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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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106 records · Page 6

Radar Wind Profiler at Santa Barbara

These data are collected as part of an observational database developed to support the floating offshore wind energy research under the ORACLE project funded by DOE Wind Energy Technologies Office (WETO). The radar wind profiler network data are collected by NOAA (https://psl.noaa.gov/data/obs/datadisplay), and only data within the state of California are part of the database.

17 WIND ENERGY↗

Radar Wind Profiler at San Nicolas Island

These data are collected as part of an observational database developed to support the floating offshore wind energy research under the ORACLE project funded by DOE Wind Energy Technologies Office (WETO). The radar wind profiler network data are collected by NOAA (https://psl.noaa.gov/data/obs/datadisplay), and only data within the state of California are part of the database.

17 WIND ENERGY↗

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↗

Reinforcement Learning-based Output Structured Feedback for Distributed Multi-Area Power System Frequency Control

Load frequency control (LFC) is a key factor to maintain the stable frequency in multi-area power systems. As the modern power systems evolve from centralized to decentralized paradigm, LFC needs to consider the decentralized scheme that considers limited information from the information-exchange graph for the generator control of each interconnected area. This paper aims to solve a data-driven constrained LQR problem with mean-variance risk constraints and output structured feedback, and applies this framework to solve the LFC problem in multi-area power systems. By reformulating the constrained optimization problem into a minimax problem, the stochastic gradient descent max-oracle (SGDmax) algorithm with zero-order policy gradient (ZOPG) is adopted to find the optimal feedback gain from the learning, while guaranteeing the convergence. In addition, to improve the adaptation of the proposed learning method to new or varying models, we construct an emulator grid that approximates the dynamics of a physical grid and performs training based on this model. Once the feedback gain is obtained from the emulator grid, it is applied to the physical grid with a robustness test to check whether the controller from the approximated emulator applies to the actual system. Numerical tests show that the obtained feedback controller can successfully control the frequency of each area, while mitigating the uncertainty from the loads, with reliable robustness that ensures the adaptability of the obtained feedback gain to the actual physical grid.

Kwon, Kyung-bin↗

Sign-OPT: A Query-Efficient Hard-label Adversarial Attack

We study the most practical problem setup for evaluating adversarial robustness of a machine learning system with limited access: the hard-label black-box attack setting for generating adversarial examples, where limited model queries are allowed and only the decision is provided to a queried data input. Several algorithms have been proposed for this problem but they typically require huge amount (>20,000) of queries for attacking one example. Among them, one of the state-of-the-art approaches (Cheng et al., 2019) showed that hard-label attack can be modeled as an optimization problem where the objective function can be evaluated by binary search with additional model queries, thereby a zeroth order optimization algorithm can be applied. In this paper, we adopt the same optimization formulation but propose to directly estimate the sign of gradient at any direction instead of the gradient itself, which enjoys the benefit of single query. Using this single query oracle for retrieving sign of directional derivative, we develop a novel query-efficient Sign-OPT approach for hard-label black-box attack. We provide a convergence analysis of the new algorithm and conduct experiments on several models on MNIST, CIFAR-10 and ImageNet. We find that Sign-OPT attack consistently requires 5X to 10X fewer queries when compared to the current state-of-the-art approaches, and usually converges to an adversarial example with smaller perturbation.

Cheng, Minhao↗

The CLoud–Aerosol–Radiation Interaction and Forcing: Year 2017 (CLARIFY-2017) measurement campaign

The representations of clouds, aerosols, and cloud–aerosol–radiation impacts remain some of the largest uncertainties in climate change, limiting our ability to accurately reconstruct past climate and predict future climate. The south-east Atlantic is a region where high atmospheric aerosol loadings and semi-permanent stratocumulus clouds are co-located, providing an optimum region for studying the full range of aerosol–radiation and aerosol–cloud interactions and their perturbations of the Earth's radiation budget. While satellite measurements have provided some useful insights into aerosol–radiation and aerosol–cloud interactions over the region, these observations do not have the spatial and temporal resolution, nor the required level of precision to allow for a process-level assessment. Detailed measurements from high spatial and temporal resolution airborne atmospheric measurements in the region are very sparse, limiting their use in assessing the performance of aerosol modelling in numerical weather prediction and climate models. CLARIFY-2017 was a major consortium programme consisting of five principal UK universities with project partners from the UK Met Office and European- and USA-based universities and research centres involved in the complementary ORACLES, LASIC, and AEROCLO-sA projects. The aims of CLARIFY-2017 were fourfold: (1) to improve the representation and reduce uncertainty in model estimates of the direct, semi-direct, and indirect radiative effect of absorbing biomass burning aerosols; (2) to improve our knowledge and representation of the processes determining stratocumulus cloud microphysical and radiative properties and their transition to cumulus regimes; (3) to challenge, validate, and improve satellite retrievals of cloud and aerosol properties and their radiative impacts; (4) to improve the impacts of aerosols in weather and climate numerical models. This paper describes the modelling and measurement strategies central to the CLARIFY-2017 deployment of the FAAM BAe146 instrumented aircraft campaign, summarizes the flight objectives and flight patterns, and highlights some key results from our initial analyses.

54 ENVIRONMENTAL SCIENCES↗

Observations of supermicron-sized aerosols originating from biomass burning in southern Central Africa

During the 3 years of the ObseRvations of Aerosols above CLouds and their intEractionS (ORACLES) campaign, the NASA Orion P-3 was equipped with a 2D stereo (2D-S) probe that imaged particles with maximum dimension (D) ranging from 10 < D < 1280 µm. The 2D-S recorded supermicron-sized aerosol particles (SAPs) outside of clouds within biomass burning plumes during flights over the southeastern Atlantic off Africa's coast. Numerous SAPs with 10 < D < 1520 µm were observed in 2017 and 2018 at altitudes between 1230 and 4000 m, 1000 km from the coastline, mostly between 7–11° S. No SAPs were observed in 2016 as flights were conducted further south and further from the coastline. Number concentrations of refractory black carbon (rBC) measured by a single particle soot photometer ranged from 200 to 1200 cm –3 when SAPs were observed. Transmission electron microscopy images of submicron particulates, collected on Holey carbon grid filters, revealed particles with potassium salts, black carbon (BC), and organics. Energy-dispersive X-ray spectroscopy spectra also detected potassium, a tracer for biomass burning. These measurements provided evidence that the submicron particles originated from biomass burning. NOAA Hybrid Single-Particle Lagrangian Integrated Trajectory (HYSPLIT) 3 d back trajectories show a source in northern Angola for times when large SAPs were observed. Fire Information for Resource Management System (FIRMS) Moderate Resolution Imaging Spectroradiometer (MODIS) 6 active fire maps showed extensive biomass burning at these locations. Given the back trajectories, the high number concentrations of rBC, and the presence of elemental tracers indicative of biomass burning, it is hypothesized that the SAPs imaged by the 2D-S are examples of BC aerosol, ash, or unburned plant material.

54 ENVIRONMENTAL SCIENCES↗

Modeled and observed properties related to the direct aerosol radiative effect of biomass burning aerosol over the southeastern Atlantic

Biomass burning smoke is advected over the southeastern Atlantic Ocean between July and October of each year. This smoke plume overlies and mixes into a region of persistent low marine clouds. Model calculations of climate forcing by this plume vary significantly in both magnitude and sign. NASA EVS-2 (Earth Venture Suborbital-2) ORACLES (ObseRvations of Aerosols above CLouds and their intEractionS) had deployments for field campaigns off the west coast of Africa in 3 consecutive years (September 2016, August 2017, and October 2018) with the goal of better characterizing this plume as a function of the monthly evolution by measuring the parameters necessary to calculate the direct aerosol radiative effect. Here, this dataset and satellite retrievals of cloud properties are used to test the representation of the smoke plume and the underlying cloud layer in two regional models (WRF-CAM5 and CNRM-ALADIN) and two global models (GEOS and UM-UKCA). The focus is on the comparisons of those aerosol and cloud properties that are the primary determinants of the direct aerosol radiative effect and on the vertical distribution of the plume and its properties. The representativeness of the observations to monthly averages are tested for each field campaign, with the sampled mean aerosol light extinction generally found to be within 20 % of the monthly mean at plume altitudes. When compared to the observations, in all models, the simulated plume is too vertically diffuse and has smaller vertical gradients, and in two of the models (GEOS and UM-UKCA), the plume core is displaced lower than in the observations. Plume carbon monoxide, black carbon, and organic aerosol masses indicate underestimates in modeled plume concentrations, leading, in general, to underestimates in mid-visible aerosol extinction and optical depth. Biases in mid-visible single scatter albedo are both positive and negative across the models. Observed vertical gradients in single scatter albedo are not captured by the models, but the models do capture the coarse temporal evolution, correctly simulating higher values in October (2018) than in August (2017) and September (2016). Uncertainties in the measured absorption Ångstrom exponent were large but propagate into a negligible (<4 %) uncertainty in integrated solar absorption by the aerosol and, therefore, in the aerosol direct radiative effect. Model biases in cloud fraction, and, therefore, the scene albedo below the plume, vary significantly across the four models. The optical thickness of clouds is, on average, well simulated in the WRF-CAM5 and ALADIN models in the stratocumulus region and is underestimated in the GEOS model; UM-UKCA simulates cloud optical thickness that is significantly too high. Overall, the study demonstrates the utility of repeated, semi-random sampling across multiple years that can give insights into model biases and how these biases affect modeled climate forcing. The combined impact of these aerosol and cloud biases on the direct aerosol radiative effect (DARE) is estimated using a first-order approximation for a subset of five comparison grid boxes. A significant finding is that the observed grid box average aerosol and cloud properties yield a positive (warming) aerosol direct radiative effect for all five grid boxes, whereas DARE using the grid-box-averaged modeled properties ranges from much larger positive values to small, negative values. It is shown quantitatively how model biases can offset each other, so that model improvements that reduce biases in only one property (e.g., single scatter albedo but not cloud fraction) would lead to even greater biases in DARE. Across the models, biases in aerosol extinction and in cloud fraction and optical depth contribute the largest biases in DARE, with aerosol single scatter albedo also making a significant contribution.

54 ENVIRONMENTAL SCIENCES↗

Light Absorption by Brown Carbon over the South-East Atlantic Ocean

Biomass burning emissions often contain brown carbon (BrC), which represents a large family of light-absorbing organics that is chemically complex and therefore difficult to estimate their absorption of incoming solar radiation, resulting in large uncertainties in the estimation of the global direct radiative effect of aerosols. Here we investigate the contribution of BrC to the total light absorption of biomass burning aerosols over the South-East Atlantic Ocean with different optical models utilizing a suite of airborne measurements from the ORACLES 2018 campaign by introducing an effective refractive index of black carbon (BC), m eBC = n eBC +ik eBC , that accounts for all possible absorbing components at 660 nm wavelength to facilitate the attribution of absorption at shorter wavelengths. Most values of the imaginary part of the refractive index, k eBC , were larger than those commonly used for BC from biomass burning emissions, suggesting contributions from absorbers beyond BC at 660 nm. The TEM-EDX single particle analysis further suggests that these long-wavelength absorbers might include iron oxides, as iron is found to be present only when large values of k eBC are derived. Using this effective BC refractive index, we find that the contribution of BrC to the total absorption at 470 nm (R BrC,470 ) ranges from ~5–15 %, with the organic aerosol mass absorption coefficient (MAC OA,470 ) at this wavelength ranging from 0.25 ± 0.34 m 2 g -1 to 0.43 ± 0.12 m 2 g -1 . The core-shell model yielded much higher estimates of MAC OA,470 and R BrC,470 than homogeneous mixing models, underscoring the importance of model treatment. Another key finding was that estimates of the BrC contribution at 470 nm from the commonly used AAE (absorption Ångström exponent) attribution method (< 5 %) are much lower than the BrC contribution estimates (R BrC,470 ) using our new methodology that accounts for contributions from both BrC and non-carbonaceous, long-wavelength absorbers, such as magnetite. Thus, it is recommended that application of any optical properties-based attribution method use absorption coefficients at the longest possible wavelength to minimize the influence of BrC at the long wavelength and to account for potential contributions from other absorbing materials.

54 ENVIRONMENTAL SCIENCES↗

Biomass burning and marine aerosol processing over the southeast Atlantic Ocean: a TEM single-particle analysis

This study characterizes single-particle aerosol composition from filters collected during the ObseRvations of Aerosols above CLouds and their intEractionS (ORACLES) and CLoud–Aerosol–Radiation Interaction and Forcing: Year 2017 (CLARIFY-2017) campaigns. In particular the study describes aged biomass burning aerosol (BBA), its interaction with the marine boundary layer and the influence of biomass burning (BB) air on marine aerosol. The study finds evidence of BBA influenced by marine boundary layer processing as well as sea salt influenced by BB air. Secondary chloride aerosols were observed in clean marine air as well as in BB-influenced air in the free troposphere. Higher-volatility organic aerosol appears to be associated with increased age of biomass burning plumes, and photolysis or oxidation may be a mechanism for the apparent increased volatility. Aqueous processing and interaction with the marine boundary layer air may be a mechanism for the presence of sodium on many aged potassium salts. By number, biomass burning potassium salts and modified sea salts are the most observed particles on filter samples. The most commonly observed BC coatings are inorganic salts. These results suggest that atmospheric processes such as photolysis, oxidation and cloud processing are key drivers in the elemental composition and morphology of aged BBA. Fresh BBA inorganic salt content, as it has an important role in the particles' ability to uptake water, may be a key driver in how aqueous processing and atmospheric aging proceed.

54 ENVIRONMENTAL SCIENCES↗

On the differences in the vertical distribution of modeled aerosol optical depth over the southeastern Atlantic

Abstract. The southeastern Atlantic is home to an expansive smoke aerosol plume overlying a large cloud deck for approximately a third of the year. The aerosol plume is mainly attributed to the extensive biomass burning activities that occur in southern Africa. Current Earth system models (ESMs) reveal significant differences in their estimates of regional aerosol radiative effects over this region. Such large differences partially stem from uncertainties in the vertical distribution of aerosols in the troposphere. These uncertainties translate into different aerosol optical depths (AODs) in the planetary boundary layer (PBL) and the free troposphere (FT). This study examines differences of AOD fraction in the FT and AOD differences among ESMs (WRF-CAM5, WRF-FINN, GEOS-Chem, EAM-E3SM, ALADIN, GEOS-FP, and MERRA-2) and aircraft-based measurements from the NASA ObseRvations of Aerosols above CLouds and their intEractionS (ORACLES) field campaign. Models frequently define the PBL as the well-mixed surface-based layer, but this definition misses the upper parts of decoupled PBLs, in which most low-level clouds occur. To account for the presence of decoupled boundary layers in the models, the height of maximum vertical gradient of specific humidity profiles from each model is used to define PBL heights. Results indicate that the monthly mean contribution of AOD in the FT to the total-column AOD ranges from 44 % to 74 % in September 2016 and from 54 % to 71 % in August 2017 within the region bounded by 25∘ S–0∘ N–S and 15∘ W–15∘ E (excluding land) among the ESMs. ALADIN and GEOS-Chem show similar aerosol plume patterns to a derived above-cloud aerosol product from the Moderate Resolution Imaging Spectroradiometer (MODIS) during September 2016, but none of the models show a similar above-cloud plume pattern to MODIS in August 2017. Using the second-generation High Spectral Resolution Lidar (HSRL-2) to derive an aircraft-based constraint on the AOD and the fractional AOD, we found that WRF-CAM5 produces 40 % less AOD than those from the HSRL-2 measurements, but it performs well at separating AOD fraction between the FT and the PBL. AOD fractions in the FT for GEOS-Chem and EAM-E3SM are, respectively, 10 % and 15 % lower than the AOD fractions from the HSRL-2. Their similar mean AODs reflect a cancellation of high and low AOD biases. Compared with aircraft-based observations, GEOS-FP, MERRA-2, and ALADIN produce 24 %–36 % less AOD and tend to misplace more aerosols in the PBL. The models generally underestimate AODs for measured AODs that are above 0.8, indicating their limitations at reproducing high AODs. The differences in the absolute AOD, FT AOD, and the vertical apportioning of AOD in different models highlight the need to continue improving the accuracy of modeled AOD distributions. These differences affect the sign and magnitude of the net aerosol radiative forcing, especially when aerosols are in contact with clouds.

54 ENVIRONMENTAL SCIENCES↗

A remote sensing algorithm for vertically resolved cloud condensation nuclei number concentrations from airborne and spaceborne lidar observations

Cloud condensation nuclei (CCN) are mediators of aerosol–cloud interactions (ACIs), contributing to the largest uncertainties in the understandings of global climate change. We present a novel remote-sensing-based algorithm that quantifies the vertically resolved CCN number concentrations (N CCN ) using aerosol optical properties measured by a multiwavelength lidar. The algorithm considers five distinct aerosol subtypes with bimodal size distributions. The inversion used the lookup tables developed in this study, based on the observations from the Aerosol Robotic Network, to efficiently retrieve optimal particle size distributions from lidar measurements. The method derives dry aerosol optical properties by implementing hygroscopic enhancement factors in lidar measurements. The retrieved optically equivalent particle size distributions and aerosol-type-dependent particle composition are utilized to calculate critical diameters using κ-Köhler theory and N CCN at six supersaturations ranging from 0.07 % to 1.0 %. Sensitivity analyses indicate that uncertainties in extinction coefficients and relative humidity greatly influence the retrieval error in N CCN . The potential of this algorithm is further evaluated by retrieving N CCN using airborne lidar from the NASA ObseRvations of Aerosols above CLouds and their intEractionS (ORACLES) campaign and is validated against simultaneous measurements from the CCN counter. The independent validation with robust correlation demonstrates promising results. Furthermore, the N CCN has been retrieved for the first time using a proposed algorithm from spaceborne lidar – Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP) – measurements. The application of this new capability demonstrates the potential for constructing a 3D CCN climatology at a global scale, which helps to better quantify ACI effects and thus reduce the uncertainty in aerosol climate forcing.

54 ENVIRONMENTAL SCIENCES↗

Undersizing of aged African biomass burning aerosol by an ultra-high-sensitivity aerosol spectrometer

The ultra-high-sensitivity aerosol spectrometer (UHSAS) differs from most other optical particle spectrometers by using a high-power infrared (IR) laser to detect small particles and reduce the sizing ambiguity due to the non-monotonicity of scattering with particle size. During the NASA ORACLES project (ObseRvations of Aerosols above CLouds and their intEractionS) over the southeast Atlantic Ocean, the UHSAS clearly undersized particles in the biomass burning plume extending from southern Africa. Since the horizontal and vertical extent of the plume was vast, the NASA P-3B research aircraft often flew through a fairly uniform biomass burning plume for periods exceeding 30 min, sufficient time to explore the details of the UHSAS response by selecting single particle sizes with a differential mobility analyzer (DMA) and passing them to the UHSAS. This was essentially an in-flight calibration of the UHSAS using the particles of interest. Two modes of responses appeared. Most particles were undersized by moderate amounts, ranging from not at all for 70 nm aerosols to 15 % for 280 nm particles. Mie scattering calculations show that composition-dependent refractive index of the particles cannot explain the pattern. Heating of brown carbon or tarballs in the beam causing evaporation and shrinking of the particles is the most plausible explanation, though mis-sizing due to non-sphericity cannot be ruled out. A small fraction (10 %–30 %) of the particles were undersized by 25 % to 35 %. Those were apparently the particles containing refractory black carbon. Laboratory calibrations confirm that black carbon is drastically undersized by the UHSAS, because particles heat to their vaporization point and shrink. A simple empirical correction equation was implemented that dramatically improves agreement with DMA distributions between 100 and 500 nm. It raised the median particle diameter by 18 nm, from 163 to 181 nm, during the August 2017 deployment and by smaller amounts during deployments with less intense pollution. Calculated scattering from UHSAS size distributions increased by about 130 %, dramatically improving agreement with scattering measured by nephelometers. The correction is only valid in polluted instances; clean marine boundary layer and free troposphere aerosols behaved more like the calibration spheres. We were unable to directly test the correction between 500 and 1000 nm, though aerodynamic particle sizer (APS) data appear to show that the correction is poor at the largest diameters, which is no surprise as the composition of those particles is likely to be quite different than that of the accumulation mode. This adds to the evidence that UHSAS data must be treated cautiously whenever the aerosol may absorb infrared light. Similar corrections may be required whenever brown carbon aerosol is present.

54 ENVIRONMENTAL SCIENCES↗

Testing convolutional neural network based deep learning systems: a statistical metamorphic approach

Machine learning technology spans many areas and today plays a significant role in addressing a wide range of problems in critical domains,i.e., healthcare, autonomous driving, finance, manufacturing, cybersecurity,etc. Metamorphic testing (MT) is considered a simple but very powerful approach in testing such computationally complex systems for which either an oracle is not available or is available but difficult to apply. Conventional metamorphic testing techniques have certain limitations in verifying deep learning-based models (i.e., convolutional neural networks (CNNs)) that have a stochastic nature (because of randomly initializing the network weights) in their training. In this article, we attempt to address this problem by using a statistical metamorphic testing (SMT) technique that does not require software testers to worry about fixing the random seeds (to get deterministic results) to verify the metamorphic relations (MRs). We propose seven MRs combined with different statistical methods to statistically verify whether the program under test adheres to the relation(s) specified in the MR(s). We further use mutation testing techniques to show the usefulness of the proposed approach in the healthcare space and test two CNN-based deep learning models (used for pneumonia detection among patients). The empirical results show that our proposed approach uncovers 85.71% of the implementation faults in the classifiers under test (CUT). Furthermore, we also propose an MRs minimization algorithm for the CUT, thus saving computational costs and organizational testing resources.

Computer Science↗

Exact and Fixed-Point Grover Search with Qudits

Grover's algorithm provides a quadratic speedup for searching unstructured databases and is traditionally implemented with qubits in Hilbert spaces whose dimensions are powers of two. With the advent of quantum platforms utilizing qudits---quantum systems with more than two levels---there is a need to generalize Grover search to these architectures, including heterogeneous systems with qudits of varying dimensions. Here, we present a unified framework for qudit-based Grover search, detailing the construction of oracles and diffusion operators with and without ancilla qubits and generalizing deterministic and fixed-point search variants that ensure exact or bounded success probabilities. We analyze phase-matching techniques and provide explicit circuit decompositions suitable for diverse hardware platforms. We also compare the corresponding trajectories on the Bloch sphere to provide an intuitive visualization of how the different phase choices amplify the target state. These results facilitate flexible, hardware-oriented protocols for implementing Grover search on qudit processors, potentially reducing circuit depth and enhancing success probabilities, thereby offering a practical toolkit for quantum computation and sensing applications leveraging multilevel quantum systems.

Roy, Tanay [Fermilab] (ORCID:000000019442862X)↗

Quantum block encoding for one-pair semiseparable matrices

Quantum block encoding (QBE) is a crucial step in the development of most quantum algorithms, as it provides an embedding of a given matrix into a suitable larger unitary matrix. Historically, the development of efficient techniques for QBE has mostly focused on sparse matrices; less effort has been devoted to data-sparse (e.g., rank-structured) matrices. In this work we examine a particular case of rank structure, namely, one-pair semiseparable matrices. We present a new block encoding approach that relies on a suitable factorization of the given matrix as the product of triangular and diagonal factors. To encode the matrix, the algorithm needs $2\log(N)+7$ ancillary qubits. Assuming that the data input oracles can be implemented with polylogarithmic depth, or that a QRAM input model is available, our proposed method requires $\mathcal{O}({\rm polylog} (N))$ time and has an error of $\mathcal{O}(N^2)$, where $N$ is the matrix size.

Antonioli, Giacomo [Pisa U.; CERN] (ORCID:00090000↗