Improved quantum algorithms for the ordered search problem via semidefinite programming

One of the most basic computational problems is the task of finding a desired item in an ordered list of N items. While the best classical algorithm for this problem uses log_2 N queries to the list, a quantum computer can solve the problem using a constant factor fewer queries. However, the precise value of this constant is unknown. By characterizing a class of quantum query algorithms for ordered search in terms of a semidefinite program, we find new quantum algorithms for small instances of the ordered search problem. Extending these algorithms to arbitrarily large instances using recursion, we show that there is an exact quantum ordered search algorithm using 4 log_{605} N \approx 0.433 log_2 N queries, which improves upon the previously best known exact algorithm.Comment: 8 pages, 4 figure

Optimal Unravellings for Feedback Control in Linear Quantum Systems

For quantum systems with linear dynamics in phase space much of classical feedback control theory applies. However, there are some questions that are sensible only for the quantum case, such as: given a fixed interaction between the system and the environment what is the optimal measurement on the environment for a particular control problem? We show that for a broad class of optimal (state-based) control problems (the stationary Linear-Quadratic-Gaussian class), this question is a semi-definite program. Moreover, the answer also applies to Markovian (current-based) feedback.Comment: 5 pages. Version published by Phys. Rev. Let

Semidefinite Representation of the kk-Ellipse

The $k$-ellipse is the plane algebraic curve consisting of all points whose sum of distances from $k$ given points is a fixed number. The polynomial equation defining the $k$-ellipse has degree $2^k$ if $k$ is odd and degree $2^k{-}\binom{k}{k/2}$ if $k$ is even. We express this polynomial equation as the determinant of a symmetric matrix of linear polynomials. Our representation extends to weighted $k$-ellipses and $k$-ellipsoids in arbitrary dimensions, and it leads to new geometric applications of semidefinite programming.Comment: 16 pages, 5 figure

Reliability of lithium dilution cardiac output in anaesthetized sheep

Background Cardiac output (CO) measurement with lithium dilution (COLD) has not been fully validated in sheep using precise ultrasonic flow probe technology (COUFP). Sheep generate important cardiovascular research models and the use of COLD has become more popular in experimental settings. Methods Ultrasonic transit-time perivascular flow probes were surgically implanted on the pulmonary artery of 13 sheep. Paired COLD readings were taken at six time points, before and after implantation of a left ventricular assist device (LVAD) and compared with COUFP recorded just after lithium injection. Results The mean COLD was 5.7 litre min−1 (range 3.8-9.6 litre min−1) and mean COUFP 5.9 litre min−1 (range 4.0-9.2 litre min−1). The bias (standard deviation) was 0.3 (1.0) litre min−1 [5.1 (16.9)%] and limits of agreement (LOA) were −1.7 to 2.3 litre min−1 (−28.8 to 39.0%) with a percentage error (PE) of 34.4%. Data to assess trending [rate (95% confidence intervals)] included a 78 (62-93)% concordance rate in the four-quadrant plot (n=27). In the half moon polar plot (n=19), the mean polar angle was +5°, the radial LOA were −49 to +35° and 68 (47-89)% of data points fell within 22.5° of the mean polar angle. Both tests indicated moderate to poor trending ability. Conclusion COLD is not precise when evaluated against COUFP in sheep based on the statistical criteria set, but the results are comparable with previously published animal studie

Computation with Polynomial Equations and Inequalities arising in Combinatorial Optimization

The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear algebra or semidefinite programming relaxations of many kinds of feasibility or optimization questions. We are particularly interested in problems arising in combinatorial optimization.Comment: 28 pages, survey pape

Quantum Detection with Unknown States

We address the problem of distinguishing among a finite collection of quantum states, when the states are not entirely known. For completely specified states, necessary and sufficient conditions on a quantum measurement minimizing the probability of a detection error have been derived. In this work, we assume that each of the states in our collection is a mixture of a known state and an unknown state. We investigate two criteria for optimality. The first is minimization of the worst-case probability of a detection error. For the second we assume a probability distribution on the unknown states, and minimize of the expected probability of a detection error. We find that under both criteria, the optimal detectors are equivalent to the optimal detectors of an ``effective ensemble''. In the worst-case, the effective ensemble is comprised of the known states with altered prior probabilities, and in the average case it is made up of altered states with the original prior probabilities.Comment: Refereed version. Improved numerical examples and figures. A few typos fixe

Distinguishing separable and entangled states

We show how to design families of operational criteria that distinguish entangled from separable quantum states. The simplest of these tests corresponds to the well-known Peres-Horodecki positive partial transpose (PPT) criterion, and the more complicated tests are strictly stronger. The new criteria are tractable due to powerful computational and theoretical methods for the class of convex optimization problems known as semidefinite programs. We successfully applied the results to many low-dimensional states from the literature where the PPT test fails. As a byproduct of the criteria, we provide an explicit construction of the corresponding entanglement witnesses.Comment: 4 pages, Latex2e. Expanded discussion of numerical procedures. Accepted for publication in Physical Review Letter

A Study of the Formation of Single- and Double-Walled Carbon Nanotubes by a CVD Method

The reduction in H2/CH4 atmosphere of aluminum-iron oxides produces metal particles small enough to catalyze the formation of single-walled carbon nanotubes. Several experiments have been made using the same temperature profile and changing only the maximum temperature (800-1070 °C). Characterizations of the catalyst materials are performed using notably 57Fe Mo¨ssbauer spectroscopy. Electron microscopy and a macroscopical method are used to characterize the nanotubes. The nature of the iron species (Fe3+, R-Fe, ç-Fe-C, Fe3C) is correlated to their location in the material. The nature of the particles responsible for the high-temperature formation of the nanotubes is probably an Fe-C alloy which is, however, found as Fe3C by postreaction analysis. Increasing the reduction temperature increases the reduction yield and thus favors the formation of surface-metal particles, thus producing more nanotubes. The obtained carbon nanotubes are mostly single-walled and double-walled with an average diameter close to 2.5 nm. Several formation mechanisms are thought to be active. In particular, it is shown that the second wall can grow inside the first one but that subsequent ones are formed outside. It is also possible that under given experimental conditions, the smallest (<2 nm) catalyst particles preferentially produce double-walled rather than single-walled carbon nanotubes

ON THE CONNECTIONS BETWEEN SEMIDEFINITE OPTIMIZATION AND VECTOR OPTIMIZATION

This paper works out connections between semidefinite optimization and vector optimization. It is shown that well-known semidefinite optimization problems are scalarized versions of a general vector optimization problem. This scalarization leads to the minimization of the trace or the maximal eigenvalue

Positive Semidefiniteness and Positive Definiteness of a Linear Parametric Interval Matrix

We consider a symmetric matrix, the entries of which depend linearly on some parameters. The domains of the parameters are compact real intervals. We investigate the problem of checking whether for each (or some) setting of the parameters, the matrix is positive definite (or positive semidefinite). We state a characterization in the form of equivalent conditions, and also propose some computationally cheap sufficient\,/\,necessary conditions. Our results extend the classical results on positive (semi-)definiteness of interval matrices. They may be useful for checking convexity or non-convexity in global optimization methods based on branch and bound framework and using interval techniques