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At least 235 records · Page 13

Effect of disorder on thermodynamic instability of binary Rare-earth – Nickel – Palladium compounds

In this work, we have investigated the thermodynamic stability of disordered rare-earth phases SmX 2 and Sm 10 X 21 (X=Ni, Pd) using machine-learning based analytical descriptor and first-principles density functional theory methods. The absence of Laves phase compounds in R-Pd binary systems is a longstanding problem of rare earth science: even though Ni and Pd belong to the same group of the periodic table and have similar electronic structure, the Pd compound crystallizes in a monoclinic (C2/m) phase with 10:21 stoichiometry, i.e., Sm 10 Pd 21 , while the Ni compound adopts a cubic Laves phase (MgCu 2 ) structure. To understand this contrasting phase stability, we performed thermodynamic convex hull analysis of Sm x Ni 1-x and Sm x Pd 1-x binary systems, which is experimentally validated using powder X-ray diffraction (PXRD) analyzes of polycrystalline Sm(Ni x Pd 1-x ) 2 samples with x=0, 0.5, and 1. A detailed electronic-structure (band-structure, charge density, and Fermi-surface) analysis of the differences between SmNi 2 /SmPd 2 and Sm 10 Ni 21 /Sm 10 Pd 21 compounds provides the quantum mechanical origin of the unfavorable mixing of Pd with Ni in cubic Laves phase. We show that the stability of Sm-Pd in 10:21 stoichiometry arises from improved intra-/inter-layer 5d-4d bonding compared to the 1:2 stoichiometry. Our work emphasizes the importance of ab-initio methods and computationally inexpensive analytical descriptors for the detailed analysis of thermodynamic and electronic properties of hard-to-prepare rare-earth compounds.

36 MATERIALS SCIENCE↗

SPACE: 3D parallel solvers for Vlasov-Maxwell and Vlasov-Poisson equations for relativistic plasmas with atomic transformations

A parallel, relativistic, three-dimensional particle-in-cell code SPACE has been developed for the simulation of electromagnetic fields, relativistic particle beams, and plasmas. In addition to the standard second-order Particle-in-Cell (PIC) algorithm, SPACE includes efficient novel algorithms to resolve atomic physics processes such as multi-level ionization of plasma atoms, recombination, and electron attachment to dopants in dense neutral gases. SPACE also contains a highly adaptive particle-based method, called Adaptive Particle-in-Cloud (AP-Cloud), for solving the Vlasov-Poisson problems. It eliminates the traditional Cartesian mesh of PIC and replaces it with an adaptive octree data structure. The code's algorithms, structure, capabilities, parallelization strategy, and performance have been discussed. Additionally, typical examples of SPACE applications to accelerator science and engineering problems are described.

43 PARTICLE ACCELERATORS↗

In-Silico Analysis of High Refractive Index Materials Through Principles of Materials Design

The intent of the paper is to use specific principles of Materials Design that were developed and applied in the electronics industry for enabling understanding and design of improved high refractive index materials. Further, by combining first-principle based ab-initio, semiempirical interatomic potential methods, and machine learning approaches in conjunction with experimental data, we identified specific determinants of high refractive index materials, which can be critically applied for informing materials design and accelerating discovery. Specifically, it was demonstrated that chalcogenides and perovskites as bulk materials can exhibit higher refractive indices with appropriate engineering of specific aspects of the materials.

36 MATERIALS SCIENCE↗

Shape Optimization for Control and Isolation of Structural Vibrations in Aerospace and Defense Applications

Among the main challenges in shape optimization is the coupling of Finite Element Method (FEM) codes in a way that facilitates efficient computation of shape derivatives. This is particularly difficult with multi-physics problems involving legacy codes, where the costs of implementing and maintaining shape derivative capabilities are prohibitive. There are two mathematically equivalent approaches to computing the shape derivative: the volume method, and the boundary method. Each has a major drawback: the boundary method is less accurate, while the volume method is more invasive to the FEM code. Prior implementations of shape derivatives at Sandia have been based on the volume method. We introduce the strip method, which computes shape derivatives on a strip adjacent to the boundary. The strip method makes code coupling simple. Like the boundary method, it queries the state and adjoint solutions at quadrature nodes, but requires no knowledge of the FEM code implementations. At the same time, it exhibits the higher accuracy of the volume method. The development of the strip method also offers us the opportunity to share some lessons learned about implementing the volume method and boundary method, to show shape optimization results on problems of interest, and to begin addressing the other main challenges at hand: constraints on optimized shapes, and their interplay with optimization algorithms.

97 MATHEMATICS AND COMPUTING↗

A Clustering-based biased Monte Carlo Approach to Protein Titration Curve Prediction

We develop and implement a novel approach to computing the ensemble averages in systems characterized by pair-wise interactions between the entities. Methods involving full enumeration of the configuration space result in exponential complexity. Sampling methods such as Markov Chain Monte Carlo (MCMC) algorithms have been proposed to tackle the exponential complexity of these problems. In certain scenarios where significant energetic coupling exists between the entities, the accuracy of the such algorithms can be diminished. We propose a strategy to improve the accuracy of the MCMC runs by taking advantage of the cluster structure in the interaction energy matrix. We propose two different schemes for performing the biased MCMC runs on the partitioned systems and show that they are valid MCMC schemes. We then apply these algorithms to the problem of computing the protonation fractions and hence the titration curves of titratable protein residues that constitute a given protein. We leverage both synthesized and real-world systems and show the improved performance of our biased MCMC methods when compared to the regular MCMC method.

Visweswara Sathanur, Arun↗

Randomized Preconditioned Solvers for Strong Constraint 4D-Var Data Assimilation

The Strong Constraint 4D Variational (SC-4DVAR) data assimilation method is widely used in climate and weather applications. SC-4DVAR involves solving a minimization problem to compute the maximum a posteriori estimate, which we tackle using the Gauss-Newton method. The computation of the descent direction is expensive since it involves the solution of a large-scale and potentially ill-conditioned linear system, solved using the preconditioned conjugate gradient (PCG) method. Here, to address this cost, we efficiently construct scalable preconditioners using three different randomization techniques, which all rely on a certain low-rank structure involving the Gauss-Newton Hessian. The proposed techniques come with theoretical guarantees on the condition number, and at the same time, are amenable to parallelization. We also develop an adaptive approach to estimate the sketch size and choose between the reuse or recomputation of the preconditioner. We demonstrate the performance and effectiveness of our methodology on two representative model problems—the Burgers and barotropic vorticity equation—showing a drastic reduction in both the number of PCG iterations and the number of Gauss-Newton Hessian products after including the preconditioner construction cost.

Gauss-Newton↗

A Flexible Quasi-Static Mooring Design Optimization Method for Floating Structures

This paper presents a flexible and efficient design method for optimizing the mooring systems of floating structures. Mooring system optimization is challenging because of the strong nonlinearity of mooring system behavior and the many technical constraints that must be satisfied. Furthermore, different mooring configurations can have very different design spaces. While some successful examples of mooring design optimization exist in the literature, developing an optimization approach that can work across various mooring design problems is a larger challenge. We present such a method based on a flexible parameterization that allows a wide variety of mooring designs to be described by a list of variables, a quasi-static mooring model that provides efficient evaluation of a mooring design without directly considering mooring system dynamics, and an optimization framework that generates, evaluates, and adjusts the mooring design while considering user-specified constraints such as offset limits, strength safety factors, and seabed contact limits. We demonstrate the design optimization framework on four mooring design problems, each for a different type of mooring system. We compare the use of different design modes to simplify the optimization problem, showing that they can reduce the computation time by up to 75%. We also compare different optimization algorithms and find that the resulting computational speed can vary by up to 51 times. We perform a sensitivity study on one design and find that the local sensitivity of anchoring radius to water depth has a positive correlation of 0.29, but the global sensitivity shows large nonlinearities. Lastly, we perform a coupled dynamic analysis on one of the optimized designs and find that the predicted mean platform motions and mooring line tensions are within 1% of dynamic results and the extreme motions and tensions are within 14%. Lastly, we show that a DEA-Chain-Polyester mooring configuration is cost-optimal for the given design problem of the demonstrations, which aligns with general industry practice.

16 TIDAL AND WAVE POWER↗

Efficient Neural Network Approaches for Conditional Optimal Transport with Applications in Bayesian Inference

In this work, we present two neural network approaches that approximate the solutions of static and dynamic conditional optimal transport (COT) problems. Both approaches enable conditional sampling and conditional density estimation, which are core tasks in Bayesian inference—particularly in the simulation-based (“likelihood-free”) setting. Our methods represent the target conditional distribution as a transformation of a tractable reference distribution. Obtaining such a transformation, chosen here to be an approximation of the COT map, is computationally challenging even in moderate dimensions. To improve scalability, our numerical algorithms use neural networks to parameterize candidate maps and further exploit the structure of the COT problem. Our static approach approximates the map as the gradient of a partially input convex neural network. It uses a novel numerical implementation to increase computational efficiency compared to state-of-the-art alternatives. Our dynamic approach approximates the conditional optimal transport via the flow map of a regularized neural ODE; compared to the static approach, it is slower to train but offers more modeling choices and can lead to faster sampling. We demonstrate both algorithms numerically, comparing them with competing state-of-the-art approaches, using benchmark datasets and simulation-based Bayesian inverse problems.

97 MATHEMATICS AND COMPUTING↗

Implementation of the Windowed Multipole Method in Shift

The windowed multipole (WMP) method has been implemented in the Shift Monte Carlo (MC) radiation transport code with support for both CPU and GPU execution. With this method, small WMP data libraries (~100 MB) can be used to accurately Doppler broaden cross sections to arbitrary temperatures “on the fly” during an MC simulation. This approach yields significant memory savings relative to traditional methods, making it ideal for high-fidelity analysis such as coupled multiphysics simulations. This document provides the exact forms of the WMP equations used by Shift, as well as a detailed description of the structure of WMP HDF5 data files provided by the Massachusetts Institute of Technology (MIT). The Shift implementation has been validated against the OpenMC radiation transport code, with excellent agreement demonstrated for 70 nuclides across an operative range of temperatures. CPU and GPU performance testing using a small module reactor (SMR) problem demonstrated that this method decreases the neutron tracking rate by a factor of ~2 on the Summitdev machine. A new set of WMP data being developed in-house will employ novel methods to improve tracking rates.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Foundations of Molecular 'Isotomics'

The naturally occurring rare isotopes are versions of common elements, such as hydrogen, carbon and oxygen, that contain a larger than usual number of neutrons in their atomic nuclei and therefore are higher in mass than the common atoms of that element. Isotopes exist for most elements and are found in most natural and synthetic materials, but are uneven in their distribution because chemical and physical processes are isotope-selective (e.g., a chemical reaction may proceed more rapidly for one isotope than for another). For this reason, abundances of isotopes in a material of interest can provide a record, or ‘signature’ of various features of that material’s origin and history. These signatures have been used in the geo, life, chemical and physical sciences in a wide variety of ways over close to 8 decades. However, many such applications struggle to reach unique interpretations of isotopic data because multiple factors combine to control a given sample’s overall isotopic content. That is, the factors controlling isotopic content are too numerous and complex to fully constrain from a simple measurement of a material’s isotope abundances. However, the distribution of isotopes within materials, at molecular scales potentially provides a vastly larger number and diversity of constraints on the chemical and physical processes that comprise a material’s history. The rare isotopes may be concentrated into one atomic position in a molecule relative to another, some proportion of molecules in a sample may contain two or more rare isotopes, and those multiply-isotope-substituted forms of molecules may also have uneven distributions of those isotopes across individual atomic sites. For these reasons, even small, seemingly simple molecules, such as sugars, amino acids or drug compounds, actually exist in a vast number of isotopically unique forms (often millions or more), and each one of those forms is in some sense an independent ‘vote’ on that sample’s history. This project has focused on opening this rich archive of information by enabling the creation of routinely and widely applicable ways of measuring and interpreting isotopic structures of molecules. This work has included the development of core technologies and analytical methods, advancing fundamental understanding of the physical and chemical properties of isotopic versions of molecules, and conducting proof of concept studies of illustrative geochemical, cosmochemical and forensic problems in order to show how these technologies, methods and principles come together to solve problems in new ways. A key to the success of this project was the adaptation of ‘Fourier transform mass spectrometry’ (FTMS) to the task of precisely measuring proportions of the rare, naturally occurring isotopic forms of molecules. FTMS is a highly specialized form of mass spectrometry that traps ions within magnetic or electrostatic cavities and, effectively, ‘listens’ (through registering of subtle electrical signals) to the harmonic signals they make while rapidly orbiting within those cavities. These signals have periods that are a function of their mass and strength (or ‘loudness’) that is proportional to their abundances. Thus, these signals constrain relative amounts of molecules that differ in their mass due to various isotopic substitutions. This technology has been essential to the identification of organic molecules in the life, chemical and environmental sciences for over 4 decades, but generally has lacked the control, stability and precision to meaningfully measure rare isotope forms of molecules. This project’s most fundamental contribution has been to modify FTMS, both in terms of hardware and methods, to enable such measurements. The raw data of molecular isotopic structure is tremendously voluminous and complex, so another important activity of this project has been developing the theoretical and data-science tools needed to interpret the data generated by this new form of isotopic measurement. A particularly challenging part of this task has been predicting molecular isotopic structure, as only through the comparison of measurements with predictions can we make progress on hypothesis driven research questions. We have attacked this this prediction task through a combination of first-principles chemical-physics models of the effects of isotope substitution on molecule properties and data-science models that permit us to generalize that chemical physics to cases that have not yet been studied by detailed chemical physics theory. The proof of concept applications we have pursued over the course of this study include biological reactions of amino acids and other biomolecules, non-biological synthesis of organic molecules in extra-terrestrial settings such as meteorites, petroleum geoscience questions concerning the origin and evolution of natural gas, oil and kerogen compounds, and forensic questions such as the sourcing of chemical weapons. The successes of these applications have laid the groundwork for the next phase of this field’s development, which will include larger scale and more ambitious studies of molecular isotopic structure as a means of diagnosing human diseases, such as cancer, and reconstructing detailed interpretations of the origin and evolution of organic molecules in modern and geological environments.

Cesar, Jaime↗

Detector alignment for X-ray crystallography using Millepede-II

I describe a method for accurately refining the geometrical parameters of segmented X-ray area detectors on the basis of serial crystallography data, using 'Millepede' – an algorithm created for a very similar problem in high-energy physics. The Millepede method for serial crystallography builds on the approach of Brewster et al. [Acta Cryst. (2018), D74, 877–894], in which the detector parameters are refined simultaneously with the parameters for each individual crystal. This accounts for the mutual dependency between the parameters and thereby avoids the bias and slow convergence problems that have afflicted older approaches in which the deviations between observed and calculated Bragg peak positions were taken directly as the updates for the detector panel positions. The Millepede method uses the special structure of the least-squares normal equations to reduce them to a much smaller form that can be solved very quickly, even compared with the sparse matrix methods used previously. This makes it practical to refine the detector geometry frequently and thereby maintain accurate calibration without specialized alignment campaigns. Tilts of detector panels out of the plane can be reliably refined, as can the overall distance of the detector in the beam direction. With a simulated test case, the new method produced panel shifts within 7% of the correct values with only one iteration, and produced almost exactly correct shifts after a second iteration. A simulated out-of-plane panel rotation was correctly determined to within 0.001°. Applied to experimental data from an X-ray free-electron laser, the method increased the indexable fraction of frames from 30% to 91% in a single iteration, and to 96% after two further iterations. Computing the geometry updates on the basis of 2060 crystals took only 0.819 s on desktop computing hardware, including the time taken to read the required data from disk. The scaling was found to be very close to linear for up to 100 980 sets of crystal parameters, which took only 78.2 s to process under the same conditions. The method has been applied as part of a real-time feedback system at a synchrotron radiation beamline, in which an out-of-plane detector tilt of 0.04° was detected and corrected. Possible further applications are also described here.

Millepede-II↗

Improving inference with matrix elements and machine learning

Particle physics processes bring together a high-energy amplitude described by quantum field theory and nonperturbative effects and detector interactions described by complex computer simulations. We review some recently developed multivariate inference techniques that leverage this structure and combine matrix-element information with machine learning. Automated by the MadMiner package, the new techniques have been applied to multiple problems in particle physics, allowing for stronger limits than traditional analysis methods and showing their potential to improve the sensitivity of the LHC legacy measurements.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Machine learning models for PDE constrained optimization

Partial differential equation (PDE)-constrained optimization problems arise in a variety of scientific and engineering applications, such as topology optimization, electrodynamics, fluid dynamics, and structural dynamics. However, these problems are often challenging and computationally expensive to solve, due to the need to solve the PDEs within the optimization loop. One approach to reducing the computational cost of these methods while providing convergence guarantees is through inexact trust region methods; this method uses lower fidelity solutions of the PDE at early stages of the optimization and adjusts the required accuracy of inexact PDE solvers as the optimization progresses. In this work, we explore the use of machine learning based surrogate models with these inexact trust region methods. We first demonstrate the potential of this approach by using Gaussian processes as the surrogate model and test this on a simple PDE-constrained optimization problem. We then document explorations into improving the computational costs of evolutional deep neural network / neural Galerkin methods, with the eventual goal of using these methods with the inexact trust region algorithms. We are able to speed up these approaches, albeit at the cost of lower accuracy.

97 MATHEMATICS AND COMPUTING↗

Stress-constrained topology optimization of lattice-like structures using component-wise reduced order models

We report lattice-like structures can provide a combination of high stiffness with light weight that is useful in many applications, but a resolved finite element mesh of such structures results in a computationally expensive discretization. This computational expense may be particularly burdensome in many-query applications, such as optimization. We develop a stress-constrained topology optimization method for lattice-like structures that uses component-wise reduced order models as a cheap surrogate, providing accurate computation of stress fields while greatly reducing run time relative to a full order model. We demonstrate the ability of our method to produce large reductions in mass while respecting a constraint on the maximum stress in a pair of test problems. The ROM methodology provides a speedup of about 150x in forward solves compared to full order static condensation and provides a relative error of less than 5% in the relaxed stress.

97 MATHEMATICS AND COMPUTING↗

Atomic-Scale Structural Mapping of Active Sites in Monolayer PGM-Free Catalysts by Low-Voltage 4D-STEM

Two-dimensional (2D) materials have attracted a large amount of attention in both basic and applied fields, and scanning transmission electron microscopy (STEM) is often uniquely well-suited for characterizing the atomic-scale structure of these materials [1-4]. As a result, STEM is poised to significantly impact progress on platinum group metal (PGM)-free catalysts, which are currently under intense development to enable low-cost, commercially viable hydrogen fuel cells [5]. While recent advancements have resulted in fuel cell performance comparable to Pt catalysts by some measures [6], cell durability remains a significant challenge, limiting practical applications [7]. Catalytically active sites in PGM-free materials are proposed to be FeN4 complexes embedded in a graphene lattice (Fig. 1b) within layered or other larger materials, but this is still under debate largely due to the range of potential actives sites predicted by computational methods and lack of methods for directly validating these models [5]. Fundamental insights into the atomic structure and resulting degradation pathways of proposed active sites are therefore needed to fully understand and control cell performance and durability [6].2D materials typically make ideal samples for STEM, but those within PGM-free catalysts present additional challenges since these materials are often defect-rich, with a high density of edges, dopant atoms, etc., which significantly increase susceptibility to beam damage at standard operating voltages. This makes analysis of potential FeN4 active sites particularly challenging, since a large proportion of Fe exists at edge sites where beam-induced atomic displacements can prohibit high-resolution structural characterization [6]. Conventional dark-field imaging compounds this problem by producing less signal for a given dose and being less sensitive to light elements than dose-efficient phase contrast imaging techniques such as those enabled by four-dimensional (4D)-STEM [8-10] (Fig. 1a). Consequently, active site structural analysis is often left to methods such as low-resolution imaging combined with quantum chemical calculations [6], which hinders accurate determination of reaction and degradation mechanisms.Here, we demonstrate direct atomic-scale structural mapping of FeN4 sites by performing low-voltage 4D-STEM on a model PGM-free catalyst system with many exposed monolayer regions. To accomplish this, we pair a 30 keV aberration-corrected probe with a fast pixelated detector that has optimal performance at low beam voltages [11]. This enables us to simultaneously image light and heavy elements with high signal-to-noise by center-of-mass analysis (Fig. 1c) while minimizing beam-induced atomic displacements at sensitive sites. The monolayer nature of these materials additionally allows for experimental validation by direct comparison with multislice simulations [12] of model structures (Fig. 1d-e). This work demonstrates how low-voltage 4D-STEM will provide new insights into the atomic-scale structure and degradation mechanisms of active sites in PGM-free catalysts, facilitating the development of low-cost hydrogen fuel cells and other energy conversion technologies in the future [13].

Zachman, Michael↗

Massive all-atom analysis of 2D materials with quantum properties (Final report)

Improvements in microscopy have enabled the acquisition of data at a scale that is difficult to process manually, making automated machine learning approaches to analyzing experimental images essential. In this project, we developed and applied machine learning (ML) workflows for atomic resolution scanning transmission electron microscopy (STEM) images. This development included improving both methodology as well as generating user-friendly codes. We developed machine learning architectures which, after training, automatically identify the location and types of defects throughout a material. We used these data to produce class-averaged images of 2D atomic coordinates with up to 0.3 pm precision, uncovering the structure and oscillations of long-range strain fields around point defects in WSe 2-2x Te 2x . We also resolved a long-standing problem in this field in the training of ML models, a lack of labeled experimental data, by developing a cycle-GAN that transformed simulated-generated labeled data into labeled data indistinguishable from experiment and therefore suitable for training. This removed the remaining parts of the ML data processing workflow where human intervention was still critical and therefore a bottleneck to working at scale. Codes have been developed and released for this full machine learning workflow. ML approaches to partially automate STEM acquisition were also developed. Finally we applied ML and other advanced data processing methods to several materials science problems in two-dimensional materials, including studying the evolution of hyperuniformity with defect concentration in WSe2, understanding phase transformations in transition metal dichalcogenides during in-situ heating in the STEM, and exploring how 2D interfaces transform from twisted into aligned structures.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Learning in continuous action space for developing high dimensional potential energy models

Reinforcement learning (RL) approaches that combine a tree search with deep learning have found remarkable success in searching exorbitantly large, albeit discrete action spaces, as in chess, Shogi and Go. Many real-world materials discovery and design applications, however, involve multi-dimensional search problems and learning domains that have continuous action spaces. Exploring high-dimensional potential energy models of materials is an example. Traditionally, these searches are time consuming (often several years for a single bulk system) and driven by human intuition and/or expertise and more recently by global/local optimization searches that have issues with convergence and/or do not scale well with the search dimensionality. Here, in a departure from discrete action and other gradient-based approaches, we introduce a RL strategy based on decision trees that incorporates modified rewards for improved exploration, efficient sampling during playouts and a “window scaling scheme" for enhanced exploitation, to enable efficient and scalable search for continuous action space problems. Using high-dimensional artificial landscapes and control RL problems, we successfully benchmark our approach against popular global optimization schemes and state of the art policy gradient methods, respectively. We demonstrate its efficacy to parameterize potential models (physics based and high-dimensional neural networks) for 54 different elemental systems across the periodic table as well as alloys. We analyze error trends across different elements in the latent space and trace their origin to elemental structural diversity and the smoothness of the element energy surface. Broadly, our RL strategy will be applicable to many other physical science problems involving search over continuous action spaces.

36 MATERIALS SCIENCE↗

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↗