Concentration Inequalities on the Multislice and for Sampling Without Replacement

We present concentration inequalities on the multislice which are based on (modified) log-Sobolev inequalities. This includes bounds for convex functions and multilinear polynomials. As an application, we show concentration results for the triangle count in the G(n, M) Erdős–Rényi model resembling known bounds in the G(n, p) case. Moreover, we give a proof of Talagrand’s convex distance inequality for the multislice. Interpreting the multislice in a sampling without replacement context, we furthermore present concentration results for n out of N sampling without replacement. Based on a bounded difference inequality involving the finite-sampling correction factor $$1 - (n / N)$$ 1 - ( n / N ) , we present an easy proof of Serfling’s inequality with a slightly worse factor in the exponent, as well as a sub-Gaussian right tail for the Kolmogorov distance between the empirical measure and the true distribution of the sample.

Efficient quantum measurement of Pauli operators in the presence of finite sampling error

Estimating the expectation value of an operator corresponding to an observable is a fundamental task in quantum computation. It is often impossible to obtain such estimates directly, as the computer is restricted to measuring in a fixed computational basis. One common solution splits the operator into a weighted sum of Pauli operators and measures each separately, at the cost of many measurements. An improved version collects mutually commuting Pauli operators together before measuring all operators within a collection simultaneously. The effectiveness of doing this depends on two factors. Firstly, we must understand the improvement offered by a given arrangement of Paulis in collections. In our work, we propose two natural metrics for quantifying this, operating under the assumption that measurements are distributed optimally among collections so as to minimise the overall finite sampling error. Motivated by the mathematical form of these metrics, we introduce SORTED INSERTION, a collecting strategy that exploits the weighting of each Pauli operator in the overall sum. Secondly, to measure all Pauli operators within a collection simultaneously, a circuit is required to rotate them to the computational basis. In our work, we present two efficient circuit constructions that suitably rotate any collection of k independent commuting n-qubit Pauli operators using at most kn−k(k+1)/2 and O(kn/log⁡k) two-qubit gates respectively. Our methods are numerically illustrated in the context of the Variational Quantum Eigensolver, where the operators in question are molecular Hamiltonians. As measured by our metrics, SORTED INSERTION outperforms four conventional greedy colouring algorithms that seek the minimum number of collections.

Coefficient-of-Determination Fourier Transform

This algorithm is designed to perform numerical transforms to convert data from the temporal domain into the spectral domain. This algorithm obtains the spectral magnitude and phase by studying the Coefficient of Determination of a series of artificial sinusoidal functions with the temporal data, and normalizing the variance data into a high-resolution spectral representation of the time-domain data with a finite sampling rate. What is especially beneficial about this algorithm is that it can produce spectral data at any user-defined resolution, and this highly resolved spectral data can be transformed back to the temporal domain.

Integrating Molecular Simulation and Experimental Data: A Bayesian/Maximum Entropy reweighting approach

We describe a Bayesian/Maximum entropy (BME) procedure and software to construct a conformational ensemble of a biomolecular system by integrating molecular simulations and experimental data. First, an initial conformational ensemble is constructed using for example Molecular Dynamics or Monte Carlo simulations. Due to potential inaccuracies in the model and finite sampling effects, properties predicted from simulations may not agree with experimental data. In BME we use the experimental data to refine the simulation so that the new conformational ensemble has the following properties: (i) the calculated averages are close to the experimental values taking uncertainty into account and (ii) it maximizes the relative Shannon entropy with respect to the original simulation ensemble. The output of this procedure is a set of optimized weights that can be used to calculate arbitrary properties and distributions. Here, we provide a practical guide on how to obtain and use such weights, how to choose adjustable parameters and discuss shortcomings of the method.

Iterative High-Accuracy Parameter Estimation of Uncooperative OFDM-LFM Radar Signals Based on FrFT and Fractional Autocorrelation Interpolation

To improve the parameter estimation performance of uncooperative Orthogonal Frequency Division Multi- (OFDM) Linear Frequency Modulation (LFM) radar signals, this paper proposes an iterative high-accuracy method, which is based on Fractional Fourier Transform (FrFT) and Fractional Autocorrelation (FA) interpolation. Two iterative estimators for rotation angle and center frequencies are derived from the analytical formulations of the OFDM-LFM signal. Both estimators are designed by measuring the residual terms between the quasi peak and the real peak in the fractional spectrum, which were obtained from the finite sampling data. Successful elimination of spectral leakage caused by multiple components of the OFDM-LFM signal is also proposed by a sequential removal of the strong coefficient in the fractional spectrum through an iterative process. The method flow is given and its superior performance is demonstrated by the simulation results.

A quantitative framework for investigating the reliability of network construction

ABSTRACTDescriptions of ecological networks typically assume that the same interspecific interactions occur each time a community is observed. This contrasts with the known stochasticity of ecological communities: community composition, species abundances, and link structure all vary in space and time. Moreover, finite sampling generates variation in the set of interactions actually observed. Here we develop the conceptual and analytical tools needed to capture uncertainty in the estimation of pairwise interactions. To define the problem, we identify the different contributions to the uncertainty of an interaction and its implications for the estimation of network properties. We then outline a framework to quantify the uncertainty around each interaction. We illustrate this framework using the most extensively sampled network to date. We found significant uncertainty in estimates for the probability of most pairwise interactions which we could, however, limit with informative priors. Through these efforts, we demonstrate the utility of our approach and the importance of acknowledging the uncertainty inherent in network studies. Most importantly, we stress that networks are best thought of as systems constructed from random variables, the stochastic nature of which must be acknowledged for an accurate representation. Doing so will fundamentally change networks analyses and yield greater realism.

Novel approaches to estimating the turbulent kinetic energy dissipation rate from low- and moderate-resolution velocity fluctuation time series

In this paper we propose two approaches to estimating the turbulent kinetic energy (TKE) dissipation rate, based on the zero-crossing method by Sreenivasan et al. (1983). The original formulation requires a fine resolution of the measured signal, down to the smallest dissipative scales. However, due to finite sampling frequency, as well as measurement errors, velocity time series obtained from airborne experiments are characterized by the presence of effective spectral cutoffs. In contrast to the original formulation the new approaches are suitable for use with signals originating from airborne experiments. The suitability of the new approaches is tested using measurement data obtained during the Physics of Stratocumulus Top (POST) airborne research campaign as well as synthetic turbulence data. They appear useful and complementary to existing methods. We show the number-of-crossings-based approaches respond differently to errors due to finite sampling and finite averaging than the classical power spectral method. Hence, their application for the case of short signals and small sampling frequencies is particularly interesting, as it can increase the robustness of turbulent kinetic energy dissipation rate retrieval.