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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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At least 1,243 records · Page 69
All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Chemical Kinetic Method for Active-Site Quantification in Fe-N-C Catalysts and Correlation with Molecular Probe and Spectroscopic Site-Counting Methods

Mononuclear Fe ions ligated by nitrogen (FeN x ) dispersed on nitrogen-doped carbon (Fe-N-C) serve as active centers for electrocatalytic O 2 reduction and thermocatalytic aerobic oxidations. Despite their promise as replacements for precious metals in a variety of practical applications, such as fuel cells, the discovery of new Fe-N-C catalysts has relied primarily on empirical approaches. In this context, the development of quantitative structure–reactivity relationships and benchmarking of catalysts prepared by different synthetic routes and by different laboratories would be facilitated by the broader adoption of methods to quantify atomically dispersed FeN x active centers. In the present study, we develop a kinetic probe reaction method that uses the aerobic oxidation of a model hydroquinone substrate to quantify the density of FeN x centers in Fe- N-C catalysts. The kinetic method is compared with low-temperature Mössbauer spectroscopy, CO pulse chemisorption, and electrochemical reductive stripping of NO derived from NO 2 – on a suite of Fe-N-C catalysts prepared by diverse routes and featuring either the exclusive presence of Fe as FeN x sites or the coexistence of aggregated Fe species in addition to FeN x . The FeN x site densities derived from the kinetic method correlate well with those obtained from CO pulse chemisorption and Mössbauer spectroscopy. The broad survey of Fe-N-C materials also reveals the presence of outliers and challenges associated with each site quantification approach. Furthermore, the kinetic method developed here does not require pretreatments that may alter active-site distributions nor specialized equipment beyond reaction vessels and analytical instrumentation (e.g., NMR).

36 MATERIALS SCIENCE↗
DOE OSTI2024

Symmetry-Breaking Charge Transfer in Metal–Organic Frameworks

High quantum-yield charge carrier generation from the initially prepared excitons defines a key step in the light-harvesting and conversion scheme. Photoinduced charge transfer in molecular electron donor—acceptor assemblies is driven by a sizable ΔG 0 , which compromises the potential of the generated carriers. Reminiscent of the special pair at the reaction center of the natural light-harvesting complex, symmetry-breaking charge transfer (SBCT) within a pair of identical struts of metal-organic framework (MOF) will facilitate the efficient generation of long-lived charge carriers with maximized potentials without incorporating any foreign redox species. We report SBCT in pyrene-based zirconium metal–organic framework (MOF) NU-1000 that leads to efficient generation of radical ions in a polar solvent and bound CT states in a low-polar solvent. Further, the probe unveils the role of the low-lying non-Franck–Condon excitonic states as intermediates in the formation of the SBCT state from the initially prepared Franck–Condon S 1 states. Ultrafast and transient spectroscopy—probed over 200 fs-30 μs time scale-evinces a k SBCT = (110 ps) -1 in polar media (ε s = 37.5) forming solvated radical ions with recombination rate k CR = (~45 ns) -1 . A slower rate with k SBCT = (203 ps) -1 was recorded in low-polar (ε s = 7.0) solvent manifesting a bound [TBAPy •+ TBAPy •– ] state with k CR ≈ (17 μs) -1 . This discovery, along with other unique photophysical features relevant to light harvesting, should define a MOF-based platform for developing heterogeneous artificial photon energy conversion systems.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

Mining for Metal–Organic Systems: Chemistry Frontiers of Th-, U-, and Zr-Materials

The conceptual framework presented in this Perspective overviews the design principles of innovative thorium-based materials that could address urgent needs of the medicinal, nuclear energy, and waste remediation sectors from the lens of zirconium and uranium analogs. We survey the intersections of Zr, Th, and U chemistry with a focus on how the intrinsic behavior of each metal translates to broader material properties, including, but not limited to, structural and topological diversity, preferential metal–ligand binding, and reactivity. On the example of several classes of materials, including organometallic complexes, polyoxometalates, and the primary focus of this Perspective, metal–organic frameworks (MOFs), the design principles that govern the preparation of Zr-, Th-, and U-compounds, including oxophilicity, variation in oxidation states, and stable coordination environments have been considered. Further, we highlight how the impact of the mentioned variables may shift throughout the progression from discrete molecular systems to extended structures. We discuss the common assumption that zirconium-organic materials are typically considered a close analog of thorium-based congeners in areas such as material design and preparation. Through consideration of fundamental chemistry principles, we shed light on the relationships between Zr-, Th-, and U-based materials and highlight how a critical analysis of their distinct properties can be used to target a desired material performance. Finally, we provide a detailed understanding of Th-based materials chemistry by anchoring their fundamental properties between two well-studied reference points, zirconium- and uranium-containing analogs.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Highly Conjugated Porphyrin Arrays Enable Optical Resolution of Ferromagnetic and Antiferromagnetic Aligned States of the Triplet Exciton and an Incorporated Stable Radical

Well-defined photogenerated molecular spin systems have potential utility in spintronics and quantum information science (QIS). Because molecular magnetic, optical, and electronic properties can be controlled by design, diverse spin systems can be prepared at modest temperatures. Photogenerated molecular spin systems often involve states prepared from the interaction of excitons and charges. Resolving the nature of electron spin alignment in photogenerated spin states described by the coupling of a triplet exciton and a stable radical commonly relies on EPR spectroscopy. Here, we describe ethyne-bridged (porphinato)metal ( PM n ) oligomers that incorporate a macrocycle-bound Cu(II) radical center. Upon photoexcitation of such PM n arrays, a singdoublet ( 2 S 1 ) state is formed; ultrafast internal conversion (IC) then produces a tripdoublet ( 2 T 1 ) state, which undergoes intersystem crossing (ISC) to produce a tripquartet ( 4 T 1 ) state, before relaxation to the ground state ( 2 S 0 ). These highly conjugated Cu(II) radical-containing PMn arrays enable direct observation of copper porphyrin 2 T 1 → 4 T 1 ISC dynamics from the biexponential decay of the near-infrared (NIR) 2,4 T 1 → 2,4 T n transient absorption manifold. Multireference n-electron valence perturbation theory (NEVPT2) computations illuminate how PM n electronic structure controls the relaxation dynamics of these long-lived (>10 ns) electronically excited multiplet states. These studies show that highly conjugated and polarizable porphyrin arrays incorporating stable spin centers provide rare π-delocalized systems where the ferromagnetic and antiferromagnetic alignment between a triplet exciton and a stable radical are both spectrally resolved and addressable using transient optical spectroscopy at wavelengths exceeding 1 μm, providing new opportunities to QIS.

14 SOLAR ENERGY↗
DOE OSTI2026

Design Principles for the Synthesis of Self-Pillared ZSM-5 Zeolite Nanosheets

The design of next-generation materials for emerging energy and environmental applications heavily relies on empirical approaches to direct nonclassical nucleation and crystal growth pathways, polymorphism, intercrystalline transformations, and seed-assisted growth processes. A long-standing obstacle to nanoporous materials design is the complexity of their crystallization, which hinders the development of predictive models and/or physical descriptors that can guide their synthesis. In this study, we use a combination of state-of-the-art synthesis, characterization, and computational design to prepare hierarchical MFI-type zeolites, which we couple with benchmark catalytic testing to assess structure-property-performance relationships. These hierarchical materials are intergrowths of two commercially relevant zeolite frameworks, MFI and MEL, prepared as self-pillared pentasil (SPP) zeolites through seed-assisted, organic-free syntheses for which little theoretical guidance existed. Here, by comparing a large library of zeolite seeds with different pore sizes, dimensions, and structural composite building units, we determined the relative impact of seed and silica source selection, among other synthesis variables. Combined experimental and computational studies are used to test several hypotheses in literature to rationalize the choice of seed structure and establish a more robust selection criteria for seed-assisted synthesis of zeolites. Specifically, we show that a data-driven approach to develop structural descriptors correlates to new, facile routes to rationally design SPP zeolites, addressing knowledge gaps in the fundamental understanding of (non)classical crystal growth mechanisms that are characteristic of nanoporous aluminosilicates.

36 MATERIALS SCIENCE↗
DOE OSTI2026

Probing Terra Incognita of Ni–P Catalysts: Operando Explorations during Hydrogen Evolution Reaction

We have developed two Ni phosphide preparation methods allowing operando XAS surface-sensitive studies of well-defined bulk systems. For Ni K-edge XAS, a Ni 2 P phase-pure powder was sintered into a high-density pellet and polished for grazing incidence XAS. Ni sites were mildly affected by the acidic electrolyte prior to the HER, while the applied cathodic potential caused the reduction of Ni surface states beyond the states of as-prepared Ni 2 P. The computed fully H-covered Ni 2 P [0001] model describes the difference in the operando Ni K-edge GIXAS spectrum well. Upon turning the applied bias off, the Ni sites became immediately oxidized, forming NiO on the surface. Thus, the active phase during the HER is covalent Ni 0 close to that in the intermetallic phosphides, and Ni 2+ oxides formed after, and not during, the HER. For P K-edge XAS, Ni foam was phosphorized to form a thin Ni 3 P layer while preserving its high surface area. Upon immersion in the acidic electrolyte, the P sites underwent removal of P 5+ phosphates and formed new P coordination, possibly due to the adsorption of protons from the electrolyte. These new P surface states were not affected by turning the cathodic current on and off as soon as the sample was immersed in the acidic electrolyte. However, the removal of the sample from the electrochemical cell and drying in air resulted in substantial depletion and oxidation of surface P. Echoing the observed Ni site chemistry during HER, XAS and XPS suggest that the in situ active P sites are different from the oxidized P states observed under ex situ conditions.

Kong, Seongyoung [Iowa State Univ., Ames, IA (Unit↗
DOE OSTI2020

Single-Crystal Polycationic Polymers Obtained by Single-Crystal-to-Single-Crystal Photopolymerization

The efficient preparation of single-crystalline ionic polymers and fundamental understanding of their structure–property relationships at the molecular level remains a challenge in chemistry and materials science. Herein, we observe the single-crystal structure of a highly ordered polycationic polymer (polyelectrolyte) and its proton conductivity. The polyelectrolyte single crystals can be prepared on a gram-scale in quantitative yield, by taking advantage of an ultraviolet/sunlight-induced topochemical polymerization, from a tricationic monomer—a self-complementary building block possessing a preorganized conformation. A single-crystal-to-single-crystal photopolymerization was revealed unambiguously by in situ single-crystal X-ray diffraction analysis, which was also employed to follow the progression of molecular structure from the monomer, to a partially polymerized intermediate, and, finally, to the polymer itself. Collinear polymer chains are held together tightly by multiple Coulombic interactions involving counterions to form two-dimensional lamellar sheets (1 nm in height) with sub-nanometer pores (5 Å). The polymer is extremely stable under 254 nm light irradiation and high temperature (above 500 K). The extraordinary mechanical strength and environmental stability—in combination with its impressive proton conductivity (~3 × 10 –4 S cm –1 )—endow the polymer with potential applications as a robust proton-conducting material. By marrying supramolecular chemistry with macromolecular science, the outcome represents a major step toward the controlled synthesis of single-crystalline polyelectrolyte materials with perfect tacticity.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Regional and Temporal Variability of Atmospheric River Seasonality: Influences of Detection Algorithms and Moisture Transport Dynamics

Abstract Understanding the regional and temporal variability of atmospheric river (AR) seasonality is crucial for preparedness and mitigation of extreme events. While ARs were thought to peak in winter, recent research shows they exhibit region‐specific seasonality and are heavily influenced by the chosen detection algorithm. This study examines the link between the year‐to‐year consistency of peak‐AR activity to the presence of a dominant seasonal pattern, considering both location and algorithm choice. Regions are categorized by their temporal characteristics: consistent patterns (e.g., East Asia), patterns with occasional outliers (e.g., British Columbia coast), and regions lacking a clear dominant peak season (e.g., South Atlantic, parts of Australia). Hence, not all regions display a consistent seasonal cycle of AR activity. This study quantifies the extent to which a region experiences a dominant peak season of AR activity (or lacks one) and offers insights to enhance decision‐making in water management, natural hazard preparedness, and forecasting. Furthermore, given our finding that detection algorithms influence the peak season of AR activity, we also examine two diagnostic variables representative of moisture transport to corroborate our results. Integrated vapor transport, which captures meridional and zonal moisture transport, and Moist Wave Activity, representing moisture intrusions from lower to higher latitudes, are examined. Our analysis indicates that inconsistencies in the seasonal cycle of AR activity are not solely due to discrepancies in detection algorithms but also arise from changes in moisture transport. Plain Language Summary Atmospheric rivers (ARs) are critical weather phenomena that can cause extreme events like heavy rainfall and flooding. Understanding when and where ARs are most likely to occur throughout the year is essential for preparing and responding to these events. Traditionally, ARs were thought to peak in winter, but recent studies show this varies by region. Our study helps address the challenge decision‐makers face in anticipating and preparing for AR events by providing insights into the consistency of peak seasonal patterns across different areas. Some regions, like East Asia, and the British Columbia coast, show a consistent peak season, while others, like the South Atlantic and parts of Australia, have significant year‐to‐year variations, making it hard to identify a dominant season. To better understand these changes over time, the study also examines how moisture moves in the atmosphere, using Integrated Vapor Transport (which looks at moisture movement in various directions) and Moist Wave Activity (which tracks moisture shifts from lower to higher latitudes). The findings suggest that inconsistencies in AR patterns are due not only to detection methods but also due to changes in moisture transport. Key Points The peak season of atmospheric river activity can change depending on the year in some areas Interannual variations in the peak season can make identifying a dominant season challenging for some regions Frequent shifts in peak season across years reflect inconsistencies tied to detection algorithms and to underlying dynamics

Kamnani, Diya↗
DOE OSTI2020

Unprecedented generation of 3D heterostructures by mechanochemical disassembly and re-ordering of incommensurate metal chalcogenides

Three-dimensional heterostructures are usually created either by assembling two-dimensional building blocks into hierarchical architectures or using stepwise chemical processes that sequentially deposit individual monolayers. Both approaches suffer from a number of issues, including lack of suitable precursors, limited reproducibility, and poor scalability of the preparation protocols. Therefore, development of alternative methods that enable preparation of heterostructured materials is desired. We create heterostructures with incommensurate arrangements of well-defined building blocks using a synthetic approach that comprises mechanical disassembly and simultaneous reordering of layered transition-metal dichalcogenides, MX 2 , and non-layered monochalcogenides, REX, where M = Ta, Nb, RE = Sm, La, and X = S, Se. We show that the discovered solid-state processes are rooted in stochastic mechanochemical transformations directed by electronic interaction between chemically and structurally dissimilar solids toward atomic-scale ordering, and offer an alternative to conventional heterostructuring. Details of composition–structure–properties relationships in the studied materials are also highlighted.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Trade off-free entanglement stabilization in a superconducting qutrit-qubit system

Abstract Quantum reservoir engineering is a powerful framework for autonomous quantum state preparation and error correction. However, traditional approaches to reservoir engineering are hindered by unavoidable coherent leakage out of the target state, which imposes an inherent trade off between achievable steady-state state fidelity and stabilization rate. In this work we demonstrate a protocol that achieves trade off-free Bell state stabilization in a qutrit-qubit system realized on a circuit-QED platform. We accomplish this by creating a purely dissipative channel for population transfer into the target state, mediated by strong parametric interactions coupling the second-excited state of a superconducting transmon and the engineered bath resonator. Our scheme achieves a state preparation fidelity of 84% with a stabilization time constant of 339 ns, leading to a 54 ns error-time product in a solid-state quantum information platform.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Fine-tuned local coordination environment of Pt single atoms on ceria controls catalytic reactivity

Constructing single atom catalysts with fine-tuned coordination environments can be a promising strategy to achieve satisfactory catalytic performance. Herein, via a simple calcination temperature-control strategy, CeO 2 supported Pt single atom catalysts with precisely controlled coordination environments are successfully fabricated. The joint experimental and theoretical analysis reveals that the Pt single atoms on Pt 1 /CeO 2 prepared at 550 °C (Pt/CeO 2 -550) are mainly located at the edge sites of CeO 2 with a Pt–O coordination number of ca. 5, while those prepared at 800 °C (Pt/CeO 2 -800) are predominantly located at distorted Ce substitution sites on CeO 2 terrace with a Pt–O coordination number of ca. 4. Pt/CeO 2 -550 and Pt/CeO 2 -800 with different Pt 1 -CeO 2 coordination environments exhibit a reversal of activity trend in CO oxidation and NH 3 oxidation due to their different privileges in reactants activation and H 2 O desorption, suggesting that the catalytic performance of Pt single atom catalysts in different target reactions can be maximized by optimizing their local coordination structures.

36 MATERIALS SCIENCE↗
DOE OSTI2023

The learnability of Pauli noise

Recently, several quantum benchmarking algorithms have been developed to characterize noisy quantum gates on today’s quantum devices. A fundamental issue in benchmarking is that not everything about quantum noise is learnable due to the existence of gauge freedom, leaving open the question what information is learnable and what is not, which is unclear even for a single CNOT gate. Here we give a precise characterization of the learnability of Pauli noise channels attached to Clifford gates using graph theoretical tools. Our results reveal the optimality of cycle benchmarking in the sense that it can extract all learnable information about Pauli noise. We experimentally demonstrate noise characterization of IBM’s CNOT gate up to 2 unlearnable degrees of freedom, for which we obtain bounds using physical constraints. In addition, we show that an attempt to extract unlearnable information by ignoring state preparation noise yields unphysical estimates, which is used to lower bound the state preparation noise.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Bottom-up evolution of perovskite clusters into high-activity rhodium nanoparticles toward alkaline hydrogen evolution

Self-reconstruction has been considered an efficient means to prepare efficient electrocatalysts in various energy transformation process for bond activation and breaking. However, developing nano-sized electrocatalysts through complete in-situ reconstruction with improved activity remains challenging. Herein, we report a bottom-up evolution route of electrochemically reducing Cs 3 Rh 2 I 9 halide-perovskite clusters on N-doped carbon to prepare ultrafine Rh nanoparticles (~2.2 nm) with large lattice spacings and grain boundaries. Various in-situ and ex-situ characterizations including electrochemical quartz crystal microbalance experiments elucidate the Cs and I extraction and Rh reduction during the electrochemical reduction. These Rh nanoparticles from Cs 3 Rh 2 I 9 clusters show significantly enhanced mass and area activity toward hydrogen evolution reaction in both alkaline and chlor-alkali electrolyte, superior to liquid-reduced Rh nanoparticles as well as bulk Cs 3 Rh 2 I 9 -derived Rh via top-down electro-reduction transformation. Theoretical calculations demonstrate water activation could be boosted on Cs 3 Rh 2 I 9 clusters-derived Rh nanoparticles enriched with multiply sites, thus smoothing alkaline hydrogen evolution.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Machine learning-assisted crystal engineering of a zeolite

Abstract It is shown that Machine Learning (ML) algorithms can usefully capture the effect of crystallization composition and conditions (inputs) on key microstructural characteristics (outputs) of faujasite type zeolites (structure types FAU, EMT, and their intergrowths), which are widely used zeolite catalysts and adsorbents. The utility of ML (in particular, Geometric Harmonics) toward learning input-output relationships of interest is demonstrated, and a comparison with Neural Networks and Gaussian Process Regression, as alternative approaches, is provided. Through ML, synthesis conditions were identified to enhance the Si/Al ratio of high purity FAU zeolite to the hitherto highest level (i.e., Si/Al = 3.5) achieved via direct (not seeded), and organic structure-directing-agent-free synthesis from sodium aluminosilicate sols. The analysis of the ML algorithms’ results offers the insight that reduced Na 2 O content is key to formulating FAU materials with high Si/Al ratio. An acid catalyst prepared by partial ion exchange of the high-Si/Al-ratio FAU (Si/Al = 3.5) exhibits improved proton reactivity (as well as specific activity, per unit mass of catalyst) in propane cracking and dehydrogenation compared to the catalyst prepared from the previously reported highest Si/Al ratio (Si/Al = 2.8).

36 MATERIALS SCIENCE↗
DOE OSTI2024

Reconfigurable droplet networks

Droplet networks stabilized by lipid interfacial bilayers or colloidal particles have been extensively investigated in recent years and are of great interest for compartmentalized reactions and biological functions. However, current design strategies are disadvantaged by complex preparations and limited droplet size. Here, by using the assembly and jamming of cucurbit[8]uril surfactants at the oil-water interface, we show a novel means of preparing droplet networks that are multi-responsive, reconfigurable, and internally connected over macroscopic distances. Openings between the droplets enable the exchange of matter, affording a platform for chemical reactions and material synthesis. Our work requires only a manual compression to construct complex patterns of droplet networks, underscoring the simplicity of this strategy and the range of potential applications.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Rare earth metals production using alternative feedstock that eliminates HF

This work reports the successful production of rare earth (RE) metal using Na-RE-F. Presently, RE metals are primarily produced using RE-fluoride due to its higher air and moisture stability compared to RE-chloride. However, its preparation requires the use of corrosive and hazardous chemicals, such as hydrofluoric acid (HF) or ammonium bifluoride (NH 4 HF 2 ). The present study demonstrates that Na-RE-F is an alternative salt to the typically used RE-fluoride. The Na-RE-F for this work is produced via a scalable hydrometallurgical approach using three different RE salts as feedstock, including acetate, nitrate, and chloride. HF is neither used nor generated during the salt preparation process. Furthermore, the Na-RE-F powder dries in air (without dry HF), and only water evolves during the drying process. Analyses of the Na-RE-F show that NaF liberates as a flux during the heating process, which lowers the salt reduction temperature to <900 °C, thus minimizing or eliminating the need for additional flux. Calciothermic reduction of the Na-RE-F salt is successfully employed to obtain RE metal. This work represents a safer, greener, and more widely deployable approach for producing the RE metals needed for permanent magnets which support the transition to a cleaner society through the decarbonization of the transportation industry.

36 MATERIALS SCIENCE↗
DOE OSTI2025

The phase diagram of quantum chromodynamics in one dimension on a quantum computer

The quantum chromodynamics (QCD) phase diagram, which reveals the state of strongly interacting matter at different temperatures and densities, is key to answering open questions in physics, ranging from the behaviour of particles in neutron stars to the conditions of the early universe. However, classical simulations of QCD face significant computational barriers, such as the sign problem at finite matter densities. Quantum computing offers a promising solution to overcome these challenges. Here, we take an important step toward exploring the QCD phase diagram with quantum devices by preparing thermal states in one-dimensional non-Abelian gauge theories. We experimentally simulate the thermal states of SU(2) and SU(3) gauge theories at finite densities on a trapped-ion quantum computer using a variational method. This is achieved by introducing two features: Firstly, we add motional ancillae to the existing qubit register to efficiently prepare thermal probability distributions. Secondly, we introduce charge-singlet measurements to enforce colour-neutrality constraints. This work pioneers the quantum simulation of QCD at finite density and temperature for two and three colours, laying the foundation to explore QCD phenomena on quantum platforms.

Quantum information↗
DOE OSTI2021

Efficient verification of anticoncentrated quantum states

I present a method for estimating the fidelity F(μ, τ) between a preparable quantum state μ and a classically specified pure target state τ=|τ> <τ|, using simple quantum circuits and on-the-fly classical calculation (or lookup) of selected amplitudes of |τ>. The method is sample efficient for anticoncentrated states (including many states that are hard to simulate classically), with approximate cost 4ϵ –2 (1 – F)dpcoll where ϵ is the desired precision of the estimate, d is the dimension of the Hilbert space, and pcoll is the collision probability of the target distribution. Furthermore, this scaling is exponentially better than that of any method based on classical sampling. I also present a more sophisticated version of the method that uses any efficiently preparable and well-characterized quantum state as an importance sampler to further reduce the number of copies of μ needed. Though some challenges remain, this work takes a significant step toward scalable verification of complex states produced by quantum processors.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗