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DOE OSTI2026

Quantum Time-Space Tradeoffs for Matrix Problems

We consider the time and space required for quantum computers to solve a wide variety of problems involving matrices, many of which have only been analyzed classically in prior work. Our main results show that for a range of linear algebra problems—including matrix-vector product, matrix inversion, matrix multiplication and powering—existing classical time-space tradeoffs, several of which are tight for every space bound, also apply to quantum algorithms with at most a constant factor loss. For example, for almost all fixed matrices 𝐴, including the discrete Fourier transform matrix, we prove that quantum circuits with at most 𝑇 input queries and 𝑆 qubits of memory require 𝑇 = Ω⁢(𝑛 2 /𝑆) to compute matrix-vector product 𝐴⁢𝑥 for 𝑥 ∈{0,1 𝑛 . We similarly prove that matrix multiplication for 𝑛 ×𝑛 binary matrices requires 𝑇 = Ω⁢(𝑛 3 /$\sqrt{𝑆}$). Because many of our lower bounds are matched by deterministic algorithms with the same time and space complexity, our results show that quantum computers cannot provide any asymptotic advantage for these problems with any space bound. We obtain matching lower bounds for the stronger notion of quantum cumulative memory complexity—the sum of the space per layer of a circuit. We also consider Boolean (i.e., AND-OR) matrix multiplication and matrix-vector products, improving the previous quantum time-space tradeoff lower bounds for 𝑛 × 𝑛 Boolean matrix multiplication to 𝑇 = Ω⁢(𝑛 2.5 /𝑆 1/4 ) from 𝑇 = Ω⁢(𝑛 2.5 /𝑆 1/2 ). Our improved lower bound for Boolean matrix multiplication is based on a new coloring argument that extracts more from the strong direct product theorem that was the basis for prior work. To obtain our tight lower bounds for linear algebra problems, we require much stronger bounds than strong direct product theorems. We obtain these bounds by adding a new bucketing method to the quantum recording-query technique of Zhandry that lets us apply classical arguments to upper bound the success probability of quantum circuits.

lower bounds↗

DOE OSTI2026

Paradigm for universal quantum information processing with integrated acousto-optic frequency beamsplitters

Frequency-bin encoding offers tremendous potential in quantum photonic information processing, in which a single waveguide can support hundreds of lightpaths in a naturally phase-stable fashion. This stability, however, comes at a cost: arbitrary unitary operations can be realized by cascaded electro-optic phase modulators and pulse shapers, but require nontrivial numerical optimization for design and have thus far been limited to discrete tabletop components. In this article, we propose, formalize, and computationally evaluate a new paradigm for universal frequency-bin quantum information processing using acousto-optic scattering processes between distinct transverse modes. We show that controllable phase matching in intermodal processes enables 2 × 2 frequency beamsplitters and transverse-mode-dependent phase shifters, which together comprise cascadable FRequency-transverse-mODe Operations (FRODOs) that can synthesize any unitary via analytical decomposition procedures. Modeling the performance of both random gates and discrete Fourier transforms, we demonstrate the feasibility of high-fidelity quantum operations with existing integrated photonics technology, highlighting prospects of parallelizable operations achieving 100% bandwidth utilization. Our approach is realizable with CMOS technology, opening the door to scalable on-chip quantum information processing in the frequency domain.

Lukens, Joseph M. [Purdue Univ., West Lafayette, I↗

DOE OSTI2021

Open Source Fault-tolerant Grid Frequency Measurement for Solar Inverters

The Discrete Fourier transform (DFT) based measurement algorithms are one of the most common measurement algorithms for grid parameter estimation such as rms, phase angle, frequency. Over the past few years, many DFT based algorithms have been developed to enhance its measurement accuracy under steady-state and/or dynamic grid conditions. For example, an adaptive band-pass filter utilizing exponential modulation filter has been proposed to reduce measurement errors at the presence of large frequency deviations. Measurement accuracy of different algorithms including FIR filter, extended Kalman filtering (EKF), and enhanced DFT method have been compared in detail under different grid conditions. Two artificial signals that have 90-degree phase difference were constructed by the Clarke transformation to address the frequency spectrum leakage of DFT. A multi-module approach was developed to enhance both steady-state and dynamic measurement accuracies, in which each module was developed to eliminate some specific errors. Besides DFT-based measurement algorithms, some signal model-based algorithms have been developed to further improve the accuracy under dynamic conditions. However, a key drawback of the state-of-the-art algorithms is that they cannot perform measurements accurately during system transient faults. In the Blue Cut Fire event, there was a phase angle jump of about 26 degrees in the voltage waveform during the transient fault. The phase angle jump fault will cause waveform discontinuity, and these algorithms will fail to provide reliable measurements during this period because they typically assume the waveform to be measured is continuous, no matter what method (DFT, PLL, EKF, FIR, or Taylor WLS) is used for estimation. In fact, the measurement errors during the system transient faults like phase-jump is not required in the IEEE Standard. As a result, although a measurement instrument can pass the strict IEEE Standard, it could still be the source of the problem in the future if we have similar system transient faults, which could happen again. Therefore, developing the fault-tolerant measurement technology is the key to solve the problem.

14 SOLAR ENERGY↗

DOE OSTI2025

Determining the Axes of a Range-Doppler Image

Synthetic aperture radar (SAR) images formed with dechirp-on-receive data collection and rectangular format processing algorithm are the result of a two-dimensional discrete Fourier transform (DFT) applied to sampled data. There are several steps required to compute the range and Doppler values associated with each pixel in a SAR range-Doppler image. This memo walks readers through the process.

47 OTHER INSTRUMENTATION↗

DOE OSTI2021

Modeling and Simulation of Inrush Currents in Harmonic Domain

Modeling and simulation capabilities are critical to the stability analysis and evaluation of power distribution systems, with respect to the emphasis on resiliency, microgrids, and distributed energy resources. In this paper, a computational method in the harmonic domain is proposed for the periodic steady-state analysis of the nonlinear inrush current phenomenon. The efficient inrush calculation facilitates the predictions of current amplitudes for the power system operation and control. To demonstrate the accuracy and efficiency, simulation results in the harmonic domain are compared with results from PSCAD in an electromagnetic timescale, as well as the authors’ previous works in the frequency-domain. Impacts of the settings of both offset flux and interested harmonic order are discussed. In addition, within the proposed harmonic-domain method, a general approach that utilizes the discrete Fourier transform to obtain the response of a nonlinear device from a stimulus represented in the frequency-domain is utilized. This method can also be extended to perform the transient analysis in future, using trapezoidal rule for the integration.

Xie, Jing↗

DOE OSTI2023

Time-Frequency Warped Waveforms for Well-Contained Massive Machine Type Communications

Herein this paper proposes a novel time-frequency warped waveform for short symbols, massive machine-type communication (mMTC), and internet of things (IoT) applications. The waveform is composed of asymmetric raised cosine (RC) pulses to increase the signal containment in time and frequency domains. The waveform has low power tails in the time domain, hence better performance in the presence of delay spread and time offsets. The time-axis warping unitary transform is applied to control the waveform occupancy in time-frequency space and to compensate for the usage of high roll-off factor pulses at the symbol edges. The paper explains a step-by-step analysis for determining the roll-off factors profile and the warping functions. Gains are presented over the conventional Zero-tail Discrete Fourier Transform-spread-Orthogonal Frequency Division Multiplexing (ZT-DFT-s-OFDM), and Cyclic prefix (CP) DFT-s-OFDM schemes in the simulations section.

42 ENGINEERING↗

DOE OSTI2025

Physics-Informed Gaussian Process Inference of Liquid Structure from Scattering Data

We present a nonparametric Bayesian framework to infer radial distribution functions from experimental scattering measurements with uncertainty quantification using nonstationary Gaussian processes. The Gaussian process prior mean and kernel functions are designed to mitigate well-known numerical challenges with the Fourier transform, including discrete measurement binning and detector windowing, while encoding fundamental yet minimal physical knowledge of the liquid structure. We demonstrate uncertainty propagation of the Gaussian process posterior to unmeasured quantities of interest. Experimental radial distribution functions of liquid argon and water with uncertainty quantification are provided as both a proof of principle for the method and a benchmark for molecular models.

Chemical structure↗

DOE OSTI2023

Warping Functions Design for Long Warped ZT-DFT-s-OFDM

Time-frequency warped waveform is a novel type of well-contained waveform composed of raised cosine (RC) pulses with roll-off factors profile customized for spectrally efficient signal containment. The waveform is suited for massive machine-type communication (mMTC), and Internet-of-Things (IoT) applications. The paper is a follow-up to the original work of time-frequency warped waveforms; here, we are proposing a warping function suited for length-independent symbols, which is missing from the previous work. The paper proposes the design steps starting from a piecewise warping function with parts of straight segments and other curved parts. A maximum Out-of-band leakage is used as a criterion to determine the warping function parameters. The roll-off factors profile determined in the previous study is used as a minimum bound for the leakage limits. Gains are presented over Windowed Zero-tail Discrete Fourier Transform-spread-Orthogonal Frequency Division Multiplexing (ZT-DFT-s-OFDM) in the simulations section.

Ibrahim, Mostafa↗

DOE OSTI2026

Enhancing photoionization rate calculations in low-temperature plasmas using spectral methods

Photoionization plays a central role in the development of streamer discharges and other non-equilibrium plasma phenomena. It creates seed electrons, which are essential for positive streamer propagation, allowing the ionization front to move forward. Because of this, accurate modeling of photoionization is very important for predicting streamer behavior and plasma evolution. The photoionization process in air (N 2 – O 2 mixture) is often described by the Zheleznyak model (1982). This model is usually solved through Helmholtz-type equations that approximate the Zheleznyak photoionization model (Zheleznyak et al. 1982) as Partial Differential Equations (PDEs). Conventional numerical methods, such as the Finite Difference Method (FDM) or Finite Volume Method (FVM), are widely used to solve these equations. Although they are prevalent, the computational cost of these methods due to their need for matrix operations and iterative solver is demanding. To address this challenge, this work develops a spectral solver based on the Fast Fourier Transform (FFT) combined with Discrete Cosine Transform (DCT) and Discrete Sine Transform (DST) to calculate the photoionization rate efficiently in an axisymmetric cylindrical domain. This method naturally satisfies the boundary conditions used in the model and converts the PDE into algebraic ones in spectral space. Thus, avoids the need for iterative matrix solvers. When compared with FDM results, it is demonstrated that the new solver not only maintains accuracy, but also reduces the computational cost, showing a performance increase of approximately 100 compared to FDM over a wide range of problem sizes. The method is parallelized using Message Passing Interface (MPI) and has been integrated into a fluid plasma model for streamer simulation. Here, this FFT-based approach provides a fast and reliable alternative for calculating photoionization in fluid models, helping large-scale plasma simulations run faster and efficiently, and allows higher-resolution simulation without extra computational cost.

Axisymmetric system↗

DOE OSTI2022

Kinetic theory of particle-in-cell simulation plasma and the ensemble averaging technique

Abstract We derive the kinetic theory of fluctuations in physically and numerically stable particle-in-cell (PIC) simulations of electrostatic plasmas. The starting point is the single-time correlation at the start of the simulation between the statistical fluctuations of the weighted densities of macroparticle centers in the plasma particle phase-space. The fluctuations are associated with different initial conditions, typically due to the random initial conditions (in velocity space) of the macroparticles/simulation plasma, assigned according to their initial distribution of probability. The single-time correlations at all time steps and in each spatial grid cell are then determined from the Laplace–Fourier transforms of the discretized Klimontovich-like equation for the macroparticles and Maxwell’s equations for the fields, as computed by modern PIC codes. We recover the expressions for the electrostatic field and the plasma particle density fluctuation autocorrelation spectra as well as the kinetic equations describing the average evolution of PIC-simulated plasma particles, first derived by Langdon (1970b Proc. 4th Conf. Numerical Simulation of Plasmas ) using a test macroparticle approach perturbing a discretized Vlasovian plasma and then averaging the obtained physical quantity over the initial macroparticle velocity distribution. We generalize and extend these results to the modern algorithms in PIC codes using arbitrary macroparticle weights. Analytical estimates of statistical fluctuation amplitudes are derived as a function of the plasma simulation parameters, using the central limit theorem in the limit of a large number of macroparticles per cell. The theory is then used to analyze the ensemble averaging technique of PIC simulations where statistical averages are performed over ensembles of PIC simulations, modeling the same plasma physics problem but using different statistical realizations of the initial distribution functions of the macroparticles. This method is illustrated by linear Landau damping uncovering (from noise, which is usually considered numerical) the physical fluctuations driven by a single small amplitude electrostatic wave perturbing a PIC simulation plasma in equilibrium.

fluctuations correlations↗

DOE OSTI2021

Spatiotemporal mapping of mesoscopic liquid dynamics

The study of liquid dynamics at mesoscopic scales is still strewn with difficulty due to limitations in theory and experiment. Historically, significant attention has been given to the analysis of space-time correlation functions and their frequency-Fourier transforms at a few discrete wave numbers. The massive computing power afforded by modern high performance computing clusters and the advent of a wide-angle neutron spin-echo spectrometer, however, have unlocked a more intuitive and fruitful approach to this problem. Using molecular dynamics simulations, here we demonstrate the benefits of spatiotemporally mapping intermediate scattering functions on a dense grid of correlation times and wave numbers. Four model systems are investigated: a Lennard-Jones liquid, a coarse-grained bead-spring polymer, a molten sodium chloride, and a poly(ethylene oxide) melt. We show that the spatiotemporal mapping approach is particularly useful for elucidating the mesoscopic dynamics in these liquids, where several underlying mechanisms, such as molecular relaxations, hydrodynamic modes, and nonhydrodynamic excitations, are potentially at play. Therefore, compared to the traditional method, direct visualization of density space-time correlation functions on two-dimensional color maps permits appraisals of complicated dynamical behavior at mesoscales in a global manner. For example, the scaling relations between space and time for different types of molecular motions can be straightforwardly identified on these plots, without any model-dependent analysis. Additionally, we show how theoretical ideas regarding collective mesoscopic dynamics, such as the classical hydrodynamic theory, the convolution approximation, and a recently proposed phenomenological model, can be discussed in terms of the global features of spatiotemporal maps of intermediate scattering functions. The new perspective offered by the spatiotemporal mapping method should prove useful for the study of liquid dynamics in general.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2022

Formation of Nano- and Micro-Scale Surface Features Induced by Long-Range Femtosecond Filament Laser Ablation

In this work, we study the characteristics of femtosecond-filament-laser–matter interactions and laser-induced periodic surface structures (LIPSS) at a beam-propagation distance up to 55 m. The quantification of the periodicity of filament-induced self-organized surface structures was accomplished by SEM and AFM measurements combined with the use of discrete two-dimensional fast Fourier transform (2D-FFT) analysis, at different filament propagation distances. The results show that the size of the nano-scale surface features increased with ongoing laser filament processing and, further, periodic ripples started to form in the ablation-spot center after irradiation with five spatially overlapping pulses. The effective number of irradiating filament pulses per spot area affected the developing surface texture, with the period of the low spatial frequency LIPSS reducing notably at a high pulse number. The high regularity of the filament-induced ripples was verified by the demonstration of the angle-of-incidence-dependent diffraction of sunlight. This work underlines the potential of long-range femtosecond filamentation for energy delivery at remote distances, with suppressed diffraction and long depth focus, which can be used in biomimetic laser surface engineering and remote-sensing applications.

36 MATERIALS SCIENCE↗

DOE OSTI2020

A discrete integral transform for rapid spectral synthesis

Accurate synthetic spectra that rely on large Line-By-Line (LBL)-databases are used in a wide range of applications such as high temperature combustion, atmospheric re-entry, planetary surveillance and laboratory plasmas. Conventionally synthetic spectra are calculated by computing a lineshape for every spectral line in the database and adding those together, which may take multiple hours for large databases. In this paper we propose a new approach for spectral synthesis based on an integral transform: the synthetic spectrum is calculated as the integral over the product of a Voigt profile and a newly proposed three-dimensional “lineshape distribution function”, which is a function of spectral position and Gaussian- & Lorentzian width coordinates. A fast discrete version of this transform based on the Fast Fourier Transform (FFT) is proposed, which improves performance compared to the conventional approach by several orders of magnitude while maintaining accuracy. Strategies that minimize the discretization error are discussed. A Python implementation of the method is compared against state-of-the-art spectral code RADIS, and is since adopted as RADIS's default synthesis method. The synthesis of a benchmark CO2 spectrum consisting of 1.8 M spectral lines and 200k spectral points took only 3.1 s using the proposed method (1011 lines × spectral points/s), a factor ~300 improvement over the state-of-the-art, with the relative improvement generally increasing for higher number of lines and/or number of spectral points. Finally, an experimental GPU-implementation of the method was also benchmarked, which demonstrated another 2~3 orders performance increase, achieving up to 5 ∙ 10 14 lines × spectral points/s.

42 ENGINEERING↗

DOE OSTI2020

Completion design improvement using a deep convolutional network

Maximizing stimulated natural and hydraulic fracture network is one of the primary hydraulic fracturing concerns for economic production from a horizontal shale gas well. Geomechanical facies and preexisting fractures in each stage are identified based on similarities in formation characteristics to optimize the locations of perforation clusters. This often requires analyzing large volumes of drilling, Logging While Drilling (LWD) and Measurement While Drilling (MWD) data. In this paper, we develop a methodology that calculates the mechanical specific energy (MSE) using real-time drill string acceleration signals directly from its definition. High resolution vibration signals have been collected using a tri-axial accerlometer, which was an auxiliary tool included in acoustic borehole imager. This technique provides a cost-efficient solution for engineered completion design. Furthermore, we adopt deep Convolutional Neural Network (CNN) with signal processing to build a data pipeline that effectively extracts patterns from dynamic acceleration signals for rock lateral MSE classification. First, we apply discrete wavelet transform and Short-Time Fourier Transform (STFT) for signal denoising and pattern recognition. Then we construct an image dataset using multi-scale image fusion at pixel level from 3 sensor channels, including axial, lateral acceleration spectrograms and zero-padded revolutions per minute (RPM). The resulted RGB image dataset includes 4,000 images of 5 MSE ranges with various rock strength conditions. Our results demonstrate that the proposed deep learning model can achieve more than 90% classification accuracy. The deep learning results, as a reference source, were applied in selected Marcellus Shale Energy and Environmental Lab (MSEEL) wells engineered completion located in the Marcellus shale gas site.

03 NATURAL GAS↗

DOE OSTI2026

A Discrete Hankel Transform Approach to Nuclear Data Processing for Fusion Applications

This study introduces advancements to the numerical solutions employed in the processing of nuclear data for fusion applications. It leverages the convolution theorem and Fourier transform techniques to enhance computational efficiency and broaden applicability. Building upon a previously reported discrete Hankel transform approach for Doppler broadening, this work refines the solution of convolution integrals central to these applications. The methodology provides a general and unified framework for evaluating any convolution operation, regardless of whether the underlying problem involves temperature effects in nuclear reactions. The applicability to the nuclear data processing for fusion is demonstrated by deriving the convolution integrals for some of the fusion-related quantities. As before, the convolution operation utilizes a Gaussian-based kernel; however, the discrete Hankel transform of order $𝛼$ = $\frac{1}{2}$ is now applied to the forward Fourier transform of the nonkernel argument, rather than the inverse Fourier transform. This modification eliminates the need for the integration of the nonkernel, cross section–based function, which is a step that posed challenges for certain pointwise cross-section representations. It also removes the requirement for cross-section linearization. Optimized for graphics processing unit architectures, the approach significantly improves computational performance. These advancements are currently under evaluation as the foundation for the next-generation thermonuclear data file processing codes being developed at Lawrence Livermore National Laboratory.

Nuclear science and engineering↗

DOE OSTI2022

Investigation of crossed-twin structure formation in magnesium and magnesium alloys

In this work, the effect of alloying addition on the propensity for twin-twin interactions to transform into crossed-twin structures in magnesium alloys is investigated. A full-field elasto-viscoplastic fast Fourier transform (EVP-FFT) framework combined with a discrete twin model and dislocation density-based hardening law for slip strengths is used to calculate the micromechanical fields in the crystalline matrix around the interacting twins. AZ31 and MgLi alloys are selected along with pure Mg to study the influence of plastic anisotropy in connection with alloying elements. These alloys were selected since their plastic anisotropy measure, which is defined as the ratio between the critical resolved shear stress for pyramidal $\langle$c +a$\rangle$ and basal $\langle$a$\rangle$ slip modes, spanned a wide range. To quantify the role of twin thicknesses, we probe a range of impinging twin thicknesses while fixing the recipient twin thickness. The analysis reveals that: (i) the local driving stress for crossed twin structure formation generated from the interaction of the two twins is lower in a low plastically anisotropic material, like a MgLi alloy, than a high plastically anisotropic material like pure Mg and (ii) the critical impinging twin thickness needed to form the crossed twin structure in pure Mg, AZ31 and MgLi alloys is ~0.5, ~0.75 and ~1.5 times the recipient twin thickness. We propose a relationship between the tendency for crossed twin structure formation and the experimentally observed higher ductility in MgLi alloys compared to pure Mg. Finally, one key implication of the findings is that crossed- twin structure formation can be hindered and the ductility of magnesium alloy thereby improved by properly choosing alloying elements that lower the slip strength for pyramidal $\langle$c + a$\rangle$ slip.

36 MATERIALS SCIENCE↗

DOE OSTI2025

Systematic Study of the Self-Renormalized Nucleon Gluon PDF in Large-Momentum Effective Theory

We present a systematic study of the nucleon gluon parton distribution function (PDF) using the self-renormalized large-momentum effective theory (LaMET) approach in lattice QCD. This work extends previous gluon-PDF extractions by performing a detailed analysis of key systematic effects, including gauge-link smearing, lattice spacing, pion mass, and nucleon boost momentum. The self-renormalization framework mitigates ultraviolet divergences associated with Wilson-line self-energy and renormalon contributions by combining lattice matrix elements with perturbative short-distance information, thereby preserving the correct infrared structure. Calculations are performed on $N_f=2+1+1$ HISQ ensembles generated by the MILC Collaboration at three lattice spacings and two pion masses, with boosted nucleon states reaching momenta up to 2.2~GeV. We determine renormalization factors from zero-momentum matrix elements and apply hybrid renormalization to suppress discretization artifacts. After extrapolating large-separation behavior and performing Fourier transforms, we reconstruct quasi-PDFs and match them to lightcone PDFs using next-to-leading order Wilson coefficients. Our results demonstrate that smearing and lattice-spacing effects are under control, and pion-mass and lattice-spacing dependence is mild relative to the current $O(10^6)$ statistics; however, momentum dependence remains a significant source of uncertainty. Future work including even larger boost momenta will be essential to reduce systematics in lattice determinations of the gluon PDF and to advance toward precision QCD phenomenology at the LHC and the future Electron-Ion Collider.

FOS: Physical sciences↗

DOE OSTI2021

Exactly unitary discrete representations of the metaplectic transform for linear-time algorithms

The metaplectic transform (MT), a generalization of the Fourier transform sometimes called the linear canonical transform, is a tool used ubiquitously in modern optics, for example, when calculating the transformations of light beams in paraxial optical systems. The MT is also an essential ingredient of the geometrical-optics modeling of caustics that we recently proposed. In particular, this application relies on the near-identity MT (NIMT); however, the NIMT approximation used so far is not exactly unitary and leads to numerical instability. Here, we develop a discrete MT that is exactly unitary, and approximate it to obtain a discrete NIMT that is also unitary and can be computed in linear time. We prove that the discrete NIMT converges to the discrete MT when iterated, thereby allowing the NIMT to compute MTs that are not necessarily near-identity. Finally, we then demonstrate the new algorithms with a series of examples.

47 OTHER INSTRUMENTATION↗