaling factor and number of quadrature nodes that minimizes the total cost for a prescribed error tolerance. We investigate the performance of our algorithm by solving several linear and semilinear time-dependent problems in 2D and 3D. The results show that our method is accurate and orders of magnitude faster than the current state-of-the-art.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Block encoding bosons by signal processing

Block Encoding (BE) is a crucial subroutine in many modern quantum algorithms, including those with near-optimal scaling for simulating quantum many-body systems, which often rely on Quantum Signal Processing (QSP). Currently, the primary methods for constructing BEs are the Linear Combination of Unitaries (LCU) and the sparse oracle approach. In this work, we demonstrate that QSP-based techniques, such as Quantum Singular Value Transformation (QSVT) and Quantum Eigenvalue Transformation for Unitary Matrices (QETU), can themselves be efficiently utilized for BE implementation. Specifically, we present several examples of using QSVT and QETU algorithms, along with their combinations, to block encode Hamiltonians for lattice bosons, an essential ingredient in simulations of high-energy physics. We also introduce a straightforward approach to BE based on the exact implementation of Linear Operators Via Exponentiation and LCU (LOVE-LCU). We find that, while using QSVT for BE results in the best asymptotic gate count scaling with the number of qubits per site, LOVE-LCU outperforms all other methods for operators acting on up to qubits, highlighting the importance of concrete circuit constructions over mere comparisons of asymptotic scalings. Using LOVE-LCU to implement the BE, we simulate the time evolution of single-site and two-site systems in the lattice theory using the Generalized QSP algorithm and compare the gate counts to those required for Trotter simulation.

Kane, Christopher F↗

DOE OSTI2023

Information loss, mixing and emergent type III 1 factors

A manifestation of the black hole information loss problem is that the two-point function of probe operators in a large Anti-de Sitter black hole decays in time, whereas, on the boundary CFT, it is expected to be an almost periodic function of time. We point out that the decay of the two-point function (clustering in time) holds important clues to the nature of observable algebras, states, and dynamics in quantum gravity. We call operators that cluster in time “mixing” and explore the necessary and sufficient conditions for mixing. The information loss problem is a special case of the statement that in type I algebras, there exists no mixing operators. We prove that, in a thermofield double state (KMS state), if mixing operators form an algebra (close under multiplication), the resulting algebra must be a von Neumann type III 1 factor. In other words, the physically intuitive requirement that all nonconserved operators should exponentially mix is so strong that it fixes the observable algebra to be an exotic algebra called a type III1 factor. More generally, for an arbitrary out-of-equilibrium state of a general quantum system (von Neumann algebra), we show that if the set of operators that mix under modular flow forms an algebra, it is a type III1 von Neumann factor.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2020

A quantum algorithm to efficiently sample from interfering binary trees

Quantum computers provide an opportunity to efficiently sample from probability distributions that include non-trivial interference effects between amplitudes. Using a simple process wherein all possible state histories can be specified by a binary tree, we construct an explicit quantum algorithm that runs in polynomial time to sample from the process once. The corresponding naive Markov Chain algorithm does not produce the correct probability distribution and an explicit classical calculation of the full distribution requires exponentially many operations. Finally, the problem can be reduced to a system of two qubits with repeated measurements, shedding light on a quantum-inspired efficient classical algorithm.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Challenging the Curse of Dimensionality in Multidimensional Numerical Integration by Using a Low-Rank Tensor-Train Format

Numerical integration is a basic step in the implementation of more complex numerical algorithms suitable, for example, to solve ordinary and partial differential equations. The straightforward extension of a one-dimensional integration rule to a multidimensional grid by the tensor product of the spatial directions is deemed to be practically infeasible beyond a relatively small number of dimensions, e.g., three or four. In fact, the computational burden in terms of storage and floating point operations scales exponentially with the number of dimensions. This phenomenon is known as the curse of dimensionality and motivated the development of alternative methods such as the Monte Carlo method. The tensor product approach can be very effective for high-dimensional numerical integration if we can resort to an accurate low-rank tensor-train representation of the integrand function. In this work, we discuss this approach and present numerical evidence showing that it is very competitive with the Monte Carlo method in terms of accuracy and computational costs up to several hundredths of dimensions if the integrand function is regular enough and a sufficiently accurate low-rank approximation is available.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Efficient Application of the Factorized form of the Unitary Coupled-Cluster Ansatz for the Variational Quantum Eigensolver Algorithm by Using Linear Combination of Unitaries

The variational quantum eigensolver is one of the most promising algorithms for near-term quantum computers. It has the potential to solve quantum chemistry problems involving strongly correlated electrons with relatively low-depth circuits, which are otherwise difficult to solve on classical computers. The variational eigenstate is constructed from a number of factorized unitary coupled-cluster terms applied onto an initial (single-reference) state. Current algorithms for applying one of these operators to a quantum state require a number of operations that scale exponentially with the rank of the operator. We exploit a hidden SU(2) symmetry to allow us to employ the linear combination of unitaries approach, Our Prepare subroutine uses n+2 ancilla qubits for a rank-n operator. Our Select (U) scheme uses O(n) Cnot gates. This results in a full algorithm that scales like the cube of the rank of the operator n 3 , a significant reduction in complexity for rank five or higher operators. This approach, when combined with other algorithms for lower-rank operators (when compared to the standard implementation), will make the factorized form of the unitary coupled-cluster approach much more efficient to implement on all types of quantum computers.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1978

Numerical method for boundary layers with blowing - The exponential box scheme

The paper describes a new numerical scheme based on exponential difference operator concepts combined with Keller's (1968) box scheme approach to produce a stable second-order accurate finite-difference scheme for convection-diffusion problems arising in boundary layer flows in the presence of massive injection through a porous surface. The technique is demonstrated by application to the self-similar boundary layer equations with massive blowing at the surface.

El-Mistikawy, T. M.↗

NASA NTRS1979

Shot-noise in resistive-diode mixers and the attenuator noise model

The representation of a pumped exponential diode, operating as a mixer, by an equivalent lossy network, is reexamined. It is shown that the model is correct provided the network has ports for all sideband frequencies at which (real) power flow can occur between the diode and its embedding. The temperature of the equivalent network is eta/2 times the physical temperature of the diode. The model is valid only if the series resistance and nonlinear capacitance of the diode are negligible. Expressions are derived for the input and output noise temperature and the noise-temperature ratio of ideal mixers. Some common beliefs concerning noise-figure and noise-temperature ratio are shown to be incorrect.

Kerr, A. R.↗

NASA NTRS2007

Reducing a Knowledge-Base Search Space When Data Are Missing

This software addresses the problem of how to efficiently execute a knowledge base in the presence of missing data. Computationally, this is an exponentially expensive operation that without heuristics generates a search space of 1 + 2n possible scenarios, where n is the number of rules in the knowledge base. Even for a knowledge base of the most modest size, say 16 rules, it would produce 65,537 possible scenarios. The purpose of this software is to reduce the complexity of this operation to a more manageable size. The problem that this system solves is to develop an automated approach that can reason in the presence of missing data. This is a meta-reasoning capability that repeatedly calls a diagnostic engine/model to provide prognoses and prognosis tracking. In the big picture, the scenario generator takes as its input the current state of a system, including probabilistic information from Data Forecasting. Using model-based reasoning techniques, it returns an ordered list of fault scenarios that could be generated from the current state, i.e., the plausible future failure modes of the system as it presently stands. The scenario generator models a Potential Fault Scenario (PFS) as a black box, the input of which is a set of states tagged with priorities and the output of which is one or more potential fault scenarios tagged by a confidence factor. The results from the system are used by a model-based diagnostician to predict the future health of the monitored system.

James, Mark↗