A body of work using the continuous wavelet transform has been growing. We provide a self-contained summary on its most relevant theoretical results, describe how such transforms can be implemented in practice, and generalize the concept of simple coherency to partial wavelet coherency and multiple wavelet coherency, moving beyond bivariate analysis. We also describe a family of wavelets, which emerges as an alternative to the popular Morlet wavelet, the generalized Morse wavelets. A user-friendly toolbox, with examples, is attached to this paper.

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The continuous wavelet transform: moving beyond uni- and bivariate analysis

Predicting properties of complex, large-scale quantum systems is essential for developing quantum technologies. We present an efficient method for constructing an approximate classical description of a quantum state using very few measurements of the state. This description, called a classical shadow, can be used to predict many different properties: order $\log M$ measurements suffice to accurately predict $M$ different functions of the state with high success probability. The number of measurements is independent of the system size, and saturates information-theoretic lower bounds. Moreover, target properties to predict can be selected after the measurements are completed. We support our theoretical findings with extensive numerical experiments. We apply classical shadows to predict quantum fidelities, entanglement entropies, two-point correlation functions, expectation values of local observables, and the energy variance of many-body local Hamiltonians. The numerical results highlight the advantages of classical shadows relative to previously known methods.

/pdf/predicting-many-properties-of-a-quantum-system-from-very-few-5853kau3z2.pdf

Predicting Many Properties of a Quantum System from Very Few Measurements

On the assessment of robustness

Like differentiability, convexity is a natural and powerful property of functions that plays a significant role in many areas of mathematics, both pure and applied. It ties together notions from topology, algebra, geometry and analysis, and is an important tool in optimization, mathematical programming and game theory. This book, which is the product of a collaboration of over 15 years, is unique in that it focuses on convex functions themselves, rather than on convex analysis. The authors explore the various classes and their characteristics and applications, treating convex functions in both Euclidean and Banach spaces. The book can either be read sequentially for a graduate course, or dipped into by researchers and practitioners. Each chapter contains a variety of specific examples, and over 600 exercises are included, ranging in difficulty from early graduate to research level.

/pdf/convex-functions-constructions-characterizations-and-3ohe487199.pdf

Convex Functions: Constructions, Characterizations and Counterexamples

We describe and analyze a 3-state one-way population protocol to compute approximate majority in the model in which pairs of agents are drawn uniformly at random to interact. Given an initial configuration of x’s, y’s and blanks that contains at least one non-blank, the goal is for the agents to reach consensus on one of the values x or y. Additionally, the value chosen should be the majority non-blank initial value, provided it exceeds the minority by a sufficient margin. We prove that with high probability n agents reach consensus in O(n log n) interactions and the value chosen is the majority provided that its initial margin is at least $${\omega(\sqrt{n} \,{\rm log}\, n)}$$
 . This protocol has the additional property of tolerating Byzantine behavior in $${o(\sqrt{n})}$$
 of the agents, making it the first known population protocol that tolerates Byzantine agents.

/pdf/a-simple-population-protocol-for-fast-robust-approximate-3652eowe15.pdf

A simple population protocol for fast robust approximate majority

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https://epubs.siam.org/doi/pdf/10.1137/1.9781611972429.fm

Sol I. Rubinow

Papers

Cbms-nsf regional conference series in applied mathematics