Finding a marked node on any graph by continuous-time quantum walk

Spatial search by discrete-time quantum walk can find a marked node on any ergodic, reversible Markov chain $P$ quadratically faster than its classical counterpart, i.e.\ in a time that is in the square root of the hitting time of $P$. However, in the framework of continuous-time quantum walks, it was previously unknown whether such general speed-up is possible. In fact, in this framework, the widely used quantum algorithm by Childs and Goldstone fails to achieve such a speedup. Furthermore, it is not clear how to apply this algorithm for searching any Markov chain $P$. In this article, we aim to reconcile the apparent differences between the running times of spatial search algorithms in these two frameworks. We first present a modified version of the Childs and Goldstone algorithm which can search for a marked element for any ergodic, reversible $P$ by performing a quantum walk on its edges. Although this approach improves the algorithmic running time for several instances, it cannot provide a generic quadratic speedup for any $P$. Secondly, using the framework of interpolated Markov chains, we provide a new spatial search algorithm by continuous-time quantum walk which can find a marked node on any $P$ in the square root of the classical hitting time. In the scenario where multiple nodes are marked, the algorithmic running time scales as the square root of a quantity known as the extended hitting time. Our results establish a novel connection between discrete-time and continuous-time quantum walks and can be used to develop a number of Markov chain-based quantum algorithms.Comment: This version deals only with new algorithms for spatial search by continuous-time quantum walk (CTQW) on ergodic, reversible Markov chains. Please see arXiv:2004.12686 for results on the necessary and sufficient conditions for the optimality of the Childs and Goldstone algorithm for spatial search by CTQ

Excess Worker Turnover and Fixed-Term Contracts: Causal Evidence in a Two-Tier System

Portuguese firms engage in intense reallocation, most employers simultaneously hire and separate from workers, resulting in high excess worker turnover flows. These flows are constrained by the employment protection gap between open-ended and fixed-term contracts. We explore a reform that increased the employment protection of open-ended contracts and generated a quasi-experiment. The causal evidence points to an increase in the share and in the excess turnover of fixed-term contracts in treated firms. The excess turnover of open-ended contracts remained unchanged. This result is consistent with a high degree of substitution between open-ended and fixed-term contracts. At the firm level, we also show that excess turnover is quite heterogeneous and quantify its association with firm, match, and worker characteristics.excess worker turnover, two-tier systems, quasi-experiment, fixed-term contracts

Weak topologies for Carath\'eodory differential equations. Continuous dependence, exponential Dichotomy and attractors

We introduce new weak topologies and spaces of Carath\'eodory functions where the solutions of the ordinary differential equations depend continuously on the initial data and vector fields. The induced local skew-product flow is proved to be continuous, and a notion of linearized skew-product flow is provided. Two applications are shown. First, the propagation of the exponential dichotomy over the trajectories of the linearized skew-product flow and the structure of the dichotomy or Sacker-Sell spectrum. Second, how particular bounded absorbing sets for the process defined by a Carath\'eodory vector field $f$ provide bounded pullback attractors for the processes with vector fields in the alpha-limit set, the omega-limit set or the whole hull of $f$. Conditions for the existence of a pullback or a global attractor for the skew-product semiflow, as well as application examples are also given.Comment: 34 page

When Supply Meets Demand: Wage Inequality in Portugal

Wage inequality in Portugal increased over the last quarter of century. The period from 1982 to 1995 witnessed strong increases in both upper- and lower-tail inequality. A shortage of skills combined with skill-biased technological changes are at the core of this evolution. Since 1995, lower-tail inequality decreased, while upper-tail inequality increased at a slower rate. The supply of high-skilled workers more than doubled during this period, contributing significantly to the slowdown. Polarization of employment demand is the more credible explanation for the more recent evolution. As in other developed economies, for instance Germany and the United States, we show that institutions played a minor role in shaping changes in inequality.inequality, polarization, supply, demand, institutions

Identifying Unemployment Insurance Income Effects with a Quasi-Natural Experiment

This paper acknowledges that UI has a non-distortionary income effect generated by easing the liquidity constraints of the unemployed. Using an exogenous increase in the entitlement period as a quasi-experimental setting, we find evidence of an important income effect. The extension of the entitlement period prolongs unemployment spells, but its effect is decreasing with the degree of liquidity constraints (indexed by wages quintiles). An exception to this pattern is the behavior of individuals in the first wages quintile. The fact that the most constrained individuals extend the least their unemployment spells conforms to the nonstationarity of the job search process. This result points to the possibility that the UI system may become regressive, benefiting significantly less those at the bottom of the wage distribution, who find it harder to benefit from extended UI entitlements.