Search NASA⌕ Search

SEARCH · Search NASA

Results for “ensemble”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 235 records · Page 13
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

In Situ/Operando Probing of Dynamic Phase Structures of Alumina‐Supported Ultrasmall Copper‐Gold Alloy Nanoparticles Under Reaction Conditions

Abstract The ability to control phase structures and surface sites of ultrasmall alloy nanoparticles under reaction conditions is essential for preparing catalysts by design. This is, however, challenging due to limited understanding of the atomic‐scale phases and their correlation with the ensemble‐averaged structures and activities of catalysts during catalytic reactions. We reveal here a dynamic structural stability of alumina‐supported ultrasmall and equiatomic copper‐gold alloy nanoparticles under reaction conditions as a model system in the in situ/operando study. In situ atomic‐scale morphological tracking under oxygen reveals temperature‐dependent dynamic crystalline‐amorphous dual‐phase structures, showing dynamic stability over an elevated temperature range. This atomic‐scale dynamic phase stability coincides with a “conversion plateau” observed for carbon monoxide oxidation on the catalyst. It is substantiated by the stable lattice ordering/disordering structures and surface sites with oscillatory characteristics shown by operando ensemble‐average structural tracking of the catalyst during the oxidation reaction. The understanding of the atomic‐scale dynamic phase structures in correlation with the ensemble‐average dynamic ordering/disordering phase structures and surface sites provides fresh insights into the unique synergy of the supported alloy nanoparticles. This understanding has implications for the design and structural tuning of active and stable ultrasmall alloy catalysts under elevated temperatures.

Chemistry↗
DOE OSTI2025

Persistent Sampling: Enhancing the Efficiency of Sequential Monte Carlo

Sequential Monte Carlo (SMC) samplers are powerful tools for Bayesian inference but suffer from high computational costs due to their reliance on large particle ensembles for accurate estimates. We introduce persistent sampling (PS), an extension of SMC that systematically retains and reuses particles from all prior iterations to construct a growing, weighted ensemble. By leveraging multiple importance sampling and resampling from a mixture of historical distributions, PS mitigates the need for excessively large particle counts, directly addressing key limitations of SMC such as particle impoverishment and mode collapse. Crucially, PS achieves this without additional likelihood evaluations-weights for persistent particles are computed using cached likelihood values. This framework not only yields more accurate posterior approximations but also produces marginal likelihood estimates with significantly lower variance, enhancing reliability in model comparison. Furthermore, the persistent ensemble enables efficient adaptation of transition kernels by leveraging a larger, decorrelated particle pool. Experiments on high-dimensional Gaussian mixtures, hierarchical models, and non-convex targets demonstrate that PS consistently outperforms standard SMC and related variants, including recycled and waste-free SMC, achieving substantial reductions in mean squared error for posterior expectations and evidence estimates, all at reduced computational cost. PS thus establishes itself as a robust, scalable, and efficient alternative for complex Bayesian inference tasks.

Karamanis, Minas↗
DOE OSTI2024

Computationally efficient and error aware surrogate construction for numerical solutions of subsurface flow through porous media

Limiting the injection rate to restrict the pressure below a threshold at a critical location can be an important goal of simulations that model the subsurface pressure between injection and extraction wells. The pressure is approximated by the solution of Darcy’s partial differential equation for a given permeability field. The subsurface permeability is modeled as a random field since it is known only up to statistical properties. This induces uncertainty in the computed pressure. Solving the partial differential equation for an ensemble of random permeability simulations enables estimating a probability distribution for the pressure at the critical location. These simulations are computationally expensive, and practitioners often need rapid online guidance for real-time pressure management. An ensemble of numerical partial differential equation solutions is used to construct a Gaussian process regression model that can quickly predict the pressure at the critical location as a function of the extraction rate and permeability realization. The Gaussian process surrogate analyzes the ensemble of numerical pressure solutions at the critical location as noisy observations of the true pressure solution, enabling robust inference using the conditional Gaussian process distribution. Our first novel contribution is to identify a sampling methodology for the random environment and matching kernel technology for which fitting the Gaussian process regression model scales as O ( n log n ) instead of the typical O ( n 3 ) rate in the number of samples n used to fit the surrogate. The surrogate model allows almost instantaneous predictions for the pressure at the critical location as a function of the extraction rate and permeability realization. Our second contribution is a novel algorithm to calibrate the uncertainty in the surrogate model to the discrepancy between the true pressure solution of Darcy’s equation and the numerical solution. Finally, although our method is derived for building a surrogate for the solution of Darcy’s equation with a random permeability field, the framework broadly applies to solutions of other partial differential equations with random coefficients.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2025

Boosting efficiency and reducing graph reliance: Basis adaptation integration in Bayesian multi-fidelity networks

The computational cost of high-fidelity numerical models makes outer-loop analysis, which requires repeated interrogation of the model such as uncertainty quantification, computationally demanding. Multi-fidelity methods, which construct a surrogate model using data from an ensemble of models of varying cost and accuracy, can substantially reduce the cost of outer-loop analysis. However, these methods can be difficult to apply when the model ensemble does not admit a clear hierarchy a priori and the correlations between models are low. Consequently, in this paper, we present a multi-fidelity method that leverages dimension reduction to enhance the correlation between models, thereby reducing the amount of data needed to train a surrogate from an unordered ensemble of models. Our method utilizes basis adaptation to build low-dimensional polynomial chaos expansions of each model and employs Multi-fidelity Networks to encode the relationships among models. We show that the resulting method exhibit two notable advantages over its counterpart: (1) enhanced accuracy (both reduced bias and variance); and (2) reduced dependency on the graph structure encoding relationships among models. We demonstrate the approach on an analytical test problem and a challenging finite element model for a spent nuclear fuel. Our method produces a surrogate model that is significantly more accurate than either a single-fidelity surrogate or a multi-fidelity surrogate constructed without basis adaptation.

42 ENGINEERING↗
DOE OSTI2025

Lectures on statistical mechanics

Presented here is a transcription of the lecture notes from Professor Allan N. Kaufman’s graduate statistical mechanics course Physics 212A and 212B at the University of California Berkeley from the 1972–1973 academic year. 212A addressed equilibrium statistical mechanics with topics: fundamentals (micro-canonical and sub-canonical ensembles, adiabatic law and action conservation, fluctuations, pressure, and virial theorem), classical fluids and other systems (equation of state, deviations from ideality, virial coefficients and van der Waals potential, canonical ensemble and partition function, quasistatic evolution, grand-canonical ensemble and partition function, chemical potential, simple model of a phase transition, quantum virial expansion, numerical simulation of equations of state, and phase transition), chemical equilibrium (systems with multiple species and chemical reactions, law of mass action, Saha equation, chemical equilibrium including ionization and excited states), and long-range interactions (including Coulomb, dipole, and gravitational interactions, Debye–Hückel theory, and shielding). 212B addressed nonequilibrium statistical mechanics with topics: fundamentals (definitions: realizations, moments, characteristic function, and discrete variables), Brownian motion (Langevin equation, fluctuation–dissipation theorem, spatial diffusion, Boltzmann’s H-theorem), Liouville and Klimontovich equations, Landau equation (derivation, elaboration, and H-theorem, and irreversibility), Markov processes and Fokker–Planck equation (derivations of the Fokker–Planck equation and a master equation), linear response and transport theory (linear Boltzmann equation, linear response theory of Kubo and Mori, relation of entropy production to electrical conductivity, transport relations and coefficients, normal mode solutions of the transport equations, sketch of a generalized Langevin equation method for transport theory), and an introduction to nonequilibrium quantum statistical mechanics.

plasma dynamics↗
DOE OSTI2025

Determination of Site Occupancy in the M–Pd–Zn (M = Cu, Ag, and Au) γ-Brass Phase by CALculation of PHAse Diagrams Modeling and Rietveld Refinement

The Pd–Zn γ-brass phase provides exciting opportunities for synthesizing site-isolated catalysts with precisely controlled Pd active site ensembles. Introducing a third metallic element into the γ-brass lattice further perturbs the catalytic active site ensembles. Here, in this work, we introduce coinage metallic elements M (M = Cu, Ag, and Au) into the Pd–Zn γ-brass phase and investigate the site occupation factors of each element in the γ-brass lattice. The CALculation of PHAse Diagrams (CALPHAD) modeling approach supported by energetics predicted by the density functional theory and X-ray and neutron diffraction with Rietveld refinement were used to identify the SOF on each Wyckoff site for various M amounts alloyed into the Pd–Zn γ-brass phase. The present analysis unveils the strong preference for Pd occupying the outer tetrahedral (OT) site in the γ-brass lattice, while the coinage metallic elements tend to substitute for Zn on the octahedral (OH) site. The determination of site occupancy in the bulk M–Pd–Zn γ-brass phase provides opportunities to investigate and tailor potential catalytically active site ensembles in the γ-brass phase materials.

36 MATERIALS SCIENCE↗
DOE OSTI2026

Scalable Quantum Monte Carlo Method for Polariton Chemistry via Mixed Block Sparsity and Tensor Hypercontraction Method

We present a reduced-scaling auxiliary-field quantum Monte Carlo (AFQMC) framework designed for large molecular systems and ensembles, with or without coupling to optical cavities. Our approach leverages the natural block sparsity of the Cholesky decomposition (CD) of electron repulsion integrals in molecular ensembles and employs tensor hypercontraction (THC) to efficiently compress low-rank Cholesky blocks. By representing the Cholesky vectors in a mixed format, keeping high-rank blocks in block-sparse form and compressing low-rank blocks with THC, we reduce the scaling of exchange-energy evaluation from quartic to robust cubic in the number of molecular orbitals N, while lowering memory from cubic toward quadratic. Benchmark analyses on one-, two-, and three-dimensional molecular ensembles (up to ∼1,200 orbitals) show that (a) the number of nonzeros in Cholesky tensors grows linearly with system size across dimensions; (b) the average numerical rank increases sublinearly and does not saturate at these sizes; and (c) rank heterogeneity─some blocks nearly full rank and many low rank, naturally motivates the proposed mixed block sparsity and THC scheme for efficient calculation of exchange energy. In conclusion, we demonstrate that the mixed scheme yields cubic wall-time scaling with favorable prefactors and preserves AFQMC accuracy.

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

Using Multiple Isotope-Labeled Infrared Spectra for the Structural Characterization of an Intrinsically Disordered Peptide

Intrinsically disordered proteins (IDPs) rapidly interconvert between conformers, requiring an ensemble description. This complicates their experimental characterization, and force field limitations pose challenges for their simulation. Here, in this work, we use isotope-labeled and unlabeled infrared (IR) spectra to reweight simulated ensembles of the elastin-like peptide GVGVPGVG, a paradigmatic disordered peptide. By comparing the results obtained with different spectra, we explicitly show that the weights are underdetermined by the ensemble averaged data. We identify which labels and frequency regions maximize structural information while minimizing sensitivity to simulation error and show that these regions report on whether the peptide makes specific interactions. Our work shows the importance of incorporating simulations and simulated spectra at the planning stages of isotope-labeled IR experiments and more generally provides a framework for interpreting IR data for IDPs.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2026

Phosphoserine Charge State Drives Ion Condensation and Spatial Polyamine Presentation in Multirepeat Silaffin

Diatom silaffins direct silica biomineralization through heavily post-translationally modified repeat domains, yet how these modifications reshape the multirepeat conformational ensemble remains unknown. We report all-atom MD simulations of a 195-residue construct spanning repeats R1–R7 of Sil1p from Cylindrotheca fusiformis, carrying the full complement of native PTMs: phosphoserine (pSer), long-chain polyamines (LCPA), dimethyllysine (MLY), and trimethylhydroxylysine phosphate (TPL). We simulated three variants (Native, P1/singly deprotonated phosphate, and P2/doubly deprotonated phosphate) at two NaCl concentrations in triplicate for 500 ns each. All systems disorder from the AlphaFold 3 starting structure. Phosphate charge state, not ionic strength, is the dominant control of ensemble compaction and ion organization. Doubly deprotonated phosphate organizes an extensive Na + condensation shell (∼100 ions, 20% of box Na + ) and a heterogeneous bridging network that integrates both pSer and TPL phosphate groups. The resulting ensemble is compact with LCPA side chains exhibiting above-median solvent accessibility in 84% of simulation frames in P2 at 300 mM. This is higher than any other condition we simulated and indicates that polyamine groups are preferentially surface-presented in the most compact, ion-organized state. A charge-neutralization control confirms that this compact state is a structured intermediate maintained by the bridging network, not a simple collapsed globule. Here, this repeat-scale spatial organization is not captured by single-repeat peptide studies. Understanding the dynamics, mechanism, and spatial organization of PTM-rich silaffin at the repeat scale is a step closer to hierarchical biomimetic materials beyond simple silica morphologies.

Amines↗
DOE OSTI2025

Single-Molecule Fluorescence Microscopy Reveals Energy Transfer Active versus Inactive Nanocrystal/Dye Conjugate Pairs

Defect-mediated energy transfer (EnT) is a radiative process that occurs between donor defect states in the forbidden bandgap of semiconductor nanocrystals (NCs) and dye molecules bound to their surfaces. The EnT efficiency depends on the number of dye molecules attached to each NC, the donor–acceptor distance, and the dipole orientation factor between the donor and acceptor, all of which vary across all individual NCs in a sample. While ensemble-level fluorescence spectroscopy measurements have provided average values for donor–acceptor distances, dye-to-NC ratios, and EnT rate constants, questions remain about the impact of donor/acceptor heterogeneity on observed EnT efficiencies. Notably, ensemble-level measurements cannot distinguish between bare NCs and EnT-active versus inactive NC/dye pairs in the same sample batch, limiting the ability to design systems with 100% EnT efficiency. To address this, we studied defect-mediated EnT between AlexaFluor 555 dye acceptors chemically bound to ZnO NC donors at the level of single molecules and single NCs. Interestingly, 20% of bound NC/dye pairs are EnT-inactive, likely contributing to residual defect photoluminescence (PL) observed in ensemble-level measurements and reducing overall EnT efficiency. Single particle-level ZnO defect PL and acceptor fluorescence trajectories exhibited distinct microfluctuations, which are absent in bare ZnO NCs. We hypothesized that our observations can be explained with a competitive dye fluorescence quenching pathway, possibly due to charge transfer between the excited state dye and the ZnO NC. Numerical simulations of single-molecule PL traces for this scenario produced microfluctuations consistent with the experimental results. These findings highlight the impact of sample heterogeneity on EnT processes and provide insights for designing light-harvesting systems with optimized EnT efficiency.

25 ENERGY STORAGE↗
DOE OSTI2025

Uncertainty quantification for neural network potential foundation models

Abstract For neural network potentials (NNPs) to gain widespread use, researchers must be able to trust model outputs. However, the blackbox nature of neural networks and their inherent stochasticity are often deterrents, especially for foundation models trained over broad swaths of chemical space. Uncertainty information provided at the time of prediction can help reduce aversion to NNPs. In this work, we detail two uncertainty quantification (UQ) methods. Readout ensembling, by finetuning the readout layers of an ensemble of foundation models, provides information about model uncertainty, while quantile regression, by replacing point predictions with distributional predictions, provides information about uncertainty within the underlying training data. We demonstrate our approach with the MACE-MP-0 model, applying UQ to the foundation model and a series of finetuned models. The uncertainties produced by the readout ensemble and quantile methods are demonstrated to be distinct measures by which the quality of the NNP output can be judged.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Distribution of centrality measures on undirected random networks via the cavity method

The Katz centrality of a node in a complex network is a measure of the node’s importance as far as the flow of information across the network is concerned. For ensembles of locally tree-like undirected random graphs, this observable is a random variable. Its full probability distribution is of interest but difficult to handle analytically because of its “global” character and its definition in terms of a matrix inverse. Leveraging a fast Gaussian Belief Propagation-Cavity algorithm to solve linear systems on tree-like structures, we show that i) the Katz centrality of a single instance can be computed recursively in a very fast way, and ii) the probability P ( K ) that a random node in the ensemble of undirected random graphs has centrality K satisfies a set of recursive distributional equations, which can be analytically characterized and efficiently solved using a population dynamics algorithm. We test our solution on ensembles of Erdős-Rényi and Scale Free networks in the locally tree-like regime, with excellent agreement. The analytical distribution of centrality for the configuration model conditioned on the degree of each node can be employed as a benchmark to identify nodes of empirical networks with over- and underexpressed centrality relative to a null baseline. We also provide an approximate formula based on a rank- 1 projection that works well if the network is not too sparse, and we argue that an extension of our method could be efficiently extended to tackle analytical distributions of other centrality measures such as PageRank for directed networks in a transparent and user-friendly way.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Engineering long-lived entanglement through dissipation in quantum hybrid solid-state platforms

Spin squeezing, a form of many-body entanglement, is a crucial resource in quantum metrology and information processing. While experimentally viable protocols for generating stable spin squeezing have been proposed in quantum optics setups, there is growing interest in quantum hybrid solid-state systems as alternative platforms for both engineering and exploring many-body quantum phenomena. In this work, we propose a scheme to generate long-lived spin squeezing in an ensemble of solid-state qubits interacting with electromagnetic noise emitted by a squeezed solid-state bath. We identify the conditions under which quantum correlations within the bath can be transferred to the qubit array, driving it into an entangled state independently of its initial configuration. To assess the experimental feasibility of our approach, we analyze the dynamics of an array of solid-state spin defects coupled to a common ferromagnetic bath, which is driven into a non-equilibrium squeezed state through its interaction with a surface acoustic wave mode. Our results demonstrate that the ensemble can exhibit steady-state spin squeezing under suitable conditions, opening new pathways for the generation of robust many-body entanglement in solid-state spin ensembles.

NV centers↗
DOE OSTI2024

Multi-scale impacts of climate change on hydropower for long-term water-energy planning in the contiguous United States

Climate change impacts on watersheds can potentially exacerbate water scarcity issues where water serves multiple purposes including hydropower. The long-term management of water and energy resources is still mostly approached in a siloed manner at different basins or watersheds, failing to consider the potential impacts that may concurrently affect many regions at once. There is a need for a large-scale hydropower modeling framework that can examine climate impacts across adjoining river basins and balancing authorities (BAs) and provide a periodic assessment at regional to national scales. Expanding from our prior assessment only for the United States (US) federal hydropower plants, we enhance and extend two regional hydropower models to cover over 85% of the total hydropower nameplate capacity and present the first contiguous US-wide assessment of future hydropower production under Coupled Model Intercomparison Project phase 6’s high-end Shared Socioeconomic Pathway 5-8.5 emission scenario using an uncertainty-aware multi-model ensemble approach. We present regional hydropower projections, using both BA regions and US Hydrologic Subregions (HUC4s), to consistently inform the energy and water communities for two future periods—the near-term (2020–2039) and the mid-term (2040–2059) relative to a historical baseline period (1980–2019). We find that the median projected changes in annual hydropower generation are typically positive—approximately 5% in the near-term, and 10% in the mid-term. However, since the risk of regional droughts is also projected to increase, future planning cannot overly rely on the ensemble median, as the potential of severe hydropower reductions could be overlooked. The assessment offers an ensemble of future hydropower generation projections, providing regional utilities and power system operators with consistent data to develop drought scenarios, design long duration storage and evaluate energy infrastructure reliability under intensified inter-annual and seasonal variability.

13 HYDRO ENERGY↗
DOE OSTI2025

Evaluation of historical precipitation interannual variability in CMIP6 over the United States

Interannual precipitation variability profoundly influences society via its effects on agriculture, water resources, infrastructure, and disaster risks. In this study, we use daily in situ precipitation observations from the global historical climatology network-daily (GHCN-D) to assess the ability of 21 Coupled Model Intercomparison Project Phase 6 (CMIP6) models, including the 50-member fifth-generation Canadian Earth System Model single model initial-condition large ensemble (CanESM5_SMILE), to realistically simulate historical interannual precipitation variability trends within 17 regions of the contiguous United States (CONUS). We assess how accurately the CMIP6 simulations align with observational data across annual, summer, and winter periods, focusing on four key hydrometeorological metrics, including interannual precipitation variability, relative interannual precipitation variability (coefficient of variation), annual mean precipitation, and annual wet day frequency. Our findings reveal that CMIP6 ensemble members generally reproduce the spatial patterns of observed trends in annual mean precipitation. In most regions, models agree well with the signs of observed changes in annual mean precipitation, though discrepancies in trend magnitude are evident. Further, observed trends in winter mean precipitation broadly exhibit a spatial pattern similar to that of the observed annual mean. However, analysis of the CanESM5_SMILE shows that trends in precipitation variability may primarily be the result of model-simulated internal variability, suggesting caution in interpreting multi-model single-realization ensemble results. Challenges in accurately simulating interannual precipitation variability underscore the need for ongoing model refinement and validation to enhance climate projections, especially in regions vulnerable to extreme precipitation events.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2026

Monitored Fluctuating Hydrodynamics

We introduce a hydrodynamic framework for describing monitored classical stochastic processes. We study the conditional ensembles for these monitored processes—i.e., we compute spacetime correlation functions conditioned on a fixed, typical measurement record. In the presence of global symmetries we show that these conditional ensembles can undergo measurement-induced “sharpening” phase transitions as a function of the monitoring rate; moreover, even weak monitoring can give rise to novel critical phases, derived entirely from a classical perspective. We give a simple hydrodynamic derivation of the known “charge-fuzzy phase” for weakly monitored diffusive many-body quantum systems. We show that although the unmonitored symmetric and asymmetric exclusion processes are in different universality classes of transport, the fluctuations in their conditional ensembles flow to the same fixed point with emergent relativistic invariance under monitoring. On the other hand, weakly monitored systems with non-Abelian symmetries enter a novel strongly coupled fixed point with nontrivial dynamical exponent, which we characterize. Our formalism naturally accounts for monitoring general observables, such as currents or density gradients, and allows for a direct calculation of information-theoretic diagnostics of sharpening transitions, including the Shannon entropy of the measurement record.

classical statistical mechanics↗
DOE OSTI2026

Coherence-Induced Deep Thermalization Transition in Random Permutation Quantum Dynamics

We report a phase transition in the projected ensemble—the collection of postmeasurement wave functions of a local subsystem obtained by measuring its complement. The transition emerges in systems undergoing random permutation dynamics, a type of quantum time evolution wherein computational basis states are shuffled without creating superpositions. It separates a phase exhibiting deep thermalization, where the projected ensemble is distributed over Hilbert space in a maximally entropic fashion (Haar random), from a phase where it is minimally entropic (“classical bit-string ensemble”). Crucially, this deep thermalization transition is invisible to the subsystem’s density matrix, which always exhibits thermalization to infinite temperature across the phase diagram. Through a combination of analytical arguments and numerical simulations, we show that the transition is tuned by the total amount of injected by the input state and the measurement basis, and is exhibited robustly across different microscopic models. Our findings represent a novel form of ergodicity-breaking universality in quantum many-body dynamics, characterized not by a failure of regular thermalization, but rather by a failure of deep thermalization.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Optimal Twirling Depth for Classical Shadows in the Presence of Noise

The classical shadows protocol is an efficient strategy for estimating properties of an unknown state p using a small number of state copies and measurements. In its original form, it involves twirling the state with unitaries from some ensemble and measuring the twirled state in a fixed basis. It was recently shown that for computing local properties, optimal sample complexity (copies of the state required) is remarkably achieved for unitaries drawn from shallow depth circuits composed of local entangling gates, as opposed to purely local (zero depth) or global twirling (infinite depth) ensembles. Here, we consider the sample complexity as a function of the depth of the circuit, in the presence of noise. We find that this noise has important implications for determining the optimal twirling ensemble. Under fairly general conditions, we (i) show that any single-site noise can be accounted for using a depolarizing noise channel with an appropriate damping parameter f, (ii) compute thresholds f th at which optimal twirling reduces to local twirling for Pauli operators, (iii) nth order Renyi entropies (n ≥2), and (iv) provide a meaningful upper bound t max on the optimal circuit depth for any finite noise strength f, which applies to observables and entanglement entropy measurements. In conclusion, these thresholds strongly constrain the search for optimal strategies to implement shadow tomography and are easily tailored to the experimental system at hand.

97 MATHEMATICS AND COMPUTING↗