Query complexity for searching multiple marked states from an unsorted database

An important and usual problem is to search all states we want from a database with a large number of states. In such, recall is vital. Grover's original quantum search algorithm has been generalized to the case of multiple solutions, but no one has calculated the query complexity in this case. We will use a generalized algorithm with higher precision to solve such a search problem that we should find all marked states and show that the practical query complexity increases with the number of marked states. In the end we will introduce an algorithm for the problem on a ``duality computer'' and show its advantage over other algorithms.Comment: 4 pages,4 figures,twocolum

Doctor of Philosophy

dissertationThis dissertation presents experimental and computational investigations of electrolyte negative differential resistance, nanoparticle dynamics in nanopores, and nanobubble formation at nanoelectrodes. Chapter 1 provides an introduction to negative differential resistance and other nonlinear electrical responses in nanopores, an overview of resistive pulse analysis of nanoparticles using nanopores, and current nanobubble research. Chapter 2 describes the first example of electrolyte negative differential resistance (NDR) discovered in nanopores, where the current decreases as the voltage is increased. The NDR turn-on voltage was found to be tunable over a ~1 V window by adjusting the applied external pressure. Finite-element simulations yielded predictions of the NDR behavior that are in qualitative agreement with the experimental observations. Chapter 3 presents the extension of NDR to an aqueous system and demonstrates the potential for chemical sensing based on NDR behavior. Solution pH and Ca2+ in the solution were separately employed as the stimulus to investigate the surface charge density dependence of the NDR behavior. The NDR turn-on voltage was found to be exceedingly sensitive to the nanopore surface charge density, suggesting possible analytical applications in detecting as few as several hundred of molecules. Chapter 4 discusses the technique of controlling the dynamics of single 8 nm diameter gold nanoparticles in nanopores, which is extended from traditional resistive pulse analysis of nanoparticles. A pressure was applied to balance electrokinetic forces acting on the charged Au nanoparticles as they translocate through a ~10 nm diameter orifice at an electric field. This force balance provides a means to vary the velocity of nanoparticles by three orders of magnitude. Finite-element simulations yielded predictions in semiquantitative agreement with the experimental results. Chapter 5 reports the electrochemical generation of individual H2 nanobubbles at Pt nanodisk electrodes immersed in a H2SO4 solution. A sudden drop in current associated with the transport-limited reduction of protons was observed in the i-V response at Pt nanodisk electrodes of radii less than 50 nm. Finite element simulation based on Fick's first law, combined with the Young-Laplace equation and Henry's Law, were employed to investigate the bubble formation and its stabilization mechanism

Small data global regularity for simplified 3-D Ericksen-Leslie's compressible hyperbolic liquid crystal model

In this article, we consider the Ericksen-Leslie's hyperbolic system for compressible liquid crystal model in three spatial dimensions. Global regularity for small and smooth initial data near equilibrium is proved for the case that the system is a nonlinear coupling of compressible Navier-Stokes equations with wave map to $\mathbb{S}^2$. Our argument is a combination of vector field method and Fourier analysis. The main strategy to prove global regularity relies on an interplay between the control of high order energies and decay estimates, which is based on the idea inspired by the method of space-time resonances. In particular the different behaviors of the decay properties of the density and velocity field for compressible fluids at different frequencies play a key role.Comment: 61 pages; all comments wellcom