Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Models based on symmetries”

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 415 records · Page 23
All sourcesNASA NTRSDOE OSTI
NASA NTRS2013

Testing Small CPAS Parachutes Using HIVAS

The High Velocity Airflow System (HIVAS) facility at the Naval Air Warfare Center (NAWC) at China Lake was successfully used as an alternative to flight test to determine parachute drag performance of two small Capsule Parachute Assembly System (CPAS) canopies. A similar parachute with known performance was also tested as a control. Realtime computations of drag coefficient were unrealistically low. This is because HIVAS produces a non-uniform flow which rapidly decays from a high central core flow. Additional calibration runs were performed to characterize this flow assuming radial symmetry from the centerline. The flow field was used to post-process effective flow velocities at each throttle setting and parachute diameter using the definition of the momentum flux factor. Because one parachute had significant oscillations, additional calculations were required to estimate the projected flow at off-axis angles. The resulting drag data from HIVAS compared favorably to previously estimated parachute performance based on scaled data from analogous CPAS parachutes. The data will improve drag area distributions in the next version of the CPAS Model Memo.

Ray, Eric S.↗
DOE OSTI2022

Less Common Topological Phenomena in Bulk Materials

"Prediction and subsequent discovery of topological insulators is considered to be one of the main results in condensed matter physics in the last decades. Not surprisingly, it has received major attention of both researchers and funding agencies. This attention is well-deserved; yet, one cannot but note that nearly all this research revolves about essentially the same concept: electronic excitations with linear dispersion, covering, of course, such diverse and intriguing phenomena as topological insulator, bulk Dirac states (or Weyl, if not spin degenerate), Mayorana fermions. In this project, we will address, mainly, other topological phenomena, such as topologically nontrivial magnetic patterns (as, for instance, topological Hall and related phenomena). Specifically, we propose three interrelated trusts: (1) Time-reversal symmetry breaking nonrelativistic antiferromagnets, called altermagnets. These are materials that break Kramers degeneracy of electronic bands, despite having zero net magnetization by symmetry and being fully collinear, and not necessarily non-centrosymmetric. The corresponding band structure is very similar to the band structure in non-centrosymmetric spin-orbital materials, but materials that we propose to study are distinctly different, first and foremost in the sense that despite sharing many aspects of their electronic properties with the latter, they break the time-reversal symmetry without either spin-orbit coupling or lack of inversion symmetry. (2) Topologically nontrivial magnetic spirals. The PI has been engaged with the experimental group of Dr. Ghimire at GMU investigating Dirac materials with helical magnetism, based on stacked magnetic Kagome layers, with a generic formula of RMn6Sn6. In particular, Y Mn6Sn6 demonstrates a component of the Hall effect that is naturally interpreted in terms of a topological spin texture, as well as linear magnetoresistance. Our calculation identify Dirac states that are robust with respect to the spiral formation, and let us derive an advance mean-field model explaining the observed phase diagram. This model predicted four distinct phases, with very distinct properties, which have now been seen in neutron experiments. The same compound is known to demonstrate topological Hall effect in a particular magnetic phase, and only at elevated temperature. Based on our understanding of the phase diagram, we have worked out a phenomenological theory of a chiral (skyrmionic) response to an external magnetic field, similar to the nematic response to external strain in Fe-based superconductors, which is possible in a centrosymmetric lattice and without interplanar Dzyaloshinskii-Moriya interaction. This phenomenological theory agrees quantitatively with the experiment. It is in our plans to research other similar materials for this effect. (3) Search for 3D analogues of Fe-based superconductors. We want to investigate materials that can be viewed as 3D analogues of FeSe. Specifically, we want materials that are good metals and host antiferromagnetism, which can be suppressed by pressure and generate an s-wave superconductivity, as in Fe-based superconductors. We have in mind some candidates already. This work will proceed in close collaboration with the experimental group of Prof. Nirmal Ghimire in the same department, whose expertise lies in sample making, magnetometry and transport measurements of materials with complex magnetic structures."

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
NASA NTRS1999

Combustion of Han-Based Monopropellant Droplets in Reduced Gravity

The objective of this research is to study combustion of monopropellant droplets and monopropellant droplet components in reduced-gravity environments so that spherical symmetry is strongly promoted. The experiments will use hydroxylammonium nitrate (HAN, chemical formula NH3OHNO3) based monopropellants. This class of monopropellant is selected for study because of its current relevance and also because it is relatively benign and safe to work with. The experimental studies will allow for accurate determination of fundamental data on deflagration rates, gas-phase temperature profiles, transient gas-phase flame behaviors, the onset of bubbling in droplets at lower pressures, and the low-pressure deflagration limit. The theoretical studies will provide rational models of deflagration mechanisms of HAN-based liquid propellants. Besides advancing fundamental knowledge, the proposed research should aid in applications (e.g., spacecraft thrusters and liquid propellant guns) of this unique class of monopropellants.

Shaw, B. D.↗
DOE OSTI2021

Low‐Density 2D Superlattices Assembled via Directional DNA Bonding

Abstract It is critical to assemble nanoparticles (NPs) into superlattices with controlled symmetries and spacings on substrates for metamaterials applications, where such structural parameters dictate their properties. Here, we use DNA to assemble anisotropic NPs of three shapes—cubes, octahedra, and rhombic dodecahedra—on substrates and investigate their thermally induced reorganization into two‐dimensional (2D) crystalline films. We report two new low‐density 2D structures, including a honeycomb lattice based on octahedral NPs. The low‐density lattices favored here are not usually seen when particles are crystallized via other bottom‐up assembly techniques. Furthermore, we show that, consistent with the complementary contact model, a primary driving force for crystallization is the formation of directional, face‐to‐face DNA bonds between neighboring NPs and between NPs and the substrate. Our results can be used to deliberately prepare crystalline NP films with novel morphologies.

Miao, Ziyi↗
DOE OSTI2023

Effective Field Theory islands from perturbative and nonperturbative four-graviton amplitudes

Theoretical data obtained from physically sensible field and string theory models suggest that gravitational Effective Field Theories (EFTs) live on islands that are tiny compared to current general bounds determined from unitarity, causality, crossing symmetry, and a good high-energy behavior. In this work, we present explicit perturbative and nonperturbative 2 → 2 graviton scattering amplitudes and their associated low-energy expansion in spacetime dimensions D ≥ 4 to support this notion. Our new results include a first example of gravity weakly coupled to a nonperturbative effective action. We show that, at energies below the mass of its nonperturbative matter, the D = 4, $\mathcal{N}$ = 1 supersymmetric field theory in the confined phase lies on the same islands identified using four-dimensional perturbative models based on string theory and minimally-coupled matter circulating a loop. Furthermore, we generalize the previous four-dimensional perturbative models based on string theory and minimally-coupled massive spin-0 and spin-1 states circulating in the loop to D dimensions. Remarkably, we again find that the low-energy EFT coefficients lie on small islands. These results offer a useful guide towards constraining possible extensions of Einstein gravity.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2024

Ligand Many-Body Expansion as a General Approach for Accelerating Transition Metal Complex Discovery

Methods that accelerate the evaluation of molecular properties are essential for chemical discovery. While some degree of ligand additivity has been established for transition metal complexes, it is underutilized in asymmetric complexes, such as the square pyramidal coordination geometries highly relevant to catalysis. To develop predictive methods beyond simple additivity, we apply a many-body expansion to octahedral and square pyramidal complexes and introduce a correction based on adjacent ligands (i.e., the cis interaction model). We first test the cis interaction model on adiabatic spin-splitting energies of octahedral Fe(II) complexes, predicting DFT-calculated values of unseen binary complexes to within an average of 1.4 kcal/mol. Uncertainty analysis reveals the optimal basis, comprising the homoleptic and mer symmetric complexes. We next show that the cis model (i.e., the cis interaction model solved for the optimal basis) infers both DFT- and CCSD(T)-calculated model catalytic reaction energies to within 1 kcal/mol on average. The cis model predicts low-symmetry complexes with reaction energies outside the range of binary complex reaction energies. We observe that trans interactions are unnecessary for most monodentate systems but can be important for some combinations of ligands, such as complexes containing a mixture of bidentate and monodentate ligands. Lastly, we demonstrate that the cis model may be combined with Δ-learning to predict CCSD(T) reaction energies from exhaustively calculated DFT reaction energies and the same fraction of CCSD(T) reaction energies needed for the cis model, achieving around 30% of the error from using the CCSD(T) reaction energies in the cis model alone.

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

Towards a Quintic Ginzburg-Landau Description of the (2,7) Minimal Model

We discuss dimensional continuation of the massless scalar field theory with the 𝑖⁢𝜙 5 interaction term. It preserves the so-called 𝒫⁢𝒯 symmetry, which acts by 𝜙 →−𝜙 accompanied by 𝑖 →−𝑖. Below its upper critical dimension 10/3, this theory has interacting infrared fixed points. We argue that the fixed point in 𝑑 = 2 describes the nonunitary minimal conformal model 𝑀⁡(2,7). We identify the operators 𝜙 and 𝜙 2 with the Virasoro primaries 𝜙 1,2 and 𝜙 1,3 , respectively, and 𝑖⁢𝜙 3 with a quasiprimary operator, which is a Virasoro descendant of 𝜙 1,3 . Our identifications appear to be consistent with the operator product expansions and with considerations based on integrability. Using constrained Padé extrapolations, we provide estimates of the critical exponents in 𝑑 = 3. We also comment on possible lattice descriptions of 𝑀⁡(2,7) and discuss RG flows to and from this conformal field theory (CFT). Finally, we conjecture that the minimal models 𝑀⁡(2,2⁢𝑛 +1) are described by the massless scalar field theories with the 𝑖⁢𝜙 2⁢𝑛−1 interaction terms.

Conformal field theory↗
DOE OSTI2020

Reconstruction and uncertainty quantification of lattice Hamiltonian model parameters from observations of microscopic degrees of freedom

The emergence of scanning probe and electron beam imaging techniques has allowed quantitative studies of atomic structure and minute details of electronic and vibrational structure on the level of individual atomic units. These microscopic descriptors, in turn, can be associated with local symmetry breaking phenomena, representing the stochastic manifestation of the underpinning generative physical model. In this work, we explore the reconstruction of exchange integrals in the Hamiltonian for a lattice model with two competing interactions from observations of microscopic degrees of freedom and establish the uncertainties and reliability of such analysis in a broad parameter-temperature space. In contrast to other approaches, we specifically specify a loss function inherent to thermodynamic systems and utilize it to estimate uncertainty in simulated realizations of different models. As an ancillary task, we develop a machine learning approach based on histogram clustering to predict phase diagrams efficiently using a reduced descriptor space. We further demonstrate that reconstruction is possible well above the phase transition and in the regions of parameter space when the macroscopic ground state of the system is poorly defined due to frustrated interactions. This suggests that this approach can be applied to the traditionally complex problems of condensed matter physics such as ferroelectric relaxors and morphotropic phase boundary systems, spin and cluster glasses, and quantum systems once the local descriptors linked to the relevant physical behaviors are known.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2021

Optical magnons with dominant bond-directional exchange interactions in the honeycomb lattice iridate α — Li 2 IrO 3

We have used resonant inelastic x-ray scattering to reveal optical magnons in a honeycomb lattice iridate α - Li 2 IrO 3 . The spectrum in the energy region 20–25 meV exhibits momentum dependence, of which energy is highest at the location of the magnetic Bragg peak, ( h , k ) = ( ± 0.32 , 0 ) , and lowered toward ( 0 , 0 ) and ( ± 1 , 0 ) . We compare our data with a linear spin-wave theory based on a generic nearest-neighbor spin model. We find that a dominant bond-directional Kitaev interaction of order 20 meV is required to explain the energy scale observed in our study. The observed excitations are understood as stemming from optical magnon modes whose intensity is modulated by a structure factor, resulting in the apparent momentum dependence. We also observed diffuse magnetic scattering arising from the short-range magnetic correlation well above T N . In contrast to Na 2 IrO 3 , this diffuse scattering lacks the C 3 rotational symmetry of the honeycomb lattice, suggesting that the bond anisotropy is far from negligible in α - Li 2 IrO 3 .

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2023

Predictive Design of Hybrid Improper Ferroelectric Double Perovskite Oxides

The computational design of suitable multiferroic double perovskite oxides requires finding materials that exhibit sizable polarization, magnetization, and coupling between them. Oxides with the chemical formula of AA'BB'O 6 with building blocks of ABO 3 single perovskite oxides in centrosymmetric Pnma symmetry are strong candidates that have been reported to satisfy such criteria. The system lowers to noncentrosymmetric, polar P2 1 symmetry if A/A' layered and B/B' rocksalt cation orderings are imposed. A detailed compositional search over a variety of chemical spaces followed by evaluating their polarization may lead to the identification of more of these compounds with ferroelectric ordering. The standard density functional theory practices to estimate polarization within the Berry phase formalism require the systems to be perfectly insulating. The number of compounds that can be evaluated using this method is therefore limited. In this work, we introduce a predictive learning strategy based on importance sampling to build a series of machine learning models using results from first-principles simulations to predict polarization and the corresponding switching barrier. The geometry-driven features related to charge states and cationic radii play key roles in predicting the switching barrier with complementary contributions from the key structural mode-based order parameters. These modes become important to draw reasonable predictions of polarization components from machine learning models. In conclusion, our predictive models identify candidates with high polarizations and low switching barriers from a pool of double perovskite oxides, suitable for future investigation for their potential applications in spintronic devices.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Interface-Driven Chiral Magnetism in Ultrathin Metallic Ferromagnets: Towards Skyrmion Spintronics

Magnetic skyrmions are localized chiral magnetic textures in the form of nanoscale vortices or bubbles that are topologically protected from being ‘unwound’. Their topological nature gives rise to rich behaviors including ordered lattice formation, emergent electrodynamics and robust current-driven displacement at remarkably low current densities. However, magnetic skyrmions have so far been restricted to just a few materials and observed only at low temperatures, limiting the experimental accessibility and technological application of these unique topological objects. This project aimed to realize magnetic skyrmions at room temperature in a new class of engineered chiral ferromagnets that exploit broken mirror symmetry at interfaces to generate helical magnetic order. Ultrathin magnetic heterostructures were engineered and patterned into laterally-confined nanostructures in which skyrmions were stabilized, manipulated, and detected for the first time under ambient conditions. Static and dynamic properties were examined using advanced x-ray imaging to provide a fundamental understanding of topological magnetic phases. The interactions between magnetic skyrmions and electron charge and spin currents were studied to identify fundamental new physics and enable device applications. These experimental studies, supported by detailed modeling, will help launch a new subfield of skyrmion-based spintronics in which individual skyrmions can be used to encode, store, and transport information in high-performance, low power memory and logic devices.

36 MATERIALS SCIENCE↗
NASA NTRS2010

The OMPS Limb Profiler Instrument: Two-Dimensional Retrieval Algorithm

The upcoming Ozone Mapper and Profiler Suite (OMPS), which will be launched on the NPOESS Preparatory Project (NPP) platform in early 2011, will continue monitoring the global distribution of the Earth's middle atmosphere ozone and aerosol. OMPS is composed of three instruments, namely the Total Column Mapper (heritage: TOMS, OMI), the Nadir Profiler (heritage: SBUV) and the Limb Profiler (heritage: SOLSE/LORE, OSIRIS, SCIAMACHY, SAGE III). The ultimate goal of the mission is to better understand and quantify the rate of stratospheric ozone recovery. The focus of the paper will be on the Limb Profiler (LP) instrument. The LP instrument will measure the Earth's limb radiance (which is due to the scattering of solar photons by air molecules, aerosol and Earth surface) in the ultra-violet (UV), visible and near infrared, from 285 to 1000 nm. The LP simultaneously images the whole vertical extent of the Earth's limb through three vertical slits, each covering a vertical tangent height range of 100 km and each horizontally spaced by 250 km in the cross-track direction. Measurements are made every 19 seconds along the orbit track, which corresponds to a distance of about 150km. Several data analysis tools are presently being constructed and tested to retrieve ozone and aerosol vertical distribution from limb radiance measurements. The primary NASA algorithm is based on earlier algorithms developed for the SOLSE/LORE and SAGE III limb scatter missions. All the existing retrieval algorithms rely on a spherical symmetry assumption for the atmosphere structure. While this assumption is reasonable in most of the stratosphere, it is no longer valid in regions of prime scientific interest, such as polar vortex and UTLS regions. The paper will describe a two-dimensional retrieval algorithm whereby the ozone distribution is simultaneously retrieved vertically and horizontally for a whole orbit. The retrieval code relies on (1) a forward 2D Radiative Transfer code (to model limb radiances within a non-uniform atmosphere and evaluate 2D analytical partial derivatives) and (2) an optimal estimator inversion routine. The algorithm uses the typically sparse nature of the kernel matrices as well as fast matrix inversion techniques to allow for fast inversion of limb data with efficient memory management (as was done for MIPAS data processing). While the method has so far only been developed in the context of Single Scatter, the paper will show how the CPU intensive Multiple Scatter modeling can be implemented using parallel CPU processing. Initial results will be presented in terms of retrieved ozone profiles and code performance.

Rault, Didier F.↗
DOE OSTI2023

Gravitational Waves from Rapid Structure Formation on Microscopic Scales before Matter-Radiation Equality

We propose a new scenario for the formation of asteroid-mass primordial black holes (PBHs). Our mechanism is based on the annihilation of the string-wall network associated with thebreaking of a U(1) global symmetry into a discrete ZN symmetry. If the potential hasmultiple local minima (N > 1) the network is stable, and the annihilation is guaranteed by a biasamong the different vacua. The collapse of the string-wall network is accompanied bycatastrogenesis, a large production of pseudo-Goldstone bosons (pGBs) — e.g. axions,ALPs, or majorons — gravitational waves, and PBHs. If pGBs rapidly decay into products thatthermalize, as predicted e.g. in the high-quality QCD axion and heavy majoron models, they do notcontribute to the dark matter population, but we show that PBHs can constitute 100% of the darkmatter. The gravitational wave background produced by catastrogenesis with heavy unstable axions,ALPs, or majorons could be visible in future interferometers.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Low-Density 2D Superlattices Assembled via Directional DNA Bonding

It is critical to assemble nanoparticles (NPs) into superlattices with controlled symmetries and spacings on substrates for metamaterials applications, where such structural parameters dictate their properties. Here, we use DNA to assemble anisotropic NPs of three shapes-cubes, octahedra, and rhombic dodecahedra-on substrates and investigate their thermally induced reorganization into two-dimensional (2D) crystalline films. We report two new low-density 2D structures, including a honeycomb lattice based on octahedral NPs. The low-density lattices favored here are not usually seen when particles are crystallized via other bottom-up assembly techniques. Furthermore, we show that, consistent with the complementary contact model, a primary driving force for crystallization is the formation of directional, face-to-face DNA bonds between neighboring NPs and between NPs and the substrate. Finally, our results can be used to deliberately prepare crystalline NP films with novel morphologies.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Generalising G 2 geometry: involutivity, moment maps and moduli

We analyse the geometry of generic Minkowski N = 1, D = 4 flux compactifications in string theory, the default backgrounds for string model building. In M-theory they are the natural string theoretic extensions of G2 holonomy manifolds. In type II theories, they extend the notion of Calabi-Yau geometry and include the class of flux backgrounds based on generalised complex structures first considered by Graña et al. (GMPT). Using E 7(7) × $\mathbb{R}$ + generalised geometry we show that these compactifications are characterised by an SU(7) ⊂ E 7(7) structure defining an involutive subbundle of the generalised tangent space, and with a vanishing moment map, corresponding to the action of the diffeomorphism and gauge symmetries of the theory. The Kähler potential on the space of structures defines a natural extension of Hitchin’s G 2 functional. Using this framework we are able to count, for the first time, the massless scalar moduli of GMPT solutions in terms of generalised geometry cohomology groups. It also provides an intriguing new perspective on the existence of G 2 manifolds, suggesting possible connections to Geometrical Invariant Theory and stability.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS2002

The Statistical Mechanics of Ideal Homogeneous Turbulence

Plasmas, such as those found in the space environment or in plasma confinement devices, are often modeled as electrically conducting fluids. When fluids and plasmas are energetically stirred, regions of highly nonlinear, chaotic behavior known as turbulence arise. Understanding the fundamental nature of turbulence is a long-standing theoretical challenge. The present work describes a statistical theory concerning a certain class of nonlinear, finite dimensional, dynamical models of turbulence. These models arise when the partial differential equations describing incompressible, ideal (i.e., nondissipative) homogeneous fluid and magnetofluid (i.e., plasma) turbulence are Fourier transformed into a very large set of ordinary differential equations. These equations define a divergenceless flow in a high-dimensional phase space, which allows for the existence of a Liouville theorem, guaranteeing a distribution function based on constants of the motion (integral invariants). The novelty of these particular dynamical systems is that there are integral invariants other than the energy, and that some of these invariants behave like pseudoscalars under two of the discrete symmetry transformations of physics, parity, and charge conjugation. In this work the 'rugged invariants' of ideal homogeneous turbulence are shown to be the only significant scalar and pseudoscalar invariants. The discovery that pseudoscalar invariants cause symmetries of the original equations to be dynamically broken and induce a nonergodic structure on the associated phase space is the primary result presented here. Applicability of this result to dissipative turbulence is also discussed.

Shebalin, John V.↗
DOE OSTI2022

Tuning between continuous time crystals and many-body scars in long-range XYZ spin chains.

Persistent oscillatory dynamics in nonequilibrium many-body systems is a tantalizing manifestation of ergodicity breakdown that continues to attract much attention. Recent works have focused on two classes of such systems: discrete time crystals and quantum many-body scars (QMBS). While both systems host oscillatory dynamics, its origin is expected to be fundamentally different: discrete time crystal is a phase of matter which spontaneously breaks the Z 2 symmetry of the external periodic drive, while QMBS span a subspace of nonthermalizing eigenstates forming an su(2) algebra representation. Here, we ask a basic question: is there a physical system that allows us to tune between these two dynamical phenomena? In contrast to much previous work, we investigate the possibility of a continuous time crystal (CTC) in undriven, energy-conserving systems exhibiting prethermalization. We introduce a long-range XYZ spin model and show that it encompasses both a CTC phase as well as QMBS. We map out the dynamical phase diagram using numerical simulations based on exact diagonalization and time-dependent variational principle in the thermodynamic limit. We identify a regime where QMBS and CTC order coexist, and we discuss experimental protocols that reveal their similarities as well as key differences.

Bull, Kieran↗
DOE OSTI2023

General quantum algorithms for Hamiltonian simulation with applications to a non-Abelian lattice gauge theory

With a focus on universal quantum computing for quantum simulation, and through the example of lattice gauge theories, we introduce rather general quantum algorithms that can efficiently simulate certain classes of interactions consisting of correlated changes in multiple (bosonic and fermionic) quantum numbers with non-trivial functional coefficients. In particular, we analyze diagonalization of Hamiltonian terms using a singular-value decomposition technique, and discuss how the achieved diagonal unitaries in the digitized time-evolution operator can be implemented. The lattice gauge theory studied is the SU(2) gauge theory in 1+1 dimensions coupled to one flavor of staggered fermions, for which a complete quantum-resource analysis within different computational models is presented. The algorithms are shown to be applicable to higher-dimensional theories as well as to other Abelian and non-Abelian gauge theories. The example chosen further demonstrates the importance of adopting efficient theoretical formulations: it is shown that an explicitly gauge-invariant formulation using loop, string, and hadron degrees of freedom simplifies the algorithms and lowers the cost compared with the standard formulations based on angular-momentum as well as the Schwinger-boson degrees of freedom. The loop-string-hadron formulation further retains the non-Abelian gauge symmetry despite the inexactness of the digitized simulation, without the need for costly controlled operations. Such theoretical and algorithmic considerations are likely to be essential in quantumly simulating other complex theories of relevance to nature.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗