Propagation and smoothing of shocks in alternative social security systems

Even with well-developed capital markets, there is no private market mechanism for trading between current and future generations. This generates a potential role for public old-age pension systems to spread economic and demographic shocks among different generations. This paper evaluates how different systems smooth and propagate shocks to productivity, fertility, mortality and migration in a realistic OLG model. We use reductions in the variance of wealth equivalents to measure performance, starting with the existing U.S. system as a unifying framework, in which we vary how much taxes and benefits adjust, and which we then compare to the existing German and Swedish systems. We find that system design and shock type are key factors. The German system and the benefit-adjustment-only U.S. system best smooth productivity shocks, which are by far the most important shocks. Overall, the German system performs best, while the Swedish system, which includes a buffer stock to relax annual budget constraints, performs rather poorly. Focusing on the U.S. system, reliance solely on tax adjustment fares best for mortality and migration shocks, while equal reliance on tax and benefit adjustments is best for fertility shocks

Binary Component Decomposition. Part I: The Positive-Semidefinite Case

This paper studies the problem of decomposing a low-rank positive-semidefinite matrix into symmetric factors with binary entries, either {±1} or {0,1}. This research answers fundamental questions about the existence and uniqueness of these decompositions. It also leads to tractable factorization algorithms that succeed under a mild deterministic condition. A companion paper addresses the related problem of decomposing a low-rank rectangular matrix into a binary factor and an unconstrained factor

Binary component decomposition. Part II: The asymmetric case

This paper studies the problem of decomposing a low-rank matrix into a factor with binary entries, either from {±1} or from {0,1}, and an unconstrained factor. The research answers fundamental questions about the existence and uniqueness of these decompositions. It also leads to tractable factorization algorithms that succeed under a mild deterministic condition. This work builds on a companion paper that addresses the related problem of decomposing a low-rank positive-semidefinite matrix into symmetric binary factors

Transition states and greedy exploration of the QAOA optimization landscape

The QAOA is a variational quantum algorithm, where a quantum computer implements a variational ansatz consisting of p layers of alternating unitary operators and a classical computer is used to optimize the variational parameters. For a random initialization the optimization typically leads to local minima with poor performance, motivating the search for initialization strategies of QAOA variational parameters. Although numerous heuristic intializations were shown to have a good numerical performance, an analytical understanding remains evasive. Inspired by the study of energy landscapes, in this work we focus on so-called transition states (TS) that are saddle points with a unique negative curvature direction that connects to local minima. Starting from a local minimum of QAOA with p layers, we analytically construct 2p + 1 TS for QAOA with p + 1 layers. These TS connect to new local minima, all of which are guaranteed to lower the energy compared to the minimum found for p layers. We introduce a Greedy procedure to effectively maneuver the exponentially increasing number of TS and corresponding local minima. The performance of our procedure matches the best available initialization strategy, and in addition provides a guarantee for the minimal energy to decrease with an increasing number of layers p. Generalization of analytic TS and the Greedy approach to other ans\"atze may provide a universal framework for initialization of variational quantum algorithms.Comment: 5 pages, 4 figures, comments are welcom

Avoiding barren plateaus using classical shadows

Variational quantum algorithms are promising algorithms for achieving quantum advantage on nearterm devices. The quantum hardware is used to implement a variational wave function and measure observables, whereas the classical computer is used to store and update the variational parameters. The optimization landscape of expressive variational ansätze is however dominated by large regions in parameter space, known as barren plateaus, with vanishing gradients, which prevents efficient optimization. In this work we propose a general algorithm to avoid barren plateaus in the initialization and throughout the optimization. To this end we define a notion of weak barren plateaus (WBPs) based on the entropies of local reduced density matrices. The presence of WBPs can be efficiently quantified using recently introduced shadow tomography of the quantum state with a classical computer. We demonstrate that avoidance of WBPs suffices to ensure sizable gradients in the initialization. In addition, we demonstrate that decreasing the gradient step size, guided by the entropies allows WBPs to be avoided during the optimization process. This paves the way for efficient barren plateau-free optimization on near-term devices

Laser linewidth determination in the sub-Megahertz range using a Brillouin fibre laser

The Brillouin fibre laser has experimentally demonstrated its suitability for the measurement of laser linewidth in the 1 kHz to 500 kHz frequency range. A major advantage of this system is to be widely independent of the source wavelength by generating the uncorrelated reference signal at a fixed frequency below the source under test. This solution is simple and affordable, since it only requires widely available standard optical supplies. The measured spectra of a compact diode-pumped Nd:YAG laser and a spectrally-narrowed semiconductor laser are show