With prevalent attacks in communication, sharing a secret between communicating parties is an ongoing challenge. Moreover, it is important to integrate quantum solutions with classical secret sharing schemes with low computational cost for the real world use. This paper proposes a novel hybrid threshold adaptable quantum secret sharing scheme, using an m-bonacci orbital angular momentum (OAM) pump, Lagrange interpolation polynomials, and reverse Huffman-Fibonacci-tree coding. To be exact, we employ entangled states prepared by m -bonacci sequences to detect eavesdropping. Meanwhile, we encode m -bonacci sequences in Lagrange interpolation polynomials to generate the shares of a secret with reverse Huffman-Fibonacci-tree coding. The advantages of the proposed scheme is that it can detect eavesdropping without joint quantum operations, and permits secret sharing for an arbitrary but no less than threshold-value number of classical participants with much lower bandwidth. Also, in comparison with existing quantum secret sharing schemes, it still works when there are dynamic changes, such as the unavailability of some quantum channel, the arrival of new participants and the departure of participants. Finally, we provide security analysis of the new hybrid quantum secret sharing scheme and discuss its useful features for modern applications.

Hybrid threshold adaptable quantum secret sharing scheme with reverse Huffman-Fibonacci-tree coding

Spectral methods have emerged as a simple yet surprisingly effective approach for extracting information from massive, noisy and incomplete data. In a nutshell, spectral methods refer to a collection of algorithms built upon the eigenvalues (resp. singular values) and eigenvectors (resp. singular vectors) of some properly designed matrices constructed from data. A diverse array of applications have been found in machine learning, data science, and signal processing. Due to their simplicity and effectiveness, spectral methods are not only used as a stand-alone estimator, but also frequently employed to initialize other more sophisticated algorithms to improve performance. 
While the studies of spectral methods can be traced back to classical matrix perturbation theory and methods of moments, the past decade has witnessed tremendous theoretical advances in demystifying their efficacy through the lens of statistical modeling, with the aid of non-asymptotic random matrix theory. This monograph aims to present a systematic, comprehensive, yet accessible introduction to spectral methods from a modern statistical perspective, highlighting their algorithmic implications in diverse large-scale applications. In particular, our exposition gravitates around several central questions that span various applications: how to characterize the sample efficiency of spectral methods in reaching a target level of statistical accuracy, and how to assess their stability in the face of random noise, missing data, and adversarial corruptions? In addition to conventional $\ell_2$ perturbation analysis, we present a systematic $\ell_{\infty}$ and $\ell_{2,\infty}$ perturbation theory for eigenspace and singular subspaces, which has only recently become available owing to a powerful "leave-one-out" analysis framework.

Spectral Methods for Data Science: A Statistical Perspective

We present a new approach to graph limit theory which unifies and generalizes the two most well developed directions, namely dense graph limits (even the more general $L^p$ limits) and Benjamini--Schramm limits (even in the stronger local-global setting). We illustrate by examples that this new framework provides a rich limit theory with natural limit objects for graphs of intermediate density. Moreover, it provides a limit theory for bounded operators (called $P$-operators) of the form $L^\infty(\Omega)\to L^1(\Omega)$ for probability spaces $\Omega$. We introduce a metric to compare $P$-operators (for example finite matrices) even if they act on different spaces. We prove a compactness result which implies that in appropriate norms, limits of uniformly bounded $P$-operators can again be represented by $P$-operators. We show that limits of operators representing graphs are self-adjoint, positivity-preserving $P$-operators called graphops. Graphons, $L^p$ graphons and graphings (known from graph limit theory) are special examples for graphops. We describe a new point of view on random matrix theory using our operator limit framework.

/pdf/action-convergence-of-operators-and-graphs-4l1gxka4za.pdf

Action convergence of operators and graphs

©2017 Frans A. Oliehoek, Matthijs T. J. Spaan, Bas Terwijn, Philipp Robbel and Joao V. Messias. This article describes the Multiagent Decision Process (MADP) Toolbox, a software library to support planning and learning for intelligent agents and multiagent systems in uncertain environments. Key features are that it supports partially observable environments and stochastic transition models; has unified support for single- and multiagent systems; provides a large number of models for decision-theoretic decision making, including one-shot and sequential decision making under various assumptions of observability and cooperation, such as Dec-POMDPs and POSGs; provides tools and parsers to quickly prototype new problems; provides an extensive range of planning and learning algorithms for single- and multiagent systems; is released under the GNU GPL v3 license; and is written in C++ and designed to be extensible via the object-oriented paradigm.

The MADP Toolbox: An Open-Source Library for Planning and Learning in (Multi-)Agent Systems.

We investigate two \emph{sparse} Sachdev-Ye-Kitaev (SYK) systems coupled by a bilinear term as a holographic quantum mechanical description of an eternal traversable wormhole in the low temperature limit. Each SYK system consists of $N$ Majorana fermions coupled by random $q$-body interactions. The degree of sparseness is captured by a regular hypergraph in such a way that the Hamiltonian contains exactly $k\,N$ independent terms. We improve on the theoretical understanding of the sparseness property by using known measures of hypergraph expansion. We show that the sparse version of the two coupled SYK model is gapped with a ground state close to a thermofield double state. Using Krylov subspace and parallelization techniques, we simulate the system for $q=4$ and $q=8.$ The sparsity of the model allows us to explore larger values of $N$ than the ones existing in the literature for the all-to-all SYK. We analyze in detail the two-point functions and the transmission amplitude of signals between the two systems. We identify a range of parameters where revivals \emph{obey the scaling predicted by holography} and signals can be interpreted as traversing the wormhole.

Sparse SYK and traversable wormholes

We prove a local law and eigenvector delocalization for general Wigner-type matrices. Our methods allow us to get the best possible interval length and optimal eigenvector delocalization in the dense case, and the first results of such kind for the sparse case down to p=g(n)log⁡nn with g(n) → ∞. We specialize our results to the case of the stochastic block model, and we also obtain a local law for the case when the number of classes is unbounded.

Sparse general Wigner-type matrices: Local law and eigenvector delocalization

We describe the non-backtracking spectrum of a stochastic block model with connection probabilities $p_{\mathrm{in}}, p_{\mathrm{out}} = \omega(\log n)/n$. In this regime we answer a question posed in Dall'Amico and al. (2019) regarding the existence of a real eigenvalue `inside' the bulk, close to the location $\frac{p_{\mathrm{in}}+ p_{\mathrm{out}}}{p_{\mathrm{in}}- p_{\mathrm{out}}}$. We also introduce a variant of the Bauer-Fike theorem well suited for perturbations of quadratic eigenvalue problems, and which could be of independent interest.

Eigenvalues of the non-backtracking operator detached from the bulk

We consider the spectral gap of a uniformly chosen random $(d_1,d_2)$-biregular bipartite graph $G$ with $|V_1|=n, |V_2|=m$, where $d_1,d_2$ could possibly grow with $n$ and $m$. Let $A$ be the adjacency matrix of $G$. Under the assumption that $d_1\geq d_2$ and $d_2=O(n^{2/3}),$ we show that $\lambda_2(A)=O(\sqrt{d_1})$ with high probability. As a corollary, combining the results from Tikhomirov and Youssef (2019), we confirm a conjecture in Cook (2017) that the second singular value of a uniform random $d$-regular digraph is $O(\sqrt{d})$ for $1\leq d\leq n/2$ with high probability. This also implies that the second eigenvalue of a uniform random $d$-regular digraph is $O(\sqrt{d})$ for $1\leq d\leq n/2$ with high probability. Assuming $d_2=O(1)$ and $d_1=O(n^2)$, we further prove that for a random $(d_1,d_2)$-biregular bipartite graph, $|\lambda_i^2(A)-d_1|=O(\sqrt{d_1(d_2-1)})$ for all $2\leq i\leq n+m-1$ with high probability. The proofs of the two results are based on the size biased coupling method introduced in Cook, Goldstein, and Johnson (2018) for random $d$-regular graphs and several new switching operations we defined for random bipartite biregular graphs.

On the second eigenvalue of random bipartite biregular graphs

Per- and polyfluoroalkyl acids (PFAAs) including polyfluoroalkyl carboxylic acids and polyfluoroalkyl sulfonic acids are a large category of crucial environmental pollutants of global concern. Besides known PFAAs, numerous unknown species may exist in the environment, urgently needing discovery and characterization. This study implemented nontarget analysis for a group of novel PFAA pollutants, viz., iodinated PFAAs (I-PFAAs) in wastewater from a fluorochemical manufacturing park by liquid chromatography-high-resolution mass spectrometry in combination with an iodine-specific data-processing algorithm. The algorithm took into account the diagnostic fragment iodine ion (I-) together with carbon and sulfur isotopologue distributions. In total, 18 I-PFAA formulas involving 21 congeners were identified. Semiquantification was conducted, and the total concentrations of I-PFAAs were 1.9-274.7 μg/L, indicating severe pollution of I-PFAAs in the wastewater. The determined concentrations along with predicted environmental behaviors and toxicities demonstrate that I-PFAAs merit further in-depth investigation. The analytical method including the instrumental analysis and data-processing algorithm can be extended to screening and identification of I-PFAAs in other matrices. Furthermore, the analysis results for the first time provide recognition on the occurrence, distribution features, and pollution status of I-PFAAs in the environment.

First Discovery of Iodinated Polyfluoroalkyl Acids by Nontarget Mass-Spectrometric Analysis and Iodine-Specific Screening Algorithm.

We consider the community detection problem in a sparse q-uniform hypergraph G, assuming that G is generated according to the Hypergraph Stochastic Block Model (HSBM). We prove that a spectral method based on the non-backtracking operator for hypergraphs works with high probability down to the generalized Kesten-Stigum detection threshold conjectured by Angelini et al. (2015). We characterize the spectrum of the non-backtracking operator for the sparse HSBM and provide an efficient dimension reduction procedure using the Ihara-Bass formula for hypergraphs. As a result, community detection for the sparse HSBM on n vertices can be reduced to an eigenvector problem of a 2n×2n non-normal matrix constructed from the adjacency matrix and the degree matrix of the hypergraph. To the best of our knowledge, this is the first provable and efficient spectral algorithm that achieves the conjectured threshold for HSBMs with r blocks generated according to a general symmetric probability tensor.

Sparse random hypergraphs: Non-backtracking spectra and community detection

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