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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 541 records · Page 30
All sourcesNASA NTRSDOE OSTI
NASA NTRS2017

A Theory of Immersion Freezing

Immersion freezing is likely involved in the initiation of precipitation and determines to large extent the phase partitioning in convective clouds. Theoretical models commonly used to describe immersion freezing in atmospheric models are based on the classical nucleation theory which however neglects important interactions near the immersed particle that may affect nucleation rates. This work introduces a new theory of immersion freezing based on two premises. First, immersion ice nucleation is mediated by the modification of the properties of water near the particle-liquid interface, rather than by the geometry of the ice germ. Second, the same mechanism that leads to the decrease in the work of germ formation also decreases the mobility of water molecules near the immersed particle. These two premises allow establishing general thermodynamic constraints to the ice nucleation rate. Analysis of the new theory shows that active sites likely trigger ice nucleation, but they do not control the overall nucleation rate nor the probability of freezing. It also suggests that materials with different ice nucleation efficiency may exhibit similar freezing temperatures under similar conditions but differ in their sensitivity to particle surface area and cooling rate. Predicted nucleation rates show good agreement with observations for a diverse set of materials including dust, black carbon and bacterial ice nucleating particles. The application of the new theory within the NASA Global Earth System Model (GEOS-5) is also discussed.

Ice Nucleating particle↗
NASA NTRS2023

A 1D Model for Nucleation of Ice From Aerosol Particles: An Application to a Mixed-Phase Arctic Stratus Cloud Layer

Mixed-phase clouds (MPCs) have been identified as significant contributors to uncertainties in climate projections, attributable to model representation of processes controlling the formation and loss of supercooled water droplets and ice particles from the atmosphere. Arctic MPCs are commonly widespread and long-lived, with sustained ice crystal formation processes that challenge current understanding. This study examines the ice-nucleating particle (INP) reservoir dynamics governing immersion-mode heterogeneous freezing in an observed case of Arctic MPCs using a simplified 1D aerosol-cloud model. The model setup includes prescribed dynamical forcings and thermodynamic profiles, and represents INPs as multicomponent and polydisperse particle size distributions. Diagnostic and prognostic approaches to immersion freezing parameterization are compared, including time-independent (singular) number- and surface area-based descriptions and a time-dependent description following classical nucleation theory (CNT). The choice of freezing parameterization defines the size of the INP reservoir. The CNT-based description yields an orders of magnitude larger INP reservoir than the singular parameterizations, which is the dominant factor for sustained ice crystal formation. The efficiency of the freezing process and cloud cooling are of secondary importance. A diagnostic treatment neglecting INP loss is only accurate when the INP reservoir size is large and INP depletion weak. Since a larger INP reservoir sustains ice crystal formation substantially longer, and ice water path scales with ice crystal concentrations for the conditions considered, resolving the source of differences in INP reservoir dynamics due to model implementation is a high priority for advancing climate model physics.

aerosol↗
NASA NTRS1995

Determination of stability and control derivatives from the NASA F/A-18 HARV from flight data using the maximum likelihood method

This report is a compilation of PID (Proportional Integral Derivative) results for both longitudinal and lateral directional analysis that was completed during Fall 1994. It had earlier established that the maneuvers available for PID containing independent control surface inputs from OBES were not well suited for extracting the cross-coupling static (i.e., C(sub N beta)) or dynamic (i.e., C(sub Npf)) derivatives. This was due to the fact that these maneuvers were designed with the goal of minimizing any lateral directional motion during longitudinal maneuvers and vice-versa. This allows for greater simplification in the aerodynamic model as far as coupling between longitudinal and lateral directions is concerned. As a result, efforts were made to reanalyze this data and extract static and dynamic derivatives for the F/A-18 HARV (High Angle of Attack Research Vehicle) without the inclusion of the cross-coupling terms such that more accurate estimates of classical model terms could be acquired. Four longitudinal flights containing static PID maneuvers were examined. The classical state equations already available in pEst for alphadot, qdot and thetadot were used. Three lateral directional flights of PID static maneuvers were also examined. The classical state equations already available in pEst for betadot, p dot, rdot and phi dot were used. Enclosed with this document are the full set of longitudinal and lateral directional parameter estimate plots showing coefficient estimates along with Cramer-Rao bounds. In addition, a representative time history match for each type of meneuver tested at each angle of attack is also enclosed.

Napolitano, Marcello R.↗
NASA NTRS1997

Searching for Biogeochemical Cycles on Mars

The search for life on Mars clearly benefits from a rigorous, yet broad, definition of life that compels us to consider all possible lines of evidence for a martian biosphere. Recent studies in microbial ecology illustrate that the classic definition of life should be expanded beyond the traditional definition of a living cell. The traditional defining characteristics of life are threefold. First, life is capable of metabolism, that is, it performs chemical reactions that utilize energy and also synthesize its cellular constituents. Second, life is capable of self-replication. Third, life can evolve in order to adapt to environmental changes. An expanded, ecological definition of life also recognizes that life is a community of organisms that must interact with their nonliving environment through processes called biogeochemical cycles. This regenerative processing maintains, in an aqueous conditions, a dependable supply of nutrients and energy for growth. In turn, life can significantly affect those processes that control the exchange of materials between the atmosphere, ocean, and upper crust. Because metabolic processes interact directly with the environment, they can alter their surroundings and thus leave behind evidence of life. For example, organic matter is produced from single-carbon-atom precursors for the biosynthesis of cellular constituents. This leads to a reservoir of reduced carbon in sediments that, in turn, can affect the oxidation state of the atmosphere. The harvesting of chemical energy for metabolism often employs oxidation-reduction reactions that can alter the chemistry and oxidation state of the redox-sensitive elements carbon, sulfur, nitrogen, iron, and manganese. Have there ever been biogeochemical cycles on Mars? Certain key planetary processes can offer clues. Active volcanism provides reduced chemical species that biota can use for organic synthesis. Volcanic carbon dioxide and methane can serve as greenhouse gases. Thus the persistence of volcanism on Mars may well have influenced the persistence of a martian biosphere. The geologic processing of the crust can affect the availability of nutrients and also control the deposition of minerals that could have served as a medium for the preservation of fossil information. Finally, the activity of liquid water is crucial to life. Was there ever an Earth-like hydrologic cycle with rainfall? Has aqueous activity instead been restricted principally to hydrothermal activity below the surface? To what extent did the inorganic chemistry driven by sunlight and hydrothermal activity influence organic chemistry (prebiotic chemical evolution)? This paper addresses these and other key questions.

DesMarais, David J.↗
NASA NTRS1987

Procedures for Computing Transonic Flows for Control of Adaptive Wind Tunnels

Numerical methods are developed for control of three dimensional adaptive test sections. The physical properties of the design problem occurring in the external field computation are analyzed, and a design procedure suited for solution of the problem is worked out. To do this, the desired wall shape is determined by stepwise modification of an initial contour. The necessary changes in geometry are determined with the aid of a panel procedure, or, with incident flow near the sonic range, with a transonic small perturbation (TSP) procedure. The designed wall shape, together with the wall deflections set during the tunnel run, are the input to a newly derived one-step formula which immediately yields the adapted wall contour. This is particularly important since the classical iterative adaptation scheme is shown to converge poorly for 3D flows. Experimental results obtained in the adaptive test section with eight flexible walls are presented to demonstrate the potential of the procedure. Finally, a method is described to minimize wall interference in 3D flows by adapting only the top and bottom wind tunnel walls.

Rebstock, Rainer↗
NASA NTRS1990

Distributed memory approaches for robotic neural controllers

The suitability is explored of two varieties of distributed memory neutral networks as trainable controllers for a simulated robotics task. The task requires that two cameras observe an arbitrary target point in space. Coordinates of the target on the camera image planes are passed to a neural controller which must learn to solve the inverse kinematics of a manipulator with one revolute and two prismatic joints. Two new network designs are evaluated. The first, radial basis sparse distributed memory (RBSDM), approximates functional mappings as sums of multivariate gaussians centered around previously learned patterns. The second network types involved variations of Adaptive Vector Quantizers or Self Organizing Maps. In these networks, random N dimensional points are given local connectivities. They are then exposed to training patterns and readjust their locations based on a nearest neighbor rule. Both approaches are tested based on their ability to interpolate manipulator joint coordinates for simulated arm movement while simultaneously performing stereo fusion of the camera data. Comparisons are made with classical k-nearest neighbor pattern recognition techniques.

Jorgensen, Charles C.↗
NASA NTRS1993

The analysis of convolutional codes via the extended Smith algorithm

Convolutional codes have been the central part of most error-control systems in deep-space communication for many years. Almost all such applications, however, have used the restricted class of (n,1), also known as 'rate 1/n,' convolutional codes. The more general class of (n,k) convolutional codes contains many potentially useful codes, but their algebraic theory is difficult and has proved to be a stumbling block in the evolution of convolutional coding systems. In this article, the situation is improved by describing a set of practical algorithms for computing certain basic things about a convolutional code (among them the degree, the Forney indices, a minimal generator matrix, and a parity-check matrix), which are usually needed before a system using the code can be built. The approach is based on the classic Forney theory for convolutional codes, together with the extended Smith algorithm for polynomial matrices, which is introduced in this article.

Mceliece, R. J.↗
NASA NTRS1988

Effects of nonlinear damping in flexible space structures

The classical Krylov-Bogoliubov "averaging" technique is used to study a class of nonlinear damping models, for which the damping force is proportional to the product of positive integer or fractional power of absolute values of displacement and that of velocity. The results are compared with linear viscous damping models. The amplitude decrement of free vibration for a single mode system with nonlinear models depends not only on damping ratio, but also on the initial amplitude, the time to measure the response, frequency of the system, and the powers of displacement and velocity. For the distributed system, the action of nonlinear damping is found to reduce energy of the system as well as to pass energy to higher modes. Experimental evidence such as in Spacecraft Control Laboratory Experiment seems to support the need for nonlinear models.

Hu, Anren↗
NASA NTRS1994

A psychologist's view of validating aviation systems

All systems, no matter what they are designed to do, have shortcomings that may make them less productive than was hoped during the initial development. Such shortcomings can arise at any stage of development: from conception to the end of the implementation life cycle. While systems failure and errors of a lesser magnitude can occur as a function of mechanical or software breakdown, the majority of such problems, in aviation are usually laid on the shoulders of the human operator and, to a lesser extent, on human factors. The operator bears the responsibility and blame even though, from a human factors perspective, error may have been designed into the system. Human factors is not a new concept in aviation. The name may be new, but the issues related to operators in the loop date back to the industrial revolution of the nineteenth century and certainly to the aviation build-up for World War I. During this first global confrontation, military services from all sides discovered rather quickly that poor selection and training led to drastically increased personnel losses. While hardware design became an issue later, the early efforts were primarily focused on increased care in pilot selection and on their training. This actually involved early labor-intensive simulation, using such devices as sticks and chairs mounted on rope networks which could be manually moved in response to control input. The use of selection criteria and improved training led to more viable person-machine systems. More pilots survived training and their first ten missions in the air, a rule of thumb arrived at by experience which predicted ultimate survival better than any other. This rule was to hold through World War II. At that time, personnel selection and training became very sophisticated based on previous standards. Also, many psychologists were drafted into Army Air Corps programs which were geared towards refining the human factor. However, despite the talent involved in these programs and the tremendous build-up of aviation during the war, there were still aircraft designs that were man killers (no sexism implied since all combat pilots were men). One classic design error that was identified fifty years ago was the multipointer altimeter, which could easily be misread especially by a pilot under considerable task load. It has led to flying fully operational aircraft into the terrain. The authors of the research which formally identified this problem put 'Human Errors' in quotes to express their dissatisfaction with the traditional approach to accident investigation. It traditionally places the burden of guilt on the operator. Some of these altimeters still exist in older aircraft to this day.

Stein, Earl S.↗
NASA NTRS2012

Approaching the Ultimate Limits of Communication Efficiency with a Photon-Counting Detector

Coherent states achieve the Holevo capacity of a pure-loss channel when paired with an optimal measurement, but a physical realization of this measurement is as of yet unknown, and it is also likely to be of high complexity. In this paper, we focus on the photon-counting measurement and study the photon and dimensional efficiencies attainable with modulations over classical- and nonclassical-state alphabets. We first review the state-of-the-art coherent on-off-keying (OOK) with a photoncounting measurement, illustrating its asymptotic inefficiency relative to the Holevo limit. We show that a commonly made Poisson approximation in thermal noise leads to unbounded photon information efficiencies, violating the conjectured Holevo limit. We analyze two binary-modulation architectures that improve upon the dimensional versus photon efficiency tradeoff achievable with conventional OOK. We show that at high photon efficiency these architectures achieve an efficiency tradeoff that differs from the best possible tradeoff--determined by the Holevo capacity--by only a constant factor. The first architecture we analyze is a coherent-state transmitter that relies on feedback from the receiver to control the transmitted energy. The second architecture uses a single-photon number-state source.

thermal noises↗
NASA NTRS1992

Thermal Control Surfaces Experiment (TCSE) materials analysis

Materials on the Thermal Control Surfaces Experiment (TCSE) underwent changes in their properties during the 5.8 years of exposure to the combined space environment. The analysis of these materials is continuing and current results are discussed. Some materials were significantly degraded such as silver Teflon, S13GLO, and Z302; while others such as Z93 and YB71 were stable. Time dependent flight data is provided, along with preflight and postflight measurements. Results are compared with other experiments, demonstrating the atomic oxygen (AO) ram effect. Atomic oxygen texturing of silver Teflon varies with incident angle. Even indirect exposure to atomic oxygen appears to be involved with surface texturing on the interior of the TCSE. Localized contamination occurred at discrete locations both inside and on exterior surfaces. Most of the visible deposits were the result of the classical photo-enhanced contamination deposition. The synergism of combined space environmental effects is demonstrated by the specific localization of contamination caused by photo-enhanced deposition versus AO removal. Optical degradation measurements of these contaminated areas, utilizing a new portable reflectometer, are presented.

Wilkes, Donald R.↗
NASA NTRS2000

High Speed Vortex Flows

A review of the research conducted at the National Aeronautics and Space Administration (NASA), Langley Research Center (LaRC) into high-speed vortex flows during the 1970s, 1980s, and 1990s is presented. The data reviewed is for flat plates, cavities, bodies, missiles, wings, and aircraft. These data are presented and discussed relative to the design of future vehicles. Also presented is a brief historical review of the extensive body of high-speed vortex flow research from the 1940s to the present in order to provide perspective of the NASA LaRC's high-speed research results. Data are presented which show the types of vortex structures which occur at supersonic speeds and the impact of these flow structures to vehicle performance and control is discussed. The data presented shows the presence of both small- and large scale vortex structures for a variety of vehicles, from missiles to transports. For cavities, the data show very complex multiple vortex structures exist at all combinations of cavity depth to length ratios and Mach number. The data for missiles show the existence of very strong interference effects between body and/or fin vortices and the downstream fins. It was shown that these vortex flow interference effects could be both positive and negative. Data are shown which highlights the effect that leading-edge sweep, leading-edge bluntness, wing thickness, location of maximum thickness, and camber has on the aerodynamics of and flow over delta wings. The observed flow fields for delta wings (i.e. separation bubble, classical vortex, vortex with shock, etc.) are discussed in the context of' aircraft design. And data have been shown that indicate that aerodynamic performance improvements are available by considering vortex flows as a primary design feature. Finally a discussing of a design approach for wings which utilize vortex flows for improved aerodynamic performance at supersonic speed is presented.

Wood, Richard M.↗
NASA NTRS2000

The Cirrus Parcel Model Comparison Project

The cirrus Parcel Model Comparison Project involves the systematic comparison of current models of ice crystal nucleation and growth for specified, typical, cirrus cloud environments. In Phase 1 of the project reported here, simulated cirrus cloud microphysical properties are compared for situations of "warm" (-40 C) and "cold" (-60 C) cirrus subject to updrafts of 4, 20 and 100 centimeters per second, respectively. Five models are participating in the project. These models employ explicit microphysical schemes wherein the size distribution of each class of particles (aerosols and ice crystals) is resolved into bins. Simulations are made including both homogeneous and heterogeneous ice nucleation mechanisms. A single initial aerosol population of sulfuric acid particles is prescribed for all simulations. To isolate the treatment of the homogeneous freezing (of haze drops) nucleation process, the heterogeneous nucleation mechanism is disabled for a second parallel set of simulations. Qualitative agreement is found amongst the models for the homogeneous-nucleation-only simulations, e.g., the number density of nucleated ice crystals increases with the strength of the prescribed updraft. However, non-negligible quantitative differences are found. Systematic bias exists between results of a model based on a modified classical theory approach and models using an effective freezing temperature approach to the treatment of nucleation. Each approach is constrained by critical freezing data from laboratory studies. This information is necessary, but not sufficient, to construct consistent formulae for the two approaches. Large haze particles may deviate considerably from equilibrium size in moderate to strong updrafts (20-100 centimeters per second) at -60 C when the commonly invoked equilibrium assumption is lifted. The resulting difference in particle-size-dependent solution concentration of haze particles may significantly affect the ice nucleation rate during the initial nucleation interval. The uptake rate for water vapor excess by ice crystals is another key component regulating the total number of nucleated ice crystals. This rate, the product of ice number concentration and ice crystal diffusional growth rate, partially controls the peak nucleation rate achieved in an air parcel and the duration of the active nucleation time period.

Lin, Ruei-Fong↗
NASA NTRS1993

Simulation of sloshing dynamics induced forces and torques actuated on dewar container driven by gravity gradient and jitter accelerations in microgravity

Some experimental spacecraft use superconducting sensors for gyro read-out and so must be maintained at a very low temperature. The boil-off from the cryogenic liquid used to cool the sensors can also be used, as the Gravity Probe B (GP-B) spacecraft does, as propellant to maintain attitude control and drag-free operation of the spacecraft. The cryogenic liquid for such spacecraft is, however, susceptible to both slosh-like motion and non-axisymmetric configurations under the influence of various kinds of gravity jitter and gravity gradient accelerations. Hence, it is important to quantify the magnitude of the liquid-induced perturbations on the spacecraft. We use the example of the GP-B to investigate such perturbations by numerical simulations. For this spacecraft disturbances can be imposed on the liquid by atmospheric drag, spacecraft attitude control maneuvers, and the earth's gravity gradient. More generally, onboard machinery vibrations and crew motion can also create disturbances. Recent studies suggest that high frequency disturbances are relatively unimportant in causing liquid motions in comparison to low frequency ones. The results presented here confirm this conclusion. After an initial calibration period, the GP-B spacecraft rotates in orbit at 0.1 rpm about the tank symmetry axis. For this rotation rate, the equilibrium liquid free surface shape is a 'doughnut' configuration for all residual gravity levels of 10(exp -6) g(sub 0) or less, as shown by experiments and by numerical simulations; furthermore, the superfluid behavior of the 1.8 K liquid helium used in GP-B eliminates temperature gradients and therefore such effects as Marangoni convection do not have to be considered. Classical fluid dynamics theory is used as the basis of the numerical simulations here, since Mason's experiments show that the theory is applicable for cryogenic liquid helium in large containers. To study liquid responses to various disturbances, we investigate and simulate three levels of gravity jitter (10(exp -6), 10(exp -7), and 10(exp -8) g(sub 0)) each at three predominant frequencies (0.1, 1.0, and 10 Hz), combined with a gravity gradient appropriate for the GP-B orbit. Dynamical evolution of sloshing dynamics excited fluid forces and torque fluctuations exerted on the dewar container driven by the combined gravity gradient and jitter accelerations are also investigated and simulated.

Hung, R. J.↗
NASA NTRS1998

Fiber-Supported Droplet Combustion Experiment-2

A major portion of the energy produced in the world today comes from the burning of liquid hydrocarbon fuels in the form of droplets. Understanding the fundamental physical processes involved in droplet combustion is not only important in energy production but also in propulsion, in the mitigation of combustion-generated pollution, and in the control of the fire hazards associated with handling liquid combustibles. Microgravity makes spherically symmetric combustion possible, allowing investigators to easily validate their droplet models without the complicating effects of gravity. The Fiber-Supported Droplet Combustion (FSDC-2) investigation was conducted in the Microgravity Glovebox facility of the shuttles' Spacelab during the reflight of the Microgravity Science Laboratory (MSL- 1R) on STS-94 in July 1997. FSDC-2 studied fundamental phenomena related to liquid fuel droplet combustion in air. Pure fuels and mixtures of fuels were burned as isolated single and duo droplets with and without forced air convection. FSDC-2 is sponsored by the NASA Lewis Research Center, whose researchers are working in cooperation with several investigators from industry and academia. The rate at which a droplet burns is important in many commercial applications. The classical theory of droplet burning assumes that, for an isolated, spherically symmetric, single-fuel droplet, the gas-phase combustion processes are much faster than the droplet surface regression rate and that the liquid phase is at a uniform temperature equal to the boiling point. Recent, more advanced models predict that both the liquid and gas phases are unsteady during a substantial portion of the droplet's burning history, thus affecting the instantaneous and average burning rates, and that flame radiation is a dominant mechanism that can extinguish flames in a microgravity environment. FSDC-2 has provided well-defined, symmetric droplet burning data including radiative emissions to validate these theoretical models for heptane, decane, ethanol, and methanol fuels. Since most commercial combustion systems burn droplets in a convective environment, data were obtained without and with convective flow over the burning droplet (see the following photos).

Colantonio, Renato O.↗
NASA NTRS1998

Phylogenetic mapping of bacterial morphology

The availability of a meaningful molecular phylogeny for bacteria provides a context for examining the historical significance of various developments in bacterial evolution. Herein, the classical morphological descriptions of selected members of the domain Bacteria are mapped upon the genealogical ancestry deduced from comparison of small-subunit rRNA sequences. For the species examined in this study, a distinct pattern emerges which indicates that the coccus shape has arisen and accumulated independently multiple times in separate lineages and typically survived as a persistent end-state morphology. At least two other morphologies persist but have evolved only once. This study demonstrates that although bacterial morphology is not useful in defining bacterial phylogeny, it is remarkably consistent with that phylogeny once it is known. An examination of the experimental evidence available for morphogenesis as well as microbial fossil evidence corroborates these findings. It is proposed that the accumulation of persistent morphologies is a result of the biophysical properties of peptidoglycan and their genetic control, and that an evolved body-plan strategy based on peptidoglycan may have been a fate-sealing step in the evolution of Bacteria. More generally, this study illustrates that significant evolutionary insights can be obtained by examining biological and biochemical data in the context of a reliable phylogenetic structure.

NASA Discipline Exobiology↗
NASA NTRS2003

Self-Supervised Dynamical Systems

Some progress has been made in a continuing effort to develop mathematical models of the behaviors of multi-agent systems known in biology, economics, and sociology (e.g., systems ranging from single or a few biomolecules to many interacting higher organisms). Living systems can be characterized by nonlinear evolution of probability distributions over different possible choices of the next steps in their motions. One of the main challenges in mathematical modeling of living systems is to distinguish between random walks of purely physical origin (for instance, Brownian motions) and those of biological origin. Following a line of reasoning from prior research, it has been assumed, in the present development, that a biological random walk can be represented by a nonlinear mathematical model that represents coupled mental and motor dynamics incorporating the psychological concept of reflection or self-image. The nonlinear dynamics impart the lifelike ability to behave in ways and to exhibit patterns that depart from thermodynamic equilibrium. Reflection or self-image has traditionally been recognized as a basic element of intelligence. The nonlinear mathematical models of the present development are denoted self-supervised dynamical systems. They include (1) equations of classical dynamics, including random components caused by uncertainties in initial conditions and by Langevin forces, coupled with (2) the corresponding Liouville or Fokker-Planck equations that describe the evolutions of probability densities that represent the uncertainties. The coupling is effected by fictitious information-based forces, denoted supervising forces, composed of probability densities and functionals thereof. The equations of classical mechanics represent motor dynamics that is, dynamics in the traditional sense, signifying Newton s equations of motion. The evolution of the probability densities represents mental dynamics or self-image. Then the interaction between the physical and metal aspects of a monad is implemented by feedback from mental to motor dynamics, as represented by the aforementioned fictitious forces. This feedback is what makes the evolution of probability densities nonlinear. The deviation from linear evolution can be characterized, in a sense, as an expression of free will. It has been demonstrated that probability densities can approach prescribed attractors while exhibiting such patterns as shock waves, solitons, and chaos in probability space. The concept of self-supervised dynamical systems has been considered for application to diverse phenomena, including information-based neural networks, cooperation, competition, deception, games, and control of chaos. In addition, a formal similarity between the mathematical structures of self-supervised dynamical systems and of quantum-mechanical systems has been investigated.

Zak, Michail↗
NASA NTRS2017

The Quantum Approximation Optimization Algorithm for MaxCut: A Fermionic View

Farhi et al. recently proposed a class of quantum algorithms, the Quantum Approximate Optimization Algorithm (QAOA), for approximately solving combinatorial optimization problems. A level-p QAOA circuit consists of steps in which a classical Hamiltonian, derived from the cost function, is applied followed by a mixing Hamiltonian. The 2p times for which these two Hamiltonians are applied are the parameters of the algorithm. As p increases, however, the parameter search space grows quickly. The success of the QAOA approach will depend, in part, on finding effective parameter-setting strategies. Here, we analytically and numerically study parameter setting for QAOA applied to MAXCUT. For level-1 QAOA, we derive an analytical expression for a general graph. In principle, expressions for higher p could be derived, but the number of terms quickly becomes prohibitive. For a special case of MAXCUT, the Ring of Disagrees, or the 1D antiferromagnetic ring, we provide an analysis for arbitrarily high level. Using a Fermionic representation, the evolution of the system under QAOA translates into quantum optimal control of an ensemble of independent spins. This treatment enables us to obtain analytical expressions for the performance of QAOA for any p. It also greatly simplifies numerical search for the optimal values of the parameters. By exploring symmetries, we identify a lower-dimensional sub-manifold of interest; the search effort can be accordingly reduced. This analysis also explains an observed symmetry in the optimal parameter values. Further, we numerically investigate the parameter landscape and show that it is a simple one in the sense of having no local optima.

quantum algorithm↗