Grover's Quantum Search Algorithm and Diophantine Approximation

In a fundamental paper [Phys. Rev. Lett. 78, 325 (1997)] Grover showed how a quantum computer can find a single marked object in a database of size N by using only O(N^{1/2}) queries of the oracle that identifies the object. His result was generalized to the case of finding one object in a subset of marked elements. We consider the following computational problem: A subset of marked elements is given whose number of elements is either M or K, M<K, our task is to determine which is the case. We show how to solve this problem with a high probability of success using only iterations of Grover's basic step (and no other algorithm). Let m be the required number of iterations; we prove that under certain restrictions on the sizes of M and K the estimation m < (2N^{1/2})/(K^{1/2}-M^{1/2}) obtains. This bound sharpens previous results and is known to be optimal up to a constant factor. Our method involves simultaneous Diophantine approximations, so that Grover's algorithm is conceptualized as an orbit of an ergodic automorphism of the torus. We comment on situations where the algorithm may be slow, and note the similarity between these cases and the problem of small divisors in classical mechanics.Comment: 8 pages, revtex, Title change

Deviation from Snell's Law for Beams Transmitted Near the Critical Angle: Application to Microcavity Lasers

We show that when a narrow beam is incident upon a dielectric interface near the critical angle for total internal reflection it will be transmitted into the far-field with an angular deflection from the direction predicted by Snell's Law, due to a phenomenon we call "Fresnel Filtering". This effect can be quite large for the parameter range relevant to dielectric microcavity lasers.Comment: 4 pages, 3 figures (eps), RevTeX 3.1, to be published in Optics Letter

Continuous atom laser with Bose-Einstein condensates involving three-body interactions

We demonstrate, through numerical simulations, the emission of a coherent continuous matter wave of constant amplitude from a Bose-Einstein Condensate in a shallow optical dipole trap. The process is achieved by spatial control of the variations of the scattering length along the trapping axis, including elastic three body interactions due to dipole interactions. In our approach, the outcoupling mechanism are atomic interactions and thus, the trap remains unaltered. We calculate analytically the parameters for the experimental implementation of this CW atom laser.Comment: 11 pages, 4 figure

A generalized light-driven model of community transitions along coral reef depth gradients

Aim: Coral reefs shift between distinct communities with depth throughout the world. Yet, despite over half a century of research on coral reef depth gradients, researchers have not addressed the driving force of these patterns. We present a theoretical, process-based model of light’s influence on the shallow to mesophotic reef transition as a single quantitative framework. We also share an interactive web application. Moving beyond depth as an ecological proxy will enhance research conducted on deeper coral reefs. Location: Global; subtropical and tropical coral reefs, oligotrophic and turbid coastal waters. Time period: Present day (2020). Major taxa: Scleractinia. Methods: We constructed ordinary differential equations representing the preferred light environments of shallow and mesophotic Scleractinia. We projected these as depth bands using light attenuation coefficients from around the world, and performed a sensitivity analysis. Results: We found light relationships alone are sufficient to capture major ecological features across coral reef depth gradients. Our model supports the depth limits currently used in coral reef ecology, predicting a global range for the shallow-upper mesophotic boundary at 36.1 m ± 5.6 and the upper-lower mesophotic boundary at 61.9 m ± 9.6. However, our model allows researchers to move past these fixed depth limits, and quantitatively predict the depths of reef zones in locations around the world. Main conclusions: The use of depth as a proxy for changes in coral reef communities offers no guidance for environmental variation between sites. We have shown it is possible to use light to predict the depth boundaries of reef zones as a continuous variable, and to accommodate this variability. Predicting the depths of reef zones in unusual light environments suggests that shallow-water turbid reefs should be considered as mesophotic coral ecosystems. Nonetheless, the current depth-based heuristics are relatively accurate at a global level

A Survey on Approximation Mechanism Design without Money for Facility Games

In a facility game one or more facilities are placed in a metric space to serve a set of selfish agents whose addresses are their private information. In a classical facility game, each agent wants to be as close to a facility as possible, and the cost of an agent can be defined as the distance between her location and the closest facility. In an obnoxious facility game, each agent wants to be far away from all facilities, and her utility is the distance from her location to the facility set. The objective of each agent is to minimize her cost or maximize her utility. An agent may lie if, by doing so, more benefit can be obtained. We are interested in social choice mechanisms that do not utilize payments. The game designer aims at a mechanism that is strategy-proof, in the sense that any agent cannot benefit by misreporting her address, or, even better, group strategy-proof, in the sense that any coalition of agents cannot all benefit by lying. Meanwhile, it is desirable to have the mechanism to be approximately optimal with respect to a chosen objective function. Several models for such approximation mechanism design without money for facility games have been proposed. In this paper we briefly review these models and related results for both deterministic and randomized mechanisms, and meanwhile we present a general framework for approximation mechanism design without money for facility games