Nonlinear Perturbation Theory Research Articles

Near-term distributed quantum computation using mean-field corrections and auxiliary qubits

Distributed quantum computation is often proposed to increase the scalability of quantum hardware, as it reduces cooperative noise and requisite connectivity by sharing quantum information between distant quantum devices. However, such exchange of quantum information itself poses unique engineering challenges, requiring high gate fidelity and costly non-local operations. To mitigate this, we propose near-term distributed quantum computing, focusing on approximate approaches that involve limited information transfer and conservative entanglement production. We first devise an approximate distributed computing scheme for the time evolution of quantum systems split across any combination of classical and quantum devices. Our procedure harnesses mean-field corrections and auxiliary qubits to link two or more devices classically, optimally encoding the auxiliary qubits to both minimize short-time evolution error and extend the approximate scheme’s performance to longer evolution times. We then expand the scheme to include limited quantum information transfer through selective qubit shuffling or teleportation, broadening our method’s applicability and boosting its performance. Finally, we build upon these concepts to produce an approximate circuit-cutting technique for the fragmented pre-training of variational quantum algorithms. To characterize our technique, we introduce a non-linear perturbation theory that discerns the critical role of our mean-field corrections in optimization and may be suitable for analyzing other non-linear quantum techniques. This fragmented pre-training is remarkably successful, reducing algorithmic error by orders of magnitude while requiring fewer iterations.

Cosmic-Eν: An- emulator for the non-linear neutrino power spectrum

ABSTRACT Cosmology is poised to measure the neutrino mass sum Mν and has identified several smaller-scale observables sensitive to neutrinos, necessitating accurate predictions of neutrino clustering over a wide range of length scales. The FlowsForTheMasses non-linear perturbation theory for the the massive neutrino power spectrum, $\Delta ^2_\nu (k)$, agrees with its companion N-body simulation at the $10~{{\ \rm per\ cent}}-15~{{\ \rm per\ cent}}$ level for k ≤ 1 h Mpc−1. Building upon the Mira-Titan IV emulator for the cold matter, we use FlowsForTheMasses to construct an emulator for $\Delta ^2_\nu (k)$, Cosmic-Eν, which covers a large range of cosmological parameters and neutrino fractions Ων, 0h2 ≤ 0.01 (Mν ≤ 0.93 eV). Consistent with FlowsForTheMasses at the 3.5 per cent level, it returns a power spectrum in milliseconds. Ranking the neutrinos by initial momenta, we also emulate the power spectra of momentum deciles, providing information about their perturbed distribution function. Comparing a Mν = 0.15 eV model to a wide range of N-body simulation methods, we find agreement to 3 per cent for k ≤ 3kFS = 0.17 h Mpc−1 and to 19 per cent for k ≤ 0.4 h Mpc−1. We find that the enhancement factor, the ratio of $\Delta ^2_\nu (k)$ to its linear-response equivalent, is most strongly correlated with Ων, 0h2, and also with the clustering amplitude σ8. Furthermore, non-linearities enhance the free-streaming-limit scaling $\partial \log (\Delta ^2_\nu /\Delta ^2_{\rm m}) / \partial \log (M_\nu)$ beyond its linear value of 4, increasing the Mν-sensitivity of the small-scale neutrino density.

Non-linear CMB lensing with neutrinos and baryons: FLAMINGO simulations versus fast approximations

ABSTRACT Weak lensing of the cosmic microwave background is rapidly emerging as a powerful probe of neutrinos, dark energy, and new physics. We present a fast computation of the non-linear CMB lensing power spectrum that combines non-linear perturbation theory at early times with power spectrum emulation using cosmological simulations at late times. Comparing our calculation with light-cones from the FLAMINGO 5.6 Gpc cube dark-matter-only simulation, we confirm its accuracy to $1{{\ \rm per\ cent}}$ ($2{{\ \rm per\ cent}}$) up to multipoles L = 3000 (L = 5000) for a νΛCDM cosmology consistent with current data. Clustering suppression due to small-scale baryonic phenomena such as feedback from active galactic nuclei can reduce the lensing power by $\sim 10{{\ \rm per\ cent}}$. To our perturbation theory and emulator-based calculation, we add SP(k), a new fitting function for this suppression, and confirm its accuracy compared to the FLAMINGO hydrodynamic simulations to $4{{\ \rm per\ cent}}$ at L = 5000, with similar accuracy for massive neutrino models. We further demonstrate that scale-dependent suppression due to neutrinos and baryons approximately factorize, implying that a careful treatment of baryonic feedback can limit biasing neutrino mass constraints.

Flows for the masses: A multi-fluid non-linear perturbation theory for massive neutrinos

Velocity dispersion of the massive neutrinos presents a daunting challenge for non-linear cosmological perturbation theory. We consider the neutrino population as a collection of non-linear fluids, each with uniform initial momentum, through an extension of the Time Renormalization Group perturbation theory. Employing recently-developed Fast Fourier Transform techniques, we accelerate our non-linear perturbation theory by more than two orders of magnitude, making it quick enough for practical use. After verifying that the neutrino mode-coupling integrals and power spectra converge, we show that our perturbation theory agrees with N-body neutrino simulations to within 10% for neutrino fractions Ω ν,0 h 2 ≤ 0.005 up to wave numbers of k = 1 h/Mpc, an accuracy consistent with ≤ 2.5% errors in the neutrino mass determination. Non-linear growth represents a > 10% correction to the neutrino power spectrum even for density fractions as low as Ω ν,0 h 2 = 0.001, demonstrating the limits of linear theory for accurate neutrino power spectrum predictions. Our code FlowsForTheMasses is avaliable online at github.com/upadhye/FlowsForTheMasses.

Resonant dynamics and the instability of the box Minkowski model

We revisit the box Minkowski model [M. Maliborski, Instability of Flat Space Enclosed in a Cavity, Phys. Rev. Lett. 109, 221101 (2012).] and provide a strong argument that, subject to the Dirichlet boundary condition, it is unstable toward black hole formation for arbitrarily small generic perturbations. Using weakly nonlinear perturbation theory, we derive the resonant system, which compared to systems with the anti--de Sitter asymptotics has extra resonant terms, and study its properties, including conserved quantities. We find that the generic solution of the resonant system becomes singular in finite time. Surprisingly, the additional resonant interactions do not significantly affect the singular evolution. Furthermore, we find that the interaction coefficients take a relatively simple form, making this a particularly attractive toy model of turbulent gravitational instability.

Fisher matrix for the one-loop galaxy power spectrum: measuring expansion and growth rates without assuming a cosmological model

We introduce a methodology to extend the Fisher matrix forecasts to mildly non-linear scales without the need of selecting a cosmological model. We make use of standard non-linear perturbation theory for biased tracers complemented by counterterms, and assume that the cosmological distances can be measured accurately with standard candles. Instead of choosing a specific model, we parametrize the linear power spectrum and the growth rate in several k and z bins. We show that one can then obtain model-independent constraints of the expansion rate E(z) = E(z)/H 0 and the growth rate f(k,z), besides the bias functions. We apply the technique to both Euclid and DESI public specifications in the range 0.6 ≤ z ≤ 1.8 and show that the gain in precision when going from k max = 0.1 to 0.2 h/Mpc is around two- to threefold, while it reaches four- to ninefold when extending to k max = 0.3 h/Mpc. In absolute terms, with k max = 0.2 h/Mpc, one can reach high precision on E(z) at each z-shell: 8–10% for DESI with Δz = 0.1, 5–6% for Euclid with Δz = 0.2–0.3. This improves to 1–2% if the growth rate f is taken to be k-independent. The growth rate itself has in general much weaker constraints, unless assumed to be k-independent, in which case the gain is similar to the one for E(z) and uncertainties around 5–15% can be reached at each z-bin. We also discuss how neglecting the non-linear corrections can have a large effect on the constraints even for k max = 0.1 h/Mpc, unless one has independent strong prior information on the non-linear parameters.

K2 Photometry on Oscillation Mode Variability: The New Pulsating Hot B Subdwarf Star EPIC 220422705

We present an analysis of oscillation mode variability in the hot B subdwarf star EPIC 220422705, a new pulsator discovered from ∼78 days of K2 photometry. The high-quality light curves provide a detection of 66 significant independent frequencies, from which we identified nine incomplete potential triplets and three quintuplets. Those g- and p-multiplets give rotation periods of ∼36 and 29 days in the core and at the surface, respectively, potentially suggesting a slightly differential rotation. We derived a period spacing of 268.5 s and 159.4 s for the sequence of dipole and quadrupole modes, respectively. We characterized the precise patterns of amplitude and frequency modulations (AM and FM) of 22 frequencies with high enough amplitude for our science. Many of them exhibit intrinsic and periodic patterns of AM and FM, with periods on a timescale of months as derived by the best fitting and Markov Chain Monte Carlo test. The nonlinear resonant mode interactions could be a natural interpretation for such AMs and FMs after other mechanisms are ruled out. Our results are the first step to building a bridge between mode variability from K2 photometry and the nonlinear perturbation theory of stellar oscillation.

Current-Limiting Virtual Synchronous Control and Stability Analysis Considering DC-Link Dynamics Under Normal and Faulty Grid Conditions

An improved nonlinear virtual synchronous control for three-phase grid-connected inverters, that can maintain a reliable operation under both normal and faulty grid conditions, i.e. balanced grid voltage sags, is proposed. The proposed controller can ensure a desired RMS current limitation at all times, provide virtual inertia and damping via the DC-link voltage and AC system frequency coupling, and realize the desired real and reactive power regulation without requiring accurate knowledge of the system parameters. Opposed to the conventional methods that use saturated PI controllers with or without anti-windup techniques to limit the reference value of the inverter current, the proposed controller includes a nonlinear bounded integrator, which limits the actual value (instead of the reference) of the inverter RMS current and leads to a fast system recovery even after significant grid voltage sags. The closed-loop stability of entire system is rigorously proven using nonlinear singular perturbation theory. Moreover, analytic conditions for the controller parameter selection to guarantee the stability of entire inverter system with the DC-link dynamics are provided. To prove the effectiveness of proposed controller and its superior performance compared to the traditional approaches, extensive Matlab/Simulink-based simulations are performed, followed by Typhoon-HIL hardware-in-the loop implementation using a TI microcontroller.

Minkowski functionals and the nonlinear perturbation theory in the large-scale structure: Second-order effects

The second-order formula of Minkowski functionals in weakly non-Gaussian fields is compared with the numerical $N$-body simulations. Recently, the weakly non-Gaussian formula of Minkowski functionals is extended to include the second-order effects of non-Gaussianity in general dimensions. We apply this formula to the three-dimensional density field in the large-scale structure of the Universe. The parameters of the second-order formula include several kinds of skewness and kurtosis parameters. We apply tree level of nonlinear perturbation theory to estimate these parameters, including novel calculations of quartic cumulants. First we compare the theoretical values with those of numerical simulations on the basis of parameter values, and next we test the performance of the analytic formula combined with the perturbation theory. The second-order formula outperforms the first-order formula in general. The performance of the perturbation theory depends on the smoothing radius applied in defining the Minkowski functionals. The quantitative comparisons are presented in detail.

Nonlinear dynamics of structure formation in protoplanetary disks

A nonlinear nonideal generalized magnetohydrodynamic (MHD) model is developed to explore the evolutionary dynamics of nonlinear wave-structural patterns naturalistically excitable in inhomogeneous protoplanetary disks (PPDs). A local nonlinear perturbation theory (multiscale analysis) over the magnetohydrostatic homogeneous equilibrium of the axisymmetric disk yields a second-order nonlinear partial differential equation in the perturbed gravitational potential curvature with a unique set of multiparametric coefficients. A numerical illustrative platform demonstrates the excitation of nonlinear coherent structures in the form of solitary lump waves (SLWs). The atypical SLW-characteristics are explored geometrically with the phase space trajectories. The SLW amplitude increases in magnitude with the Alfvenic Mach number, and vice-versa. It is interestingly inferred that the Mach number (reference frame velocity and density) acts as a destabilizing (stabilizing) agency to the PPD-dynamics against the nonlocal self-gravity effects, and so forth. It enlightens the long-standing gravitational Laplace 1796 conjecture having a real astronomic significance. At the last, the nontrivial applicability of our findings in the SLW-driven bounded planetary structure formation in varied astroenvirons is actualized.

Sine-Gordon on a wormhole

In an attempt to understand the soliton resolution conjecture, we consider the sine-Gordon equation on a spherically symmetric wormhole spacetime. We show that within each topological sector (indexed by a positive integer degree n) there exists a unique linearly stable soliton, which we call the n-kink. We give numerical evidence that the n-kink is a global attractor in the evolution of any smooth, finite energy solutions of degree n. When the radius of the wormhole throat a is large enough, the convergence to the n-kink is shown to be governed by internal modes that slowly decay due to the resonant transfer of energy to radiation. We compute the exact asymptotics of this relaxation process for the one-kink using the Soffer–Weinstein weakly nonlinear perturbation theory.

Countable zipper fractal interpolation and some elementary aspects of the associated nonlinear zipper fractal operator

This note aims to extend the notion of affine zipper fractal interpolation function from the case of a finite data set to an infinite sequence of data points. We work with a slightly more general setting wherein the assumption of affinity on the functions involved in the construction of the zipper fractal interpolant is dropped. Invoking the iterative functional equation for the countable zipper fractal interpolant, its stability with a perturbation of data points and sensitivity to perturbations in the maps that define the zipper are examined. In the second part of this note, the countable zipper fractal interpolation is used to obtain a parameterized family of zipper fractal functions corresponding to a prescribed real-valued Lipschitz continuous function on a closed bounded interval in $$\mathbb {R}$$ . An operator obtained by associating each Lipschitz continuous function to its fractal counterpart is approached from the standpoint of nonlinear functional analysis and perturbation theory of operators.

Nonlinear perturbation theory extension of the Boltzmann code CLASS

We present a new open-source code that calculates one-loop power auto- and cross-power spectra for matter fields and biased tracers in real and redshift space. These spectra incorporate all ingredients required for a direct application to data: non-linear bias, redshift-space distortions, infra-red resummation, counterterms, and the Alcock-Paczynski effect. Our code is based on the Boltzmann solver CLASS and inherits its advantage: user friendliness, ease of modification, high speed, and simple interface with other software. We present detailed descriptions of the theoretical model, the code structure, approximations, and accuracy tests. A typical end-to-end run for one cosmology takes $\sim 0.3$ seconds, which is sufficient for Markov Chain Monte Carlo parameter extraction. As an example, we apply the code to data from the Baryon Oscillation Spectroscopic Survey (BOSS) and infer cosmological parameters from the shape of the galaxy power spectrum.

Velocity bias and the nonlinear perturbation theory of peaks

The biasing in the large-scale structure of the universe is a crucial problem in cosmological applications. The peaks model of biasing predicts a linear velocity bias of halos, which is not present in a simple model of local bias. We investigate the origin of the velocity bias in the peaks model from the viewpoint of the integrated perturbation theory, which is a nonlinear perturbation theory in the presence of general Lagrangian bias. The presence of the velocity bias in the peaks model is a consequence of the "flat constraint," ${\nabla}\delta = 0$; i.e., all the first spatial derivatives should vanish at the locations of peaks. We show that the velocity bias in the peaks model is systematically derived in the framework of the integrated perturbation theory, and then develop a formal theory to perturbatively trace the nonlinear evolution of biased objects with the flat constraint. A formula for the nonlinear velocity dispersion of peaks with the one-loop approximation is also derived.

Gravitational perturbations in a cavity: Nonlinearities

Motivated by recent studies of nonlinear perturbations of asymptotically anti-de Sitter spacetimes, we study gravitational perturbations of $(n+2)$ dimensional Minkowski spacetime with a spherical Dirichlet wall. By considering the tensor, vector and scalar perturbations on the $n$ sphere, we present simplified nonhomogeneous equations at arbitrary order in weakly nonlinear perturbation theory for each sector. A suitable choice of perturbative variables is required at higher orders to simplify the expression for the boundary conditions and to expand the variables in terms of linear order eigenfunctions. Finally we comment on the nonlinear stability of the system. Some of the tools used can easily be generalized to study nonlinear perturbations of anti-de Sitter spacetime.

Nonlinear perturbations of Reissner-Nordström black holes

We develop a nonlinear perturbation theory of Reissner-Nordstr\"om black holes. We show that, at each perturbation level, Einstein-Maxwell equations can be reduced to four inhomogeneous wave equations, two for polar and two for axial sector. Gravitational part of these equations is similar to Regge-Wheeler and Zerilli equations with source and additional coupling to the electromagnetic sector. We construct solutions to the inhomogeneous part of wave equations in terms of sources for Einstein-Maxwell equations. We discuss $\ell=0$ and $ \ell=1$ cases separately.

On the road to percent accuracy: non-linear reaction of the matter power spectrum to dark energy and modified gravity

ABSTRACT We present a general method to compute the non-linear matter power spectrum for dark energy (DE) and modified gravity scenarios with per cent-level accuracy. By adopting the halo model and non-linear perturbation theory, we predict the reaction of a lambda cold dark matter (ΛCDM) matter power spectrum to the physics of an extended cosmological parameter space. By comparing our predictions to N-body simulations we demonstrate that with no-free parameters we can recover the non-linear matter power spectrum for a wide range of different w0–wa DE models to better than 1 per cent accuracy out to k ≈ 1 $h \,{\rm Mpc}^{-1}$. We obtain a similar performance for both DGP and f(R) gravity, with the non-linear matter power spectrum predicted to better than 3 per cent accuracy over the same range of scales. When including direct measurements of the halo mass function from the simulations, this accuracy improves to 1 per cent. With a single suite of standard ΛCDM N-body simulations, our methodology provides a direct route to constrain a wide range of non-standard extensions to the concordance cosmology in the high signal-to-noise non-linear regime.

Neutrino mass and dark energy constraints from redshift-space distortions

Cosmology in the near future promises a measurement of the sum of neutrino masses ∑ mν, a fundamental Standard Model parameter, as well as substantially-improved constraints on the dark energy. We use the shape of the BOSS redshift-space galaxy power spectrum, in combination with CMB and supernova data, to constrain the neutrino masses and the dark energy. Essential to this calculation are several recent advances in non-linear cosmological perturbation theory, including fast Fourier transform methods, redshift space distortions, and scale-dependent growth. Our 95% confidence upper bound ∑ mν < 180 meV degrades substantially to ∑ mν < 540 meV when the dark energy equation of state and its first derivative are also allowed to vary, representing a significant challenge to current constraints. We also study the impact of additional galaxy bias parameters, finding that a greater allowed range of scale-dependent bias only slightly shifts the preferred ∑ mν, weakens its upper bound by ≈ 20%, and has a negligible effect on the other cosmological parameters.

Stability Analysis for Incremental Nonlinear Dynamic Inversion Control

As a sensor-based control method, incremental nonlinear dynamic inversion (INDI) has been applied to various aerospace systems and has shown desirable robust performance against aerodynamic model uncertainties. However, its previous derivation based on the time scale separation principle has some limitations. There is also a need for stability and robustness analysis for INDI. Therefore, this paper reformulates the INDI control law without using the time scale separation principle and generalizes it for systems with arbitrary relative degree, with consideration of the internal dynamics. The stability of the closed-loop system in the presence of external disturbances is analyzed using Lyapunov methods and nonlinear system perturbation theory. Moreover, the robustness of the closed-loop system against regular and singular perturbations is analyzed. Finally, this reformulated INDI control law is verified by a Monte Carlo simulation for an aircraft command tracking problem in the presence of external disturbances and model uncertainties.

Flexible Aircraft Gust Load Alleviation with Incremental Nonlinear Dynamic Inversion

This paper designs an incremental nonlinear dynamic inversion control law for free-flying flexible aircraft, which can regulate rigid-body motions, alleviate gust loads, reduce the wing root bending moment, and suppress elastic modes. By fully exploring the sensor measurements, the model dependency of the proposed control law can be reduced while maintaining desirable robustness, which simplifies the implementation process and reduces the onboard computational load. The elastic states are observed online from accelerometer measurements, with a Padé approximation to model the pure time delay. Theoretical analyses based on the Lyapunov methods and the nonlinear system perturbation theory show that the proposed control has inherent robustness to model uncertainties, external disturbances, and sudden actuator faults. These merits are demonstrated by time-domain simulations in various spatial turbulence and gust fields, as well as by a Monte Carlo study.