Environment-assisted analog quantum search

Two main obstacles for observing quantum advantage in noisy intermediate-scale quantum computers (NISQ) are the finite precision effects due to control errors, or disorders, and decoherence effects due to thermal fluctuations. It has been shown that dissipative quantum computation is possible in presence of an idealized fully-engineered bath. However, it is not clear, in general, what performance can be achieved by NISQ when internal bath degrees of freedom are not controllable. In this work, we consider the task of quantum search of a marked node on a complete graph of $n$ nodes in the presence of both static disorder and non-zero coupling to an environment. We show that, given fixed and finite levels of disorder and thermal fluctuations, there is an optimal range of bath temperatures that can significantly improve the success probability of the algorithm. Remarkably for a fixed disorder strength $\sigma$, the system relaxation time decreases for higher temperatures within a robust range of parameters. In particular, we demonstrate that for strong disorder, the presence of a thermal bath increases the success probability from $1/(n \sigma^2)$ to at least $1/2$. While the asymptotic running time is approximately maintained, the need to repeat the algorithm many times and issues associated with unitary over-rotations can be avoided as the system relaxes to an absorbing steady state. Furthermore, we discuss for what regimes of disorder and bath parameters quantum speedup is possible and mention conditions for which similar phenomena can be observed in more general families of graphs. Our work highlights that in the presence of static disorder, even non-engineered environmental interactions can be beneficial for a quantum algorithm

Optimal Transportation Theory with Repulsive Costs

This paper intents to present the state of art and recent developments of the optimal transportation theory with many marginals for a class of repulsive cost functions. We introduce some aspects of the Density Functional Theory (DFT) from a mathematical point of view, and revisit the theory of optimal transport from its perspective. Moreover, in the last three sections, we describe some recent and new theoretical and numerical results obtained for the Coulomb cost, the repulsive harmonic cost and the determinant cost.Comment: Survey for the special volume for RICAM (Special Semester on New Trends in Calculus of Variations

Network connectivity tracking for a team of unmanned aerial vehicles

Algebraic connectivity is the second-smallest eigenvalue of the Laplacian matrix and can be used as a metric for the robustness and efficiency of a network. This connectivity concept applies to teams of multiple unmanned aerial vehicles (UAVs) performing cooperative tasks, such as arriving at a consensus. As a UAV team completes its mission, it often needs to control the network connectivity. The algebraic connectivity can be controlled by altering edge weights through movement of individual UAVs in the team, or by adding and deleting edges. The addition and deletion problem for algebraic connectivity, however, is NP-hard. The contributions of this work are 1) a comparison of four heuristic methods for modifying algebraic connectivity through the addition and deletion of edges, 2) a rule-based algorithm for tracking a connectivity profile through edge weight modification and the addition and deletion of edges, 3) a new, hybrid method for selecting the best edge to add or remove, 4) a distributed method for estimating the eigenvectors of the Laplacian matrix and selecting the best edge to add or remove for connectivity modification and tracking, and 5) an implementation of the distributed connectivity tracking using a consensus controller and double-integrator dynamics

Non-convex regularization in remote sensing

In this paper, we study the effect of different regularizers and their implications in high dimensional image classification and sparse linear unmixing. Although kernelization or sparse methods are globally accepted solutions for processing data in high dimensions, we present here a study on the impact of the form of regularization used and its parametrization. We consider regularization via traditional squared (2) and sparsity-promoting (1) norms, as well as more unconventional nonconvex regularizers (p and Log Sum Penalty). We compare their properties and advantages on several classification and linear unmixing tasks and provide advices on the choice of the best regularizer for the problem at hand. Finally, we also provide a fully functional toolbox for the community.Comment: 11 pages, 11 figure

Efficient and Globally Robust Causal Excursion Effect Estimation

Causal excursion effect (CEE) characterizes the effect of an intervention under policies that deviate from the experimental policy. It is widely used to study effect of time-varying interventions that have the potential to be frequently adaptive, such as those delivered through smartphones. We study the semiparametric efficient estimation of CEE and we derive a semiparametric efficiency bound for CEE with identity or log link functions under working assumptions. We propose a class of two-stage estimators that achieve the efficiency bound and are robust to misspecified nuisance models. In deriving the asymptotic property of the estimators, we establish a general theory for globally robust Z-estimators with either cross-fitted or non-cross-fitted nuisance parameters. We demonstrate substantial efficiency gain of the proposed estimator compared to existing ones through simulations and a real data application using the Drink Less micro-randomized trial