Finite Ensemble Research Articles

QCD deconfinement transition line up to μB=400 MeV from finite volume lattice simulations

The QCD crossover line in the temperature (T)-baryochemical potential (μB) plane has been computed by several lattice groups by calculating the chiral order parameter and its susceptibility at finite values of μB. In this work we focus on the deconfinement aspect of the transition between hadronic and quark gluon plasma phases. We define the deconfinement temperature as the peak position of the static quark entropy [SQ(T,μB)] in T, which is based on the renormalized Polyakov loop. We extrapolate SQ(T,μB) based on high statistics finite temperature ensembles on a 163×8 lattice to finite density by means of a Taylor expansion to eighth order in μB (NNNLO) along the strangeness neutral line. For the simulations the 4HEX staggered action was used with 2+1 flavors at physical quark masses. In this setup the phase diagram can be drawn up to unprecedentedly high chemical potentials. Our results for the deconfinement temperature are in rough agreement with phenomenological estimates of the freeze-out curve in relativistic heavy ion collisions. In addition, we study the width of the deconfinement crossover. We show that up to μB≈400 MeV, the deconfinement transition gets broader at higher densities, disfavoring the existence of a deconfinement critical endpoint in this range. Finally, we examine the transition line without the strangeness neutrality condition and observe a hint for the narrowing of the crossover towards large μB. Published by the American Physical Society 2024

Corrected generalized cross-validation for finite ensembles of penalized estimators

Abstract Generalized cross-validation (GCV) is a widely used method for estimating the squared out-of-sample prediction risk that employs scalar degrees of freedom adjustment (in a multiplicative sense) to the squared training error. In this paper, we examine the consistency of GCV for estimating the prediction risk of arbitrary ensembles of penalized least-squares estimators. We show that GCV is inconsistent for any finite ensemble of size greater than one. Towards repairing this shortcoming, we identify a correction that involves an additional scalar correction (in an additive sense) based on degrees of freedom adjusted training errors from each ensemble component. The proposed estimator (termed CGCV) maintains the computational advantages of GCV and requires neither sample splitting, model refitting, or out-of-bag risk estimation. The estimator stems from a finer inspection of the ensemble risk decomposition and two intermediate risk estimators for the components in this decomposition. We provide a non-asymptotic analysis of the CGCV and the two intermediate risk estimators for ensembles of convex penalized estimators under Gaussian features and a linear response model. Furthermore, in the special case of ridge regression, we extend the analysis to general feature and response distributions using random matrix theory, which establishes model-free uniform consistency of CGCV.

On questions of uniqueness for the vacant set of Wiener sausages and Brownian interlacements

We consider connectivity properties of the vacant set of (random) ensembles of Wiener sausages in Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^d$$\\end{document} in the transient dimensions d≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 3$$\\end{document}. We prove that the vacant set of Brownian interlacements contains at most one infinite connected component almost surely. For finite ensembles of Wiener sausages, we provide sharp polynomial bounds on the probability that their vacant set contains at least 2 connected components in microscopic balls. The main proof ingredient is a sharp polynomial bound on the probability that several Brownian motions visit jointly all hemiballs of the unit ball while avoiding a slightly smaller ball.

Optimal artificial boundary conditions based on second-order correctors for three dimensional random elliptic media

We are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale l in an infinite heterogeneous medium, in a situation where the medium is only known in a box of diameter L ≫ l around the support of the charge. We propose a boundary condition that with overwhelming probability is (near) optimal with respect to scaling in terms of l and L, in the setting where the medium is a sample from a stationary ensemble with a finite range of dependence (set to be unity and with the assumption that l ≫ 1 ). The boundary condition is motivated by quantitative stochastic homogenization that allows for a multipole expansion. This work extends, the algorithm in which is optimal in two dimension, and thus we need to take quadrupoles, next to dipoles, into account. This in turn relies on stochastic estimates of second-order, next to first-order, correctors. These estimates are provided for finite range ensembles under consideration, based on an extension of the semi-group approach of Gloria and Otto.

Tracking optimal feedback control under uncertain parameters

Optimal control problems of tracking type for a class of linear systems with uncertain parameters in the dynamics are investigated. An affine tracking feedback control input is obtained by considering the minimization of an energy-like functional depending on a finite ensemble of training/sample parameters. It is computed from the solution of an associated differential Riccati equation. Simulations are presented showing the tracking performance of the computed input for trained as well as untrained parameters.

Robust Extraction of Thermal Observables from State Sampling and Real-Time Dynamics on Quantum Computers

Simulating properties of quantum materials is one of the most promising applications of quantum computation, both near- and long-term. While real-time dynamics can be straightforwardly implemented, the finite temperature ensemble involves non-unitary operators that render an implementation on a near-term quantum computer extremely challenging. Recently, Lu, Bañuls and Cirac \cite{Lu2021} suggested a "time-series quantum Monte Carlo method" which circumvents this problem by extracting finite temperature properties from real-time simulations via Wick's rotation and Monte Carlo sampling of easily preparable states. In this paper, we address the challenges associated with the practical applications of this method, using the two-dimensional transverse field Ising model as a testbed. We demonstrate that estimating Boltzmann weights via Wick's rotation is very sensitive to time-domain truncation and statistical shot noise. To alleviate this problem, we introduce a technique that imposes constraints on the density of states, most notably its non-negativity, and show that this way, we can reliably extract Boltzmann weights from noisy time series. In addition, we show how to reduce the statistical errors of Monte Carlo sampling via a reweighted version of the Wolff cluster algorithm. Our work enables the implementation of the time-series algorithm on present-day quantum computers to study finite temperature properties of many-body quantum systems.

Emergence of super-Poissonian light from indistinguishable single-photon emitters

Photon indistinguishability constitutes an essential resource in modern physics. At the scale of individual atoms and photons, it is a diverse concept that causes different coherent phenomena. We present the experimental characterization of indistinguishable photons detected in a single mode from a finite and stable ensemble of single-photon emitters: trapped ions. We conclusively observe an increase of second-order correlation and super-Poissonian statistics of these photons with a number of contributing phase-incoherent independent emissions from the atoms, ranging from a single to up to several hundred. This optical emission regime provides insight into the emergence of super-Poissonian light at the atomic scale by photon indistinguishability. It constitutes a unique toolbox for its generation and control at the most microscopic level.

Degenerate parametric down-conversion facilitated by exciton-plasmon polariton states in a nonlinear plasmonic cavity

We study the effect of degenerate parametric down-conversion (DPDC) in an ensemble of two-level quantum emitters (QEs) coupled via near-field interactions to a single surface plasmon (SP) mode of a nonlinear plasmonic cavity. For this purpose, we develop a quantum driven-dissipative model capturing non-equilibrium dynamics of the system in which incoherently pumped QEs have transition frequency tuned near the second-harmonic response of the SPs. Considering the strong coupling regime, i.e. the SP-QE interaction rate exceeds system dissipation rates, we find a critical SP-QE coupling attributed to the phase transition between normal and lasing steady states. Examining fluctuations above the system’s steady states, we predict new elementary excitations, namely, the exciton-plasmon polaritons formed by the two-SP quanta and single-exciton states of QEs. The contribution of two-SP quanta results in the linear scaling of the SP-QE interaction rate with the number of QEs, , as opposed to the -scaling known for the Dicke and Tavis–Cummings models. We further examine how SP-QE interaction scaling affects the polariton dispersions and power spectra in the vicinity of the critical coupling. For this purpose, we compare the calculation results assuming a finite ensemble of QEs and the model thermodynamic limit. The calculated power spectra predict an interplay of coherent photon emission by QEs near the second-harmonic frequency and correlated photon-pair emission at the fundamental frequency by the SPs (i.e. the photonic DPDC effect).

Quantum neural networks force fields generation

Accurate molecular force fields are of paramount importance for the efficient implementation of molecular dynamics techniques at large scales. In the last decade, machine learning (ML) methods have demonstrated impressive performances in predicting accurate values for energy and forces when trained on finite size ensembles generated with ab initio techniques. At the same time, quantum computers have recently started to offer new viable computational paradigms to tackle such problems. On the one hand, quantum algorithms may notably be used to extend the reach of electronic structure calculations. On the other hand, quantum ML is also emerging as an alternative and promising path to quantum advantage. Here we follow this second route and establish a direct connection between classical and quantum solutions for learning neural network (NN) potentials. To this end, we design a quantum NN architecture and apply it successfully to different molecules of growing complexity. The quantum models exhibit larger effective dimension with respect to classical counterparts and can reach competitive performances, thus pointing towards potential quantum advantages in natural science applications via quantum ML.

Towards an Information Description of Space-Time

We attempt to describe geometry in terms of informational quantities for the universe considered as a finite ensemble of correlated quantum particles. As the main dynamical principle, we use the conservation of the sum of all kinds of entropies: thermodynamic, quantum and informational. The fundamental constant of speed is interpreted as the information velocity for the world ensemble and also connected with the gravitational potential of the universe on a particle. The two postulates, which are enough to derive the whole theory of Special Relativity, are re-formulated as the principles of information entropy universality and finiteness of information density.

Multiobjective Risk-Aware Path Planning in Uncertain Transient Currents: An Ensemble-Based Stochastic Optimization Approach

A multiobjective, risk-aware framework is developed for optimal path planning of autonomous underwater vehicles operating in uncertain current. The uncertainty in the current is described in terms of a finite ensemble of flow realizations. A new optimization framework is proposed that accounts for the full variability of the ensemble in a single optimization problem whose solution may not necessarily coincide with any of the optimal deterministic paths corresponding to individual ensemble members. We formulate stochastic problemsthat aim to minimize a risk measure of the travel time or energy consumption, using a flexible methodology that enables the user to seamlessly explore various objectives, ranging from risk neutral to risk averse. We illustrate the application of the proposed approach using two case studies based on synthetic 2-D settings, and one case involving a real-world problem in the Gulf of Aden. The results are analyzed to assess the value of stochastic solution, guide the selection of suitable risk measures, and demonstrate the impact of the risk measures on the resulting path and on the distribution of travel times.

A New Scheme of Adaptive Covariance Inflation for Ensemble Filtering Data Assimilation

Due to the model and sampling errors of the finite ensemble, the background ensemble spread becomes small and the error covariance is underestimated during filtering for data assimilation. Because of the constraint of computational resources, it is difficult to use a large ensemble size to reduce sampling errors in high-dimensional real atmospheric and ocean models. Here, based on Bayesian theory, we explore a new spatially and temporally varying adaptive covariance inflation algorithm. To increase the statistical presentation of a finite background ensemble, the prior probability of inflation obeys the inverse chi-square distribution, and the likelihood function obeys the t distribution, which are used to obtain prior or posterior covariance inflation schemes. Different ensemble sizes are used to compare the assimilation quality with other inflation schemes within both the perfect and biased model frameworks. With two simple coupled models, we examined the performance of the new scheme. The results show that the new inflation scheme performed better than existing schemes in some cases, with more stability and fewer assimilation errors, especially when a small ensemble size was used in the biased model. Due to better computing performance and relaxed demand for computational resources, the new scheme has more potential applications in more comprehensive models for prediction initialization and reanalysis. In a word, the new inflation scheme performs well for a small ensemble size, and it may be more suitable for large-scale models.

About the computation of finite temperature ensemble averages of hybrid quantum-classical systems with molecular dynamics

Molecular or condensed matter systems are often well approximated by hybrid quantum-classical models: the electrons retain their quantum character, whereas the ions are considered to be classical particles. We discuss various alternative approaches for the computation of equilibrium (canonical) ensemble averages for observables of these hybrid quantum-classical systems through the use of molecular dynamics (MD)-i.e. by performing dynamics in the presence of a thermostat and computing time-averages over the trajectories. Often, in classical or ab initio MD, the temperature of the electrons is ignored and they are assumed to remain at the instantaneous ground state given by each ionic configuration during the evolution. Here, however, we discuss the general case that considers both classical and quantum subsystems at finite temperature canonical equilibrium. Inspired by a recent formal derivation for the canonical ensemble for quantum classical hybrids, we discuss previous approaches found in the literature, and provide some new formulas.

On construction of finite averaging sets for via its Cartan decomposition

Averaging physical quantities over Lie groups appears in many contexts across the rapidly developing branches of physics like quantum information science or quantum optics. Such an averaging process can be always represented as averaging with respect to a finite number of elements of the group, called a finite averaging set. In the previous research such sets, known as t-designs, were constructed only for the case of averaging over unitary groups (hence the name unitary t -designs). In this work we investigate the problem of constructing finite averaging sets for averaging over general non-compact matrix Lie groups, which is much more subtle task due to the fact that the the uniform invariant measure on the group manifold (the Haar measure) is infinite. We provide a general construction of such sets based on the Cartan decomposition of the group, which splits the group into its compact and non-compact components. The averaging over the compact part can be done in a uniform way, whereas the averaging over the non-compact one has to be endowed with a suppressing weight function, and can be approached using generalized Gauss quadratures. This leads us to the general form of finite averaging sets for semisimple matrix Lie groups in the product form of finite averaging sets with respect to the compact and non-compact parts. We provide an explicit calculation of such sets for the group , although our construction can be applied to other cases. Possible applications of our results cover finding finite ensembles of random operations in quantum information science and quantum optics, which can be used in constructions of randomised quantum algorithms, including optical interferometric implementations.

Applicability of Squeezed- and Coherent-State Continuous-Variable Quantum Key Distribution over Satellite Links.

We address the applicability of quantum key distribution with continuous-variable coherent and squeezed states over long-distance satellite-based links, considering low Earth orbits and taking into account strong varying channel attenuation, atmospheric turbulence and finite data ensemble size effects. We obtain tight security bounds on the untrusted excess noise on the channel output, which suggest that substantial efforts aimed at setup stabilization and reduction of noise and loss are required, or the protocols can be realistically implemented over satellite links once either individual or passive collective attacks are assumed. Furthermore, splitting the satellite pass into discrete segments and extracting the key from each rather than from the overall single pass allows one to effectively improve robustness against the untrusted channel noise and establish a secure key under active collective attacks. We show that feasible amounts of optimized signal squeezing can substantially improve the applicability of the protocols allowing for lower system clock rates and aperture sizes and resulting in higher robustness against channel attenuation and noise compared to the coherent-state protocol.

Spatial Temporal Anisotropic Transport of the Microalgae Chlorella Vulgaris in the Microfluidic Channel

Transport of biomaterials in confined geometry exhibit complex dynamics, both in spatial and temporal scales. In this work, we examined the transport behavior of microalgae Chlorella Vulgaris inside the microfluidic channel under pressure-driven Poiseuille flow environment. The microalgae system is treated as the spherical naturally buoyant particles. The algae particles trajectories are visually traced using particle imaging techniques and the sample paths are resolved in stream wise (flow) and perpendicular directions. In order to understand boundary wall effects on the flow, we partitioned the microfluidic channel into three different regions, namely the center region and two near-wall boundary regions based on the velocity flow profile. Time-averaged mean square displacement (MSD), probability density function (PDF), skewness, and kurtosis of finite ensembles of particle trajectories are determined for these regions. In addition to the spatial dependencies, we also examined the transient characteristics of the algae transport at early-time and long-time. We found the existence of the mixed types for transport dynamics irrespective of flow region separation often assumed in many laminar flow simulations. This finding will be useful for optimization of mixing of algae culture in micro-scale photobiore actor as the productivity of cell growth depends critically on the cell dispersion or transport

Correlated dynamics of fermionic impurities induced by the counterflow of an ensemble of fermions

We investigate the nonequilibrium quantum dynamics of a single and two heavy fermionic impurities being harmonically trapped and repulsively interacting with a finite ensemble of majority fermions. A quench of the potential of the majority species from a double-well to a harmonic trap is applied, enforcing its counterflow which in turn perturbs the impurities. For weak repulsions it is shown that the mixture undergoes a periodic mixing and demixing dynamics, while stronger interactions lead to a more pronounced dynamical spatial separation. In the presence of correlations the impurity exhibits an expansion dynamics which is absent in the Hartree-Fock case resulting in an enhanced degree of miscibility. We generalize our results to different impurity masses and demonstrate that the expansion amplitude of the impurity reduces for a larger mass. Furthermore, we showcase that the majority species is strongly correlated and a phase separation occurs on the two-body level. Most importantly, signatures of attractive impurity-impurity induced interactions mediated by the majority species are identified in the time-evolution of the two-body correlations of the impurities, a result that is supported by inspecting their spatial size.

Operator limit of the circular beta ensemble

We provide a precise coupling of the finite circular beta ensembles and their limit process via their operator representations. We prove explicit bounds on the distance of the operators and the corresponding point processes. We also prove an estimate on the beta-dependence of the $\operatorname{Sine}_{\beta }$ process.

Spectral distributions of periodic random matrix ensembles

Koloğlu, Kopp and Miller compute the limiting spectral distribution of a certain class of real random matrix ensembles, known as [Formula: see text]-block circulant ensembles, and discover that it is exactly equal to the eigenvalue distribution of an [Formula: see text] Gaussian unitary ensemble. We give a simpler proof that under very general conditions which subsume the cases studied by Koloğlu–Kopp–Miller, real-symmetric ensembles with periodic diagonals always have limiting spectral distribution equal to the eigenvalue distribution of a finite Hermitian ensemble with Gaussian entries which is a ‘complex version’ of a [Formula: see text] submatrix of the ensemble. We also prove an essentially algebraic relation between certain periodic finite Hermitian ensembles with Gaussian entries, and the previous result may be seen as an asymptotic version of this for real-symmetric ensembles. The proofs show that this general correspondence between periodic random matrix ensembles and finite complex Hermitian ensembles is elementary and combinatorial in nature.

Twisted States in a System of Nonlinearly Coupled Phase Oscillators

We study the dynamics of the ring of identical phase oscillators with nonlinear nonlocal coupling. Using the Ott–Antonsen approach, the problem is formulated as a system of partial derivative equations for the local complex order parameter. In this framework, we investigate the existence and stability of twisted states. Both fully coherent and partially coherent stable twisted states were found (the latter ones for the first time for identical oscillators). We show that twisted states can be stable starting from a certain critical value of the medium length, or on a length segment. The analytical results are confirmed with direct numerical simulations in finite ensembles.