Grover's search with faults on some marked elements

Grover's algorithm is a quantum query algorithm solving the unstructured search problem of size $N$ using $O(\sqrt{N})$ queries. It provides a significant speed-up over any classical algorithm \cite{Gro96}. The running time of the algorithm, however, is very sensitive to errors in queries. It is known that if query may fail (report all marked elements as unmarked) the algorithm needs $\Omega(N)$ queries to find a marked element \cite{RS08}. \cite{AB+13} have proved the same result for the model where each marked element has its own probability to be reported as unmarked. We study the behavior of Grover's algorithm in the model where the search space contains both faulty and non-faulty marked elements. We show that in this setting it is indeed possible to find one of non-faulty marked items in $O(\sqrt{N})$ queries. We also analyze the limiting behavior of the algorithm for a large number of steps and show the existence and the structure of limiting state $\rho_{lim}$.Comment: 17 pages, 6 figure

Expected lengths and distribution functions for Young diagrams in the hook

AbstractWe consider β-Plancherel measures [J. Baik, E. Rains, The asymptotics of monotone subsequences of involutions, Duke Math. J. 109 (2001) 205–281] on subsets of partitions—and their asymptotics. These subsets are the Young diagrams contained in a (k,ℓ)-hook, and we calculate the asymptotics of the expected shape of these diagrams, relative to such measures. We also calculate the asymptotics of the distribution function of the lengths of the rows and the columns for these diagrams. This might be considered as the restriction to the (k,ℓ)-hook of the fundamental work of Baik, Deift and Johansson [J. Baik, P. Deift, K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc. 12 (1999) 1119–1178]. The above asymptotics are given here by ratios of certain Selberg-type multi-integrals

Effects of dissipation in an adiabatic quantum search algorithm

We consider the effect of two different environments on the performance of the quantum adiabatic search algorithm, a thermal bath at finite temperature, and a structured environment similar to the one encountered in systems coupled to the electromagnetic field that exists within a photonic crystal. While for all the parameter regimes explored here, the algorithm performance is worsened by the contact with a thermal environment, the picture appears to be different when considering a structured environment. In this case we show that, by tuning the environment parameters to certain regimes, the algorithm performance can actually be improved with respect to the closed system case. Additionally, the relevance of considering the dissipation rates as complex quantities is discussed in both cases. More particularly, we find that the imaginary part of the rates can not be neglected with the usual argument that it simply amounts to an energy shift, and in fact influences crucially the system dynamics.Comment: 18 pages, 9 figure

Hydrodynamic response of rotationally supported flows in the Small Shearing Box model

The hydrodynamic response of the inviscid small shearing box model of a midplane section of a rotationally supported astrophysical disk is examined. An energy functional ${\cal E}$ is formulated for the general nonlinear problem. It is found that the fate of disturbances is related to the conservation of this quantity which, in turn, depends on the boundary conditions utilized: ${\cal E}$ is conserved for channel boundary conditions while it is not conserved in general for shearing box conditions. Linearized disturbances subject to channel boundary conditions have normal-modes described by Bessel Functions and are qualitatively governed by a quantity $\Sigma$ which is a measure of the ratio between the azimuthal and vertical wavelengths. Inertial oscillations ensue if $\Sigma >1$ - otherwise disturbances must in general be treated as an initial value problem. We reflect upon these results and offer a speculation.Comment: 6 pages, resubmitted to Astronomy and Astrophysics, shortened with references adde

What are the factors that contribute to road accidents? An assessment of law enforcement views, ordinary drivers’ opinions, and road accident records

What are the main contributing factors to road accidents? Factors such as inexperience, lack of skill, and risk-taking behaviors have been associated with the collisions of young drivers. In contrast, visual, cognitive, and mobility impairment have been associated with the collisions of older drivers. We investigated the main causes of road accidents by drawing on multiple sources: expert views of police officers, lay views of the driving public, and official road accident records. In Studies 1 and 2, police officers and the public were asked about the typical causes of road traffic collisions using hypothetical accident scenarios. In Study 3, we investigated whether the views of police officers and the public about accident causation influence their recall accuracy for factors reported to contribute to hypothetical road accidents. The results show that both expert views of police officers and lay views of the driving public closely approximated the typical factors associated with the collisions of young and older drivers, as determined from official accident records. The results also reveal potential underreporting of factors in existing accident records, identifying possible inadequacies in law enforcement practices for investigating driver distraction, drug and alcohol impairment, and uncorrected or defective eyesight. Our investigation also highlights a need for accident report forms to be continuously reviewed and updated to ensure that contributing factor lists reflect the full range of factors that contribute to road accidents. Finally, the views held by police officers and the public on accident causation influenced their memory recall of factors involved in hypothetical scenarios. These findings indicate that delay in completing accident report forms should be minimised, possibly by use of mobile reporting devices at the accident scene

Cell-type-specific resolution epigenetics without the need for cell sorting or single-cell biology.

High costs and technical limitations of cell sorting and single-cell techniques currently restrict the collection of large-scale, cell-type-specific DNA methylation data. This, in turn, impedes our ability to tackle key biological questions that pertain to variation within a population, such as identification of disease-associated genes at a cell-type-specific resolution. Here, we show mathematically and empirically that cell-type-specific methylation levels of an individual can be learned from its tissue-level bulk data, conceptually emulating the case where the individual has been profiled with a single-cell resolution and then signals were aggregated in each cell population separately. Provided with this unprecedented way to perform powerful large-scale epigenetic studies with cell-type-specific resolution, we revisit previous studies with tissue-level bulk methylation and reveal novel associations with leukocyte composition in blood and with rheumatoid arthritis. For the latter, we further show consistency with validation data collected from sorted leukocyte sub-types