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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 73 records · Page 4
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

The second-generation Shifted Boundary Method and its numerical analysis

Recently, the Shifted Boundary Method (SBM) was proposed within the class of unfitted (or immersed, or embedded) finite element methods. By reformulating the original boundary value problem over a surrogate (approximate) computational domain, the SBM avoids integration over cut cells and the associated problematic issues regarding numerical stability and matrix conditioning. Accuracy is maintained by modifying the original boundary conditions using Taylor expansions. Hence the name of the method, that shifts the location and values of the boundary conditions. In this article, we present enhanced variational SBM formulations for the Poisson and Stokes problems with improved flexibility and robustness. These simplified variational forms allow to relax some of the assumptions required by the mathematical proofs of stability and convergence of earlier implementations. First, we show that these new SBM implementations can be proved asymptotically stable and convergent even without the rather restrictive assumption that the inner product between the normals to the true and surrogate boundaries is positive. Second, we show that it is not necessary to introduce a stabilization term involving the tangential derivatives of the solution at Dirichlet boundaries, therefore avoiding the calibration of an additional stabilization parameter. Finally, we prove enhanced L 2 -estimates without the cumbersome assumption – of earlier proofs – that the surrogate domain is convex. Instead we rely on a conventional assumption that the boundary of the true domain is smooth, which can also be replaced by requiring convexity of the true domain. The aforementioned improvements open the way to a more general and efficient implementation of the Shifted Boundary Method, particularly in complex three-dimensional geometries. We complement these theoretical developments with numerical experiments in two and three dimensions.

42 ENGINEERING↗
DOE OSTI2021

An embedded variable step IMEX scheme for the incompressible Navier–Stokes equations

This report presents a series of implicit–explicit (IMEX) variable stepsize algorithms for the incompressible Navier–Stokes equations (NSE). Here, IMEX means the nonlinear term is treated fully explicitly while the remaining terms are treated implicitly. With the advent of new computer architectures there has been growing demand for low memory solvers of this type. The addition of time adaptivity improves the accuracy and greatly enhances the efficiency of the algorithm. We prove energy stability of an embedded first–second order IMEX pair. For the first order member of the pair, we prove stability for variable stepsizes, and analyze convergence. We believe this to be the first proof of this type for a variable stepsize IMEX scheme for the incompressible NSE. We then define and test a variable stepsize, variable order IMEX scheme using these methods. Our work contributes several firsts for IMEX NSE schemes, including an energy argument and error analysis of a two-step, variable stepsize method, and embedded error estimation for an IMEX multi-step method.

42 ENGINEERING↗
DOE OSTI2023

Pore-scale simulation of multiphase flow and reactive transport processes involved in geologic carbon sequestration

Multiphase flow and reactive transport are two essential physicochemical processes that govern the effectiveness of geological carbon sequestration (GCS). The interaction and feedback among different phases and components during intricate physicochemical processes hold great significance in understanding CO 2 sequestration. Pore-scale simulations can account for multiphase flow and reactive transport processes in porous media and obtain spatial distributions of parameters (density, velocity, concentration, etc.) in the pore space as well as their temporal evolutions. This proves especially valuable considering that experiments can be hindered by constraints in spatial and temporal resolution. The comprehensive insights garnered from pore-scale research can be leveraged for continuum modeling using the representative elementary volume (REV) concept. In this contribution, four sequential mechanisms of CO 2 -brine-rock interaction in three zones delineated by CO 2 saturation are elaborated to elucidate complicated physicochemical processes involved in GCS, which are followed by general descriptions of mathematical equations and pore-scale numerical methods. In addition, as interested and commonly encountered processes, leakage risks during GCS and CO 2 -enhanced oil recovery (CO 2 -EOR) processes are presented. The existing challenges and future directions are discussed for both the performance of the pore-scale models and the current gaps in the field of GCS. Importantly, we expect that this review will prove beneficial for researchers interested in pore-scale simulations, GCS, and related disciplines.

58 GEOSCIENCES↗
DOE OSTI2022

Towards sharp error analysis of extended Lagrangian molecular dynamics

The extended Lagrangian molecular dynamics (XLMD) method provides a useful framework for reducing the computational cost of a class of molecular dynamics simulations with constrained latent variables. The XLMD method relaxes the constraints by introducing a fictitious mass ε for the latent variables and solving a set of singularly perturbed ordinary differential equations. While favorable numerical performance of XLMD has been demonstrated in several different contexts in the past decade, mathematical analysis of the method remains scarce. Here, we propose the first error analysis of the XLMD method in the context of a classical polarizable force field model. While the dynamics with respect to the atomic degrees of freedom are general and nonlinear, the key mathematical simplification of the polarizable force field model is that the constraints on the latent variables are given by a linear system of equations. We prove that when the initial value of the latent variables is compatible in a sense that we define, XLMD converges as the fictitious mass ε is made small with $\mathscr{O}$(ε) error for the atomic degrees of freedom and with $\mathscr{O}$($\sqrt{ε}$) error for the latent variables, when the dimension of the latent variable d' is 1. Furthermore, when the initial value of the latent variables is improved to be optimally compatible in a certain sense, we prove that the convergence rate can be improved to $\mathscr{O}$(ε) for the latent variables as well. Numerical results verify that both estimates are sharp not only for d'=1, but also for arbitrary d'. In the setting of general d', we do obtain convergence, but with the non-sharp rate of $\mathscr{O}$($\sqrt{ε}$) for both the atomic and latent variables.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2021

Thermal runaway mechanism of lithium-ion battery with LiNi 0.8 Mn 0.1 Co 0.1 O 2 cathode materials

Battery safety is critical to the application of lithium-ion batteries, especially for high energy density battery applied in electric vehicles. In this paper, the thermal runaway mechanism of LiNi 0.8 Co 0.1 Mn 0.1 O 2 based lithium-ion battery is illustrated. And the reaction between cathode and flammable electrolyte is proved as the trigger of the thermal runaway accident. In detail, with differential scanning calorimeter tests for battery components, the material combination contributing to thermal runaway was decoupled. Characterization with synchrotron X-ray diffraction and transmission electron microscopy with in-situ heating proved that the vigorous exothermic reaction is initiated by the liberated oxygen species. The pulse of highly active oxygen species reacted quickly with the electrolyte, accompanied with tremendous heat release, which accelerated the phase transformation of charged cathode. Also, the mechanism is verified by a confirmatory experiment when the highly active oxygen species were trapped by anion receptor, the phase transformation of the charged cathode was inhibited. Clarifying the thermal runaway mechanism of LiNi 0.8 Co 0.1 Mn 0.1 based lithium-ion battery may light the way to battery chemistries of both high energy density and high safety.

25 ENERGY STORAGE↗
DOE OSTI2021

Central vortex steady states and dynamics of Bose–Einstein condensates interacting with a microwave field

Here we study central vortex steady states and dynamics of a two-dimensional (2D), two-component, Gross–Pitaevskii equation (CGPE) system for two pseudo-spinor Bose Einstein condensates (BECs) interacting with an electromagnetic field (microwave) analytically and numerically. For the central vortex steady state at any given winding number S, we prove its existence in a reduced, single component detuning limit when contact-interaction strength β < β b , and nonexistence when β > β b , respectively, where β b is a threshold value, whose value is given in Theorem 3.1 in the paper. We extend the existence and nonexistence result to the general two pseudo-spinor case and prove that a central vortex steady state exists for any given S if β ≤ β b while it does not exist when β > 2β b . We then derive dynamical equations for some observables (expectations of the matter wave function) such as the center-of-mass, position of dispersion, the linear and angular momentum of the two pseudo-spinor CGPEs. Finally, numerical computations are brought in to validate and extend the existence of vortex steady state results to β b ≤ β < 2β b for the two pseudo-spinor case and to explore transient dynamics of the observables.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Projective embedding of dynamical systems: Uniform mean field equations

Herein we study embeddings of continuous dynamical systems in larger dimensions via projector operators. We call this technique PEDS, projective embedding of dynamical systems, as the stable fixed point of the original system dynamics are recovered via projection from the higher dimensional space. In this paper we provide a general definition and prove that for a particular type of rank-1 projector operator, the uniform mean field projector, the equations of motion become a mean field approximation of the dynamical system. While in general the embedding depends on a specified variable ordering, the same is not true for the uniform mean field projector. We prove a variety of results on the relationship between the spectrum of the Jacobian for fixed points in the original and in the embedded system. Direct applications of PEDS can be non-convex optimization and machine learning.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Characterization and interaction of geometric and contact/impact nonlinearities in dynamical systems

Here, we study how a contact/impact nonlinearity interacts with a geometric cubic nonlinearity in an oscillator system. Specific focus is shown to the effects on bifurcation behavior and secondary resonances (i.e., super- and sub-harmonic resonances). The effects of the individual nonlinearities are first explored for comparison, and then the influences of the combined nonlinearities, varying one parameter at a time, are analyzed and discussed. Nonlinear characterization is then performed on an arbitrary system configuration to study super- and sub-harmonic resonances and grazing contacts or bifurcations. Both the cubic and contact nonlinearities cause a drop in amplitude and shift up in frequency for the primary resonance, and they activate high-amplitude subharmonic resonance regions. The nonlinearities seem to never destructively interfere. The contact nonlinearity generally affects the system’s superharmonic resonance behavior more, particularly with regard to the occurrence of grazing contacts and the activation of many bifurcations in the system’s response. The subharmonic resonance behavior is more strongly affected by the cubic nonlinearity and is prone to multistable behavior. Perturbation theory proved useful for determining when the cubic nonlinearity would be dominant compared to the contact nonlinearity. The limiting behaviors of the contact stiffness and freeplay gap size indicate the cubic nonlinearity is dominant overall. It is demonstrated that the presence of contact may result in the activation of several bifurcations. In addition, it is proved that the system’s subharmonic resonance region is prone to multistable dynamical responses having distinct magnitudes.

42 ENGINEERING↗
DOE OSTI2022

Role of Eu-Doping in the Electron Transport Behavior in the Zintl Thermoelectric Ca 5-x-y Yb x Eu y Al 2 Sb 6 System

A series of Eu-doped Zintl compounds belonging to theCa 5-x-y Yb x Eu y Al 2 Sb 6 (x = 0, 1.12; 0 ≤ y ≤ 0.63(2)) system have been successfully synthesized by both the arc-melting and the molten Pb-flux methods. All of the five title compounds initially crystallized in the Ca 5 Ga 2 As 6 -type phase (space group Pbam, Z = 2, Pearson code oP26) and maintained their original structure even after the post-heat treatment, unlike the recently reported n-type Zintl analogues in the Ca 5-x-y Yb x RE y Al 2 Sb 6 (RE = Pr, Nd, Sm) systems, which underwent a phase transition from the Ca 5 Ga 2 As 6 -type to the Ca 5 Al 2 Bi 6 -type phase after annealing. This research aimed to understand the origin of the structural preference of the title Ca 5-x-y Yb x Eu y Al 2 Sb 6 system, whether it was affected by the valence electron count or the cationic size. Electrical transport property measurements showed an increase in electrical conductivities and a decrease of Seebeck coefficients for Ca 4.89(1) Eu 0.11 Al 2 Sb 6 , Ca 4.82(1) Eu 0.18 Al 2 Sb 6 , and Ca 4.62(1) Eu 0.38 Al 2 Sb 6 , compared to the parental compound Ca 5 Al 2 Sb 6 . Hole effect measurements proved that these changes should be attributed to the reduced carrier concentration and enhanced carrier mobility. The comprehensive density functional theory calculations including electron density map analysis for the hypothetical model Ca 4.5 Eu 0.5 Al 2 Sb 6 revealed that the polarity between Al and Sb forming the anionic frameworks decreased as the Eu-dopants were introduced, which eventually affected the carrier mobility in the anionic frameworks. Thermal conductivity measurements proved that the Eu-doping successfully lowered the lattice thermal conductivity because of the enhanced atomic disordering. In conclusion, the magnetization measurements for Ca 4.37(2) Eu 0.63 Al 2 Sb 6 showed a typical Curie–Weiss behavior with weak antiferromagnetic nearest-neighbor interactions with θ p = -5.07 K.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Tuning the Radius Ratio to Enhance Thermoelectric Properties in the Zintl Compounds AM 2 Sb 2 (A = Ba, Sr; M = Zn, Cd)

Five novel Zintl phase solid solutions in the Ba 1–x Sr x Zn 2–y Cd y Sb 2 (0 ≤ x ≤ 0.13(1); 0 ≤ y ≤ 0.32(2)) system were successfully synthesized by the molten Pb metal-flux method, and the powder X-ray diffraction and single-crystal X-ray diffraction analyses proved that all five title compounds adopted the BaCu 2 S 2 -type phase having the orthorhombic Pnma space group (Z = 4, Pearson code oP20) with five crystallographically independent atomic sites. The previously studied BaCu 2 S 2 -type antimonides demonstrated a limited tolerance for doping in contrast to the CaAl 2 Si 2 -type antimonides. To understand the relatively narrower phase width and limited dopability of the title BaCu 2 S 2 -type phase than the CaAl 2 Si 2 -type phase in the overall Ba 1–x Sr x Zn 2–y Cd y Sb 2 system, the radius ratio of cations and anionic elements r + /r – for two structure types were thoroughly investigated. For the first time, the r + /r – ratio was identified as a critical factor for the phase selectivity: (1) r + /r – > 1 favored the BaCu 2 S 2 -type phase, and (2) r + /r – < 1 favored the CaAl 2 Si 2 -type phase. Further, we also revealed the structural transformation mechanism from the more widely observed CaAl 2 Si 2 -type phase to the title BaCu 2 S 2 -type phase as the relatively larger cationic elements were introduced to the system. A series of DFT calculations using the three hypothetical models indicated that a resonance peak near EF in the density of states curves was descended from the relatively flat band structure at several special symmetry points rationalizing the enhanced Seebeck coefficients of Ba 0.94(1) Sr 0.06 Zn 1.86(3) Cd 0.14 Sb 2 and Ba 0.96(1) Sr 0.04 Zn 1.68(2) Cd 0.32 Sb 2 . Electron localization function analysis rationalized the correlation between the polarity change of anionic Zn/Cd–Sb bonds and the charge carrier mobility on the anionic frameworks. Temperature-dependent thermoelectric properties were studied for the four title compounds, and the results proved that the Sr and Cd doping in the title Ba 1–x Sr x Zn 2–y Cd y Sb 2 system successfully enhanced the ZT values through the increased Seebeck coefficients and the reduced total thermal conductivities.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Gas Production from Hot Water Circulation through Hydraulic Fractures in Methane Hydrate-Bearing Sediments: THC-Coupled Simulation of Production Mechanisms

Methane hydrates, widely found in permafrost and deep marine sediments, have great potential as a future energy 6 source. Conventional production schemes perform poorly for challenging hydrate reservoirs with low permeability. We propose an 7 efficient production scheme by combining hydraulic fracturing from horizontal wells and hot water circulation through fractures. A 8 fully coupled thermo-hydro-chemical (THC) model is developed to simulate the key physical processes during gas production from 9 a hydrate reservoir representative of typical geological settings in Shenhu, South China Sea. We found that the gas production 10 process has two distinct stages divided by thermal breakthrough: a relatively short prebreakthrough stage and a postbreakthrough 11 stage yielding stable gas production. Heat advection along and near the hydraulic fracture dominates the prebreakthrough stage, 12 whereas conduction-driven thermal recovery in the volume around fractures dominates the postbreakthrough stage. We identified 13 that the steady-state injection temperature has a strong effect on the performance of the proposed scheme while the fluid mass 14 circulation rate has a moderate impact beyond a threshold. The proposed scheme proves to be efficient and robust over a range of 15 reservoir conditions with respect to initial hydrate saturation and intrinsic permeability, including their spatial heterogeneities, 16 thereby offering a promising solution for challenging reservoir conditions.

gas hydrate, fracture flow, gas production, numeri↗
DOE OSTI2025

Thermal Oxo-degradation and Catalytic Upgrading of Plastic Waste to Light Olefins for a Circular Economy

The commercialization of conventional pyrolysis of plastic in an inert atmosphere has been hindered by large thermal requirements stemming from long reaction rates. The rate of thermal depolymerization of waste plastics can be accelerated by the addition of oxygen in a process known as thermal oxo-degradation (TOD). This study offers the prospect of TOD to upcycle postconsumer waste rapidly and efficiently. Using moderate temperatures and small amounts of air in a fluidized bed reactor, we demonstrated that waste high-density polyethylene and polypropylene are rapidly deconstructed to condensable products. These condensable products were catalytically upgraded in a micropyrolysis reactor using commercially available zeolite (HZSM-5) to monomeric olefins. The olefin yields proved to be greater than those achieved through the catalytic upgrading of condensable products from (nonoxidative) the pyrolysis of the same plastic wastes. The coupling of TOD with catalytic upgrading proves to be an energy-efficient pathway in a plastics circular economy for the production of light olefins from wastes.

Enthalpy↗
DOE OSTI2023

Yb Substitution and Ultralow Thermal Conductivity of the Ca 3– x Yb x AlSb 3 (0 ≤ x ≤ 0.81(1)) System

Here, a series of Yb-substituted Zintl phases in the Ca 3–x Yb x AlSb 3 (0 ≤ x ≤ 0.81(1)) system has been synthesized by initial arc melting and post-heat treatment, and their isotypic crystal structures were characterized by both powder and single crystal X-ray diffraction analysis. All four title compounds adopted the Ca 3 AlAs 3 -type structure (space group Pnma, Pearson code oP28, Z = 4). The overall structure can be described as a combination of the 1-dimensional (1D) infinite chain of ∞ 1 [Al(Sb 2 Sb 2/2 )] formed by two vertices sharing [AlSb 4 ] tetrahedral moieties and three Ca 2+ /Yb 2+ mixed sites located in between these 1D chains. The charge balance and the resultant independency of the 1D chains in the title system were explained by the Zintl-Klemm formalism [Ca 2+ /Yb 2+ ] 3 [(4b-Al 1– )(1b-Sb 2– ) 2 (2b-Sb 1– ) 2/2 ]. A series of DFT calculations proved that (1) the band overlap between the d-orbital states from two types of cations and the p-orbital states from Sb at the high symmetry Γ point implied a heavily doped degenerate semiconducting behavior of the quaternary Ca 2 YbAlSb 3 model and (2) the site preference of Yb for the M1 site was due to the electronic-factor criterion based on the Q values of each atomic site. The electron localization function calculations also proved that the two different shapes of lone pairs of the Sb atoms—the “umbrella-shape” and the “C-shape”—are determined by local geometry and the coordination environment on the anionic frameworks. Thermoelectric measurements of the quaternary title compound Ca 2.19(1) Yb 0.81 AlSb 3 showed an approximately two times larger ZT value than that of ternary Ca 3 AlSb 3 at 623 K due to increased electrical conductivity and ultralow thermal conductivity originated from Yb substitution for Ca.

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

Atomic Layer Deposition of Ruthenium Films Using Ruthenium Diketonates and O 2 , H 2 , or N 2 O: The Role of Ruthenium Etching

In this work, the evolution of the surface during the steps that comprise the atomic layer deposition (ALD) of ruthenium films on a nickel substrate using tris(2,2,6,6-tetramethyl-3,5-heptanedionato)ruthenium(III) (Ru(tmhd) 3 ) and molecular oxygen was characterized using a combination of X-ray photoelectron (XPS) and reflection–absorption infrared (RAIRS) spectroscopies. The uptake of the Ru metalorganic precursor was determined to be activated, involving the average loss of two out of the three ligands (and the retention of the third, in molecular form, at the surface–vacuum interface), and self-limited, as required in ALD. The reaction of the resulting layer from that first half of the ALD cycle with O 2 proved to be more complex: in addition to the desired removal of the carbon-containing material, the Ni substrate becomes oxidized, and some Ru is etched away in the form of the volatile RuO 4 gas. By testing different combinations of exposures and temperatures it was determined that Ru film growth was possible, but tuning and optimizing the ALD or atomic layer etching (ALE) process conditions for maximum film growth or removal rate per cycle proved difficult. Replacing O 2 with H 2 was shown not to be viable either, as such an agent can reduce the nickel oxide formed after O 2 treatments (in fact, Ru(tmhd) 3 can do this as well) but not remove the carbonaceous material deposited on the surface by the Ru precursor. The use of N 2 O, on the other hand, showed much promise, being capable of removing most of the surface carbon without affecting the Ru film or the Ni substrate.

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

Double Perovskite Interlayer Stabilized Highly Efficient Perovskite Solar Cells

Metal halide perovskite solar cell (PSC) technology has an impressive power conversion efficiency (PCE) exceeding 26.1% and demonstrates cost-effective manufacturing. However, the stability of these PSCs poses a significant challenge, hindering their widespread manufacturing and commercialization. To tackle the degradation issue inherent in PSCs, surface passivation techniques, particularly employing a thin layer of two-dimensional (2D) perovskites, create a 2D/3D heterostructure. Beyond this, the exploration of metal halide double perovskites adds a new dimension to the chemical and band gap phase space of materials for optoelectronic applications. In this study, we leverage a wide band gap double perovskite interlayer to enhance the stability of 3D metal halide perovskite. Specifically, the double perovskite nanoparticle Cs 2 AgBiBr 6 , with its substantial band gap of 2.2 eV and exceptional air stability, is utilized. Through optimization, a Cs 2 AgBiBr 6 -treated PSC achieves an open-circuit voltage of 1.12 V and an impressive PCE of 19.52%. Additionally, the Cs 2 AgBiBr 6 passivation layer proves to be effective in bolstering the stability of PSCs. This work demonstrates an additional strategy and design motif to simultaneously increase the PCE of PSCs along with achieving improved stability.

14 SOLAR ENERGY↗
DOE OSTI2021

Robust Solid/Electrolyte Interphase (SEI) formation on Si Anodes Using Glyme-based Electrolytes

Silicon (Si) is the most naturally abundant element possessing 10-fold theoretical capacity than graphitebased anodes. The practicality of implementing Si anodes is, however limited by the unstable solidelectrolyte interphase (SEI) and anode fracturing during continuous lithiation/delithiation. We demonstrate that glyme-based electrolytes (GlyEls) assure a conformal SEI on Si and keep the Si ‘fracture-free’. Benchmarking against the optimal, commonly-used carbonate electrolyte with the fluoroethylene carbonate (FEC) additive, Si anode cycled in a GlyEl exhibits reduced early parasitic current (by 62.5%) and interfacial resistance (by 72.8%), while the cell capacity retention is promoted by >7% over a course of 110 cycles. The mechanistic investigation by X-ray photoelectron spectroscopy (XPS) and energy-dispersive X-ray spectroscopy (EDX) indicates GlyEl enriches Si SEI with elastic polyether but diminishes its carbonate species. Glyme-based electrolytes prove viable in stabilizing the SEI on silicon for future high energy density lithium-ion batteries.

Yang, Guang↗
DOE OSTI2021

Noise-induced barren plateaus in variational quantum algorithms

Abstract Variational Quantum Algorithms (VQAs) may be a path to quantum advantage on Noisy Intermediate-Scale Quantum (NISQ) computers. A natural question is whether noise on NISQ devices places fundamental limitations on VQA performance. We rigorously prove a serious limitation for noisy VQAs, in that the noise causes the training landscape to have a barren plateau (i.e., vanishing gradient). Specifically, for the local Pauli noise considered, we prove that the gradient vanishes exponentially in the number of qubits n if the depth of the ansatz grows linearly with n . These noise-induced barren plateaus (NIBPs) are conceptually different from noise-free barren plateaus, which are linked to random parameter initialization. Our result is formulated for a generic ansatz that includes as special cases the Quantum Alternating Operator Ansatz and the Unitary Coupled Cluster Ansatz, among others. For the former, our numerical heuristics demonstrate the NIBP phenomenon for a realistic hardware noise model.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

The complexity of NISQ

Abstract The recent proliferation of NISQ devices has made it imperative to understand their power. In this work, we define and study the complexity class , which encapsulates problems that can be efficiently solved by a classical computer with access to noisy quantum circuits. We establish super-polynomial separations in the complexity among classical computation, , and fault-tolerant quantum computation to solve some problems based on modifications of Simon’s problems. We then consider the power of for three well-studied problems. For unstructured search, we prove that cannot achieve a Grover-like quadratic speedup over classical computers. For the Bernstein-Vazirani problem, we show that only needs a number of queries logarithmic in what is required for classical computers. Finally, for a quantum state learning problem, we prove that is exponentially weaker than classical computers with access to noiseless constant-depth quantum circuits.

42 ENGINEERING↗