SEARCH · Engineering Papers

Results for “Randomized methods”

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.

SourceResearch connectionFrom yearThrough year

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

At least 235 records · Page 13

All sources NASA NTRS DOE OSTI

DOE OSTI2023

Efficient QAOA Optimization using Directed Restarts and Graph Lookup

Variational Quantum Algorithms (VQA) aim to enhance the capabilities of Noisy Intermediate-Scale Quantum (NISQ) devices. These algorithms utilize parameterized circuits and classical optimizers to iteratively execute circuits with varying parameters. However, VQA faces computational overheads due to repeated iterations and random restarts. Prior work suggests using basic sub-graphs to transfer parameters for the input graph, reducing optimizer overheads but limiting applicability to structured regular graphs. In real-world applications, random irregular graphs are common, and existing methods are not scalable or practical for such graphs. This paper presents a framework that aims to improve random irregular graphs in VQA. The framework uses graph similarity and important features like total edge counts, average edge counts, and variance. It follows an iterative process to choose basis sub-graphs from a small database and adjust parameters accordingly. Classical optimizers then utilize these parameters to determine when to restart and perform gradient descent. This approach increases the chances of reaching global maximum points.

Wang, Meng↗

DOE OSTI2024

Improved Subseasonal Forecasting of Extreme Polar Vortices Using Machine Learning

Our research was focused on forecasting the position and shape of the winter stratospheric polar vortex at a subseasonal timescale of 15 days in advance. To achieve this, we employed both statistical and neural network machine learning techniques. The analysis was performed on 42 winter seasons of reanalysis data provided by NASA giving us a total of 6,342 days of data. The state of the polar vortex for determined by using geometric moments to calculate the centroid latitude and the aspect ratio of an ellipse fit onto the vortex. Timeseries for thirty additional precursors were calculated to help improve the predictive capabilities of the algorithm. Feature importance of these precursors was performed using random forest to measure the predictive importance and the ideal number of precursors. Then, using the precursors identified as important, various statistical methods were tested for predictive accuracy with random forest and nearest neighbor performing the best. An echo state network, a type of recurrent neural network that features sparsely connected hidden layer and a reduced number of trainable parameters that allows for rapid training and testing, was also implemented for the forecasting problem. Hyperparameter tuning was performed for each methods using a subset of the training data. The algorithms were trained and tuned on the first 41 years of data, then tested for accuracy on the final year. In general, the centroid latitude of the polar vortex proved easier to predict than the aspect ratio across all algorithms. Random forest outperformed other statistical forecasting algorithms overall but struggled to predict extreme values. Forecasting from echo state network suggested a strong predictive capability past 15 days, but further work is required to fully realize the potential of recurrent neural network approaches.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2022

Effective Field Theory of Random Quantum Circuits

Quantum circuits have been widely used as a platform to simulate generic quantum many-body systems. In particular, random quantum circuits provide a means to probe universal features of many-body quantum chaos and ergodicity. Some such features have already been experimentally demonstrated in noisy intermediate-scale quantum (NISQ) devices. On the theory side, properties of random quantum circuits have been studied on a case-by-case basis and for certain specific systems, and a hallmark of quantum chaos—universal Wigner–Dyson level statistics—has been derived. This work develops an effective field theory for a large class of random quantum circuits. The theory has the form of a replica sigma model and is similar to the low-energy approach to diffusion in disordered systems. The method is used to explicitly derive the universal random matrix behavior of a large family of random circuits. In particular, we rederive the Wigner–Dyson spectral statistics of the brickwork circuit model by Chan, De Luca, and Chalker [Phys. Rev. X 8, 041019 (2018)] and show within the same calculation that its various permutations and higher-dimensional generalizations preserve the universal level statistics. Finally, we use the replica sigma model framework to rederive the Weingarten calculus, which is a method of evaluating integrals of polynomials of matrix elements with respect to the Haar measure over compact groups and has many applications in the study of quantum circuits. The effective field theory derived here provides both a method to quantitatively characterize the quantum dynamics of random Floquet systems (e.g., calculating operator and entanglement spreading) and a path to understanding the general fundamental mechanism behind quantum chaos and thermalization in these systems.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2023

Posiform planting: generating QUBO instances for benchmarking

We are interested in benchmarking both quantum annealing and classical algorithms for minimizing quadratic unconstrained binary optimization (QUBO) problems. Such problems are NP-hard in general, implying that the exact minima of randomly generated instances are hard to find and thus typically unknown. While brute forcing smaller instances is possible, such instances are typically not interesting due to being too easy for both quantum and classical algorithms. In this contribution, we propose a novel method, called posiform planting , for generating random QUBO instances of arbitrary size with known optimal solutions, and use those instances to benchmark the sampling quality of four D-Wave quantum annealers utilizing different interconnection structures (Chimera, Pegasus, and Zephyr hardware graphs) and the simulated annealing algorithm. Posiform planting differs from many existing methods in two key ways. It ensures the uniqueness of the planted optimal solution, thus avoiding groundstate degeneracy, and it enables the generation of QUBOs that are tailored to a given hardware connectivity structure, provided that the connectivity is not too sparse. Posiform planted QUBOs are a type of 2-SAT boolean satisfiability combinatorial optimization problems. Our experiments demonstrate the capability of the D-Wave quantum annealers to sample the optimal planted solution of combinatorial optimization problems with up to 5, 627 qubits.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

A spatially regularized detector for emergent/re-emergent disease outbreaks

Early detection of outbreaks caused by emergent pathogens, using epidemiological surveillance data i.e., daily case counts, is difficult. This is because the data tend to be noisy during the early epoch of the outbreak. In contrast, the spread-rate of the disease tends to be well-behaved, as it depends only on the mixing patterns of the population and the characteristics of the pathogen, neither of which behave erratically in space-time. In this report, we explore whether the spread-rate can be used for epidemiological surveillance, conditional on case count data. Estimating the spread-rate from case count data allows us to exploit exogenous information, e.g., incubation period distributions etc., which can considerably smooth out any erratic temporal behavior. Further, epidemiological dynamics are spatially correlated, and if case counts are available for multiple areal units e.g., counties, these correlations could potentially be used to suppress noise in the early epoch data. These exogenous information and structure are not exploited by conventional syndromic surveillance detectors to extract a well-behaved latent variable for monitoring purposes. The technical challenge lies in the estimation of the spread-rate field jointly over a collection of areal units; further, the spread-rate varies over time. We develop a method based on mean-field variational inference to approximately estimate the spread-rate field, using a Gaussian Random Field Model for spatial regularization. The method is tested on the estimation of spread-rate in the thirty-three counties of New Mexico and detect the arrival of the Fall 2020 COVID-19 wave in September 2020. We find that the method is scalable, but underestimates the uncertainty in the estimated spread-rate field. We detect the arrival of the Fall 2020 wave a week ahead of conventional syndromic surveillance algorithms, but our simplistic detection algorithm, based on simple anomaly detection, suffers from a high false positive rate, similar to conventional detectors.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2025

Denoising Autoencoder for Reconstructing Sensor Observation Data and Predicting Evapotranspiration: Noisy and Missing Values Repair and Uncertainty Quantification

Abstract Machine learning (ML) methods applied in scientific research often deal with interrelated features in high‐dimensional data. Reducing data noise and redundancy is needed to increase prediction accuracy and efficiency especially when dealing with data from field sensors. We explored an unsupervised learning method, the denoising autoencoder (DAE), to extract the underlying data structure from noisy raw data in the context of predicting hydrologic quantities from multiple field sensors. These sensors have intrinsic instrumental noise and occasional malfunctions that cause missing values. Our DAE neural network reconstructed meteorological sensor data containing noise and missing values to predict evapotranspiration in a mountainous watershed. The DAE reconstructed the sensor variables with a mean coefficient of determination value of 0.77 across 15 dimensions representing individual sensors. It reduced variance and bias uncertainties compared to a classical autoencoder model. The reconstruction quality varied across dimensions depending on their cross‐correlation and alignment with the underlying data structure. Uncertainties arising from the model structure were overall higher than those resulting from data corruption. We attached the DAE structure to a downstream ET‐prediction neural network in three formats and achieved reasonably accurate ET predictions . The use of the DAE notably reduced variance uncertainty in ET prediction. However, excessive variance reduction may be accompanied by an increase in bias due to the intrinsic bias‐variance tradeoff. Our method of evaluating and reducing uncertainties in aggregated data from different sources can be used to improve predictive models, process understanding, and uncertainty quantification for better water resource management. Plain Language Summary We present a machine learning method, namely the denoising autoencoder, which reduces the effects of data noise and missing values typically present in scientific data sets collected through sensor measurements. This method selects the most relevant information from noisy raw data collected by the instruments and fills in missing values. To demonstrate the effectiveness of our method, we applied it to predict evapotranspiration, a hydrologic variable that represents the water moved from the land surface to the atmosphere through a combination of evaporation and plant water use (transpiration). We also used a random sampling technique (the Monte Carlo method) to compare the uncertainty in the predictions when using the raw and noisy data versus the reconstructed data. The denoising process produced more accurate predictions of evapotranspiration with less uncertainty. Improved predictions of evapotranspiration can lead to a better understanding and accounting of water budgets. This ML approach is broadly suitable for a wide variety of applications that involve noisy sensor data with missing values. Key Points We used a denoising autoencoder (DAE) neural network to reduce noise in meteorological and soil sensor observations by on average We used Monte Carlo sampling to estimate the bias and variance of all model outputs, including uncertainty sources from data and the model We attached the DAE component to a downstream neural network to predict ET with the variance reduced by , compared to that without the DAE

denoising autoencoder↗

DOE OSTI2023

Unifying and benchmarking state-of-the-art quantum error mitigation techniques

Error mitigation is an essential component of achieving a practical quantum advantage in the near term, and a number of different approaches have been proposed. In this work, we recognize that many state-of-the-art error mitigation methods share a common feature: they are data-driven, employing classical data obtained from runs of different quantum circuits. For example, Zero-noise extrapolation (ZNE) uses variable noise data and Clifford-data regression (CDR) uses data from near-Clifford circuits. We show that Virtual Distillation (VD) can be viewed in a similar manner by considering classical data produced from different numbers of state preparations. Observing this fact allows us to unify these three methods under a general data-driven error mitigation framework that we call UNIfied Technique for Error mitigation with Data (UNITED). In certain situations, we find that our UNITED method can outperform the individual methods (i.e., the whole is better than the individual parts). Specifically, we employ a realistic noise model obtained from a trapped ion quantum computer to benchmark UNITED, as well as other state-of-the-art methods, in mitigating observables produced from random quantum circuits and the Quantum Alternating Operator Ansatz (QAOA) applied to Max-Cut problems with various numbers of qubits, circuit depths and total numbers of shots. We find that the performance of different techniques depends strongly on shot budgets, with more powerful methods requiring more shots for optimal performance. For our largest considered shot budget (10 10 ), we find that UNITED gives the most accurate mitigation. Hence, our work represents a benchmarking of current error mitigation methods and provides a guide for the regimes when certain methods are most useful.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2025

Informing Plant Asset Reliability and Availability Through AI-Driven Analysis of Operator Logs

The availability and reliability of nuclear power plant (NPP) structures, systems, and components (SSCs) are critical parameters for NPP safety. Tracking these parameters is necessary but costly and labor-intensive, requiring the collection and evaluation of SSC event data such as shutdowns, startups, and failures. To show how these events are needed for the parameters an example is given: one measure of reliability is based on the number of equipment failure events and the number of run hours (i.e., the time from a startup event to a shutdown event). Here, this work investigates using artificial intelligence (AI) to mine NPP operator log entry texts for SSC event data. Four AI approaches were explored for identifying these events, including natural language processing (NLP) methods, generative AI, generative AI combined with NLP, and topic modeling. A key challenge addressed with all four approaches is the brevity of operator log entries. Among these four a neural network–based NLP method was shown to be the most promising for this application, achieving F1 scores of 86.0% for shutdowns, 92.2% for startups, and 80.4% for failures on a subject-matter-expert-curated dataset from NPP operator logs, compared to a baseline of 66.6% for a random classifier. This shows that NLP methods can perform better than generative AI. Additionally, the NLP methods combined with generative AI were shown to perform better than generative AI alone. Generative AI was most successful at providing the background information for the NLP methods to use. This work demonstrates the potential to use AI to automate parameter collection from NPP operator log entries and other records.

97 - MATHEMATICS AND COMPUTING↗

DOE OSTI2021

From Anderson localization on random regular graphs to many-body localization

Highlights: • Anderson transition from ergodicity to localization on random regular graphs (RRG). • Analytical, pool method, and exact-diagonalization study of correlations on RRG. • Many-body localization (MBL) to ergodicity transition: quantum dots and spin chains. • Dynamical eigenstate correlation functions in RRG and MBL problems. • Anderson localization on RRG as a toy model for MBL. The article reviews the physics of Anderson localization on random regular graphs (RRG) and its connections to many-body localization (MBL) in disordered interacting systems. Properties of eigenstate and energy level correlations in delocalized and localized phases, as well at criticality, are discussed. In the many-body part, models with short-range and power-law interactions are considered, as well as the quantum-dot model representing the limit of the “most long-range” interaction. Central themes – which are common to the RRG and MBL problems – include ergodicity of the delocalized phase, localized character of the critical point, strong finite-size effects, and fractal scaling of eigenstate correlations in the localized phase.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Particulate Fuel Modeling of MC 2 -3 using Iterative Local Spatial Self-shielding Method

We report a new spatial self-shielding method for particulate fuels has been developed based on disadvantage factors and implemented in the MC 2 -3 code. This method named the iterative local spatial self-shielding (ILSS) method considers the shadowing effect of randomly distributed particles on spatial self-shielding in particles through a homogenized composition region added outside the particle of interest at the center. The self-shielded cross sections of the central particle are determined iteratively since they are used in determining the cross sections of the homogenized composition region. The ILSS method was verified for infinite stochastic medium problems of single and multiple types of particles, VHTR unit cell problems, and HTTR assembly problems. The verification test results show that the ILSS method accurately predicts the stochastic particle shadowing effect and reaction rates in particles, whereas the regular array model and the stochastic collision probability method underpredict the particle shadowing effect and overestimate reaction rates in particles.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

DOE OSTI2025

Randomized Algorithms for Low-Rank Matrix and Tensor Decompositions

This paper surveys randomized algorithms in numerical linear algebra for low-rank decompositions of matrices and tensors. The survey begins with a review of classical matrix algorithms that can be accelerated by randomized dimensionality reduction, such as the singular value decomposition (SVD) or interpolative (ID) and CUR decompositions. Recent advances in randomized dimensionality reduction are discussed, including new methods of fast matrix sketching and sampling techniques, which are incorporated into classical matrix algorithms for fast low-rank matrix approximations. The extension of randomized matrix algorithms to tensors is then explored for several low-rank tensor decompositions in the CP and Tucker formats, including the higher-order SVD, ID, and CUR decomposition.

Pearce, Katherine J. [The University of Texas at A↗

DOE OSTI2025

Microscopic calculations with noniterative finite amplitude methods and the application to neutron radiative captures and inelastic scatterings

We derive the fully self-consistent quasiparticle random-phase approximation (QRPA) equations with noniterative finite amplitude methods and calculate the transition strengths of giant resonances. Then, we apply the QRPA results to both neutron radiative capture calculations based on the statistical Hauser-Feshbach theory and inelastic scattering calculations based on distorted-wave Born approximation (DWBA). We compare the calculated results with available experimental data and demonstrate how our approach can reproduce giant resonances and various nuclear reactions.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2022

Random strain induced correlations in materials with intertwined nematic and magnetic orders

Electronic nematicity is rarely observed as an isolated instability of a correlated electron system. Instead, in iron pnictides and in certain cuprates and heavy-fermion materials, nematicity is intertwined with an underlying spin-stripe or charge-stripe state. As a result, random strain, ubiquitous in any real crystal, creates both random-field disorder for the nematic degrees of freedom and random-bond disorder for the spin or charge ones. Here, we put forward an Ashkin-Teller model with random Baxter fields to capture the dual role of random strain in nematic systems for which nematicity is a composite order arising from a stripe state. Using Monte Carlo to simulate this random Baxter-field model, we find not only the expected break-up of the system into nematic domains, but also the emergence of nontrivial disorder-promoted magnetic correlations. Such correlations enhance and tie up the fluctuations associated with the two degenerate magnetic stripe states from which nematicity arises, leaving characteristic signatures in the spatial profile of the magnetic domains, in the configurational space of the spin variables, and in the magnetic noise spectrum. We discuss possible experimental manifestations of these effects in iron-pnictide superconductors. Furthermore, our work establishes the random Baxter-field model as a more complete alternative to the random-field Ising model to describe complex electronic nematic phenomena in the presence of disorder.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2022

Proposal and application of ROM-Lasso method for sensitivity coefficient evaluation

We propose a novel method for evaluating sensitivity coefficients of neutronics parameters to cross sections, so-called the reduced-order modeling technique ROM-Lasso. In this method, cross sections of interest are randomly sampled, and corresponding perturbed core analyses are performed. Then, the sensitivity coefficient vector of the higher-level model is expanded via the active subspace bases obtained with the lower-level model whose dimensional complexity is smaller than that of the higher-level model, and the expansion coefficients are estimated by the Lasso regression. A unique feature of the ROM-Lasso method allows the use of different bases optimized for each neutronics parameter. We conducted a verification calculation for an accelerator-driven system and demonstrated that the ROM-Lasso method can reproduce the sensitivity coefficients with a much smaller number of forward calculations than the direct method. The proposed method can be used to practically evaluate sensitivity coefficients. (authors)

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

DOE OSTI2020

Structure–activity relationship-based chemical classification of highly imbalanced Tox21 datasets

Abstract The specificity of toxicant-target biomolecule interactions lends to the very imbalanced nature of many toxicity datasets, causing poor performance in Structure–Activity Relationship (SAR)-based chemical classification. Undersampling and oversampling are representative techniques for handling such an imbalance challenge. However, removing inactive chemical compound instances from the majority class using an undersampling technique can result in information loss, whereas increasing active toxicant instances in the minority class by interpolation tends to introduce artificial minority instances that often cross into the majority class space, giving rise to class overlapping and a higher false prediction rate. In this study, in order to improve the prediction accuracy of imbalanced learning, we employed SMOTEENN, a combination of Synthetic Minority Over-sampling Technique (SMOTE) and Edited Nearest Neighbor (ENN) algorithms, to oversample the minority class by creating synthetic samples, followed by cleaning the mislabeled instances. We chose the highly imbalanced Tox21 dataset, which consisted of 12 in vitro bioassays for > 10,000 chemicals that were distributed unevenly between binary classes. With Random Forest (RF) as the base classifier and bagging as the ensemble strategy, we applied four hybrid learning methods, i.e., RF without imbalance handling (RF), RF with Random Undersampling (RUS), RF with SMOTE (SMO), and RF with SMOTEENN (SMN). The performance of the four learning methods was compared using nine evaluation metrics, among which F 1 score, Matthews correlation coefficient and Brier score provided a more consistent assessment of the overall performance across the 12 datasets. The Friedman’s aligned ranks test and the subsequent Bergmann-Hommel post hoc test showed that SMN significantly outperformed the other three methods. We also found that a strong negative correlation existed between the prediction accuracy and the imbalance ratio (IR), which is defined as the number of inactive compounds divided by the number of active compounds. SMN became less effective when IR exceeded a certain threshold (e.g., > 28). The ability to separate the few active compounds from the vast amounts of inactive ones is of great importance in computational toxicology. This work demonstrates that the performance of SAR-based, imbalanced chemical toxicity classification can be significantly improved through the use of data rebalancing.

Idakwo, Gabriel↗

DOE OSTI2021

Solving inverse problems in stochastic models using deep neural networks and adversarial training

Inverse problems associated with stochastic models constitute a significant portion of scientific and engineering applications. In such cases the unknown quantities are distributions. The applicability of traditional methods is limited because of their demanding assumptions or prohibitive computational consumption; for example, maximum likelihood methods require closed-form density functions, and Markov Chain Monte Carlo needs a large number of simulations. We propose a new method that estimates the unknown distribution by matching the statistical properties between observed and simulated random processes. We leverage the expressive power of neural networks to approximate the unknown distribution and use a discriminative neural network for computing the statistical discrepancies between the observed and simulated random processes. Here we demonstrated numerically that the proposed methods can estimate both the model parameters and learn complicated unknown distributions.

42 ENGINEERING↗

DOE OSTI2020

Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks

One of the open problems in scientific computing is the long-time integration of nonlinear stochastic partial differential equations (SPDEs), especially with arbitrary initial data. We address this problem by taking advantage of recent advances in scientific machine learning and the spectral dynamically orthogonal (DO) and borthogonal (BO) methods for representing stochastic processes. The recently introduced DO/BO methods reduce the SPDE to solving a system of deterministic PDEs and a system of stochastic ordinary differential equations. Specifically, we propose two new physics-informed neural networks (PINNs) for solving time-dependent SPDEs, namely the neural network (NN)-DO/BO methods. The proposed methods incorporate the DO/BO constraints into the loss function (along with the modal decomposition of the SPDE) with an implicit form instead of generating explicit expressions for the temporal derivatives of the DO/BO modes. Hence, the NN-DO/BO methods can overcome some of the drawbacks of the original DO/BO methods. For example, we do not need the assumption that the covariance matrix of the random coefficients is invertible as in the original DO method, and we can remove the assumption of no eigenvalue crossing as in the original BO method. Moreover, the NN-DO/BO methods can be used to solve time-dependent stochastic inverse problems with the same formulation and same computational complexity as for forward problems. Furthermore, we demonstrate the capability of the proposed methods via several numerical examples, namely: (1) A linear stochastic advection equation with deterministic initial condition: we obtain good results with the proposed methods, while the original DO/BO methods cannot be applied directly in this case. (2) Long-time integration of the stochastic Burgers' equation: we show the good performance of NN-DO/BO methods, especially the effectiveness of the NN-BO approach for such problems with many eigenvalue crossings during the whole time evolution, while the original BO method fails. (3) Nonlinear reaction diffusion equation: we consider both the forward problem and the inverse problems, including very noisy initial point values, to investigate the flexibility of the NN-DO/BO methods in handling inverse and mixed type problems. Taken together, these simulation results demonstrate that the NN-DO/BO methods can be employed to effectively quantify uncertainty propagation in a wide range of physical problems, but future work should address the efficiency issue of PINNs for forward problems.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Efficient Unitary Designs from Random Sums and Permutations

A unitary k-design is an ensemble of unitaries that matches the first k moments of the Haar measure. In this work, we provide two efficient constructions of k-designs on n-qubits using new random matrix theory techniques. Our first construction is based on exponentiating sums of random i.i.d. Hermitian matrices and uses O(k2n2)-many gates. In the spirit of central limit theorems, we show that this random sum approximates the Gaussian Unitary Ensemble (GUE). We then show that the product of just two exponentiated GUE matrices is already approximately Haar random. Our second construction is based on products of exponentiated sums of random permutations and uses Õ(k poly (n)) many gates. The k dependence is optimal (up to polylogarithmic factors) and is inherited from the efficiency of existing k-wise independent permutations. Furthermore, replacing random permutations with quantum-secure pseudorandom permutations (PRPs), we also obtain a pseudorandom unitary (PRU) ensemble that is secure under nonadaptive queries. A central feature of both proofs is a new connection between the polynomial method in quantum query complexity and the large-dimension (N) expansion in random matrix theory. In particular, the first construction uses the polynomial method to control high moments of certain random matrix ensembles without requiring delicate Weingarten calculations. In doing so, we define and solve a moment problem on the unit circle, asking whether a finite number of equally weighted points can reproduce a given set of moments. In our second construction, the key step is to exhibit an orthonormal basis for irreducible representations of the partition algebra that has a low-degree large-N expansion. This allows us to show that the distinguishing probability is a low-degree rational polynomial of the dimension N.

algebra↗