A geometric approach to quantum circuit lower bounds

What is the minimal size quantum circuit required to exactly implement a specified n-qubit unitary operation, U, without the use of ancilla qubits? We show that a lower bound on the minimal size is provided by the length of the minimal geodesic between U and the identity, I, where length is defined by a suitable Finsler metric on the manifold SU(2(n)). The geodesic curves on these manifolds have the striking property that once an initial position and velocity are set, the remainder of the geodesic is completely determined by a second order differential equation known as the geodesic equation. This is in contrast with the usual case in circuit design, either classical or quantum, where being given part of an optimal circuit does not obviously assist in the design of the rest of the circuit. Geodesic analysis thus offers a potentially powerful approach to the problem of proving quantum circuit lower bounds. In this paper we construct several Finsler metrics whose minimal length geodesics provide lower bounds on quantum circuit size. For each Finsler metric we give a procedure to compute the corresponding geodesic equation. We also construct a large class of solutions to the geodesic equation, which we call Pauli geodesics, since they arise from isometries generated by the Pauli group. For any unitary U diagonal in the computational basis, we show that: (a) provided the minimal length geodesic is unique, it must be a Pauli geodesic; (b) finding the length of the minimal Pauli geodesic passing from I to U is equivalent to solving an exponential size instance of the closest vector in a lattice problem (CVP); and (c) all but a doubly exponentially small fraction of such unitaries have minimal Pauli geodesics of exponential length.

Fate of hairpin transcript components during RNA silencing and its suppression in transgenic virus-resistant tobacco

Transgenic tobacco plants, carrying a Potato virus Y (PVY)-NIa hairpin sequence separated by a unique unrelated spacer sequence were specifically silenced and highly resistant to PVY infection. In such plants neither PVY-NIa nor spacer transgene transcripts were detectable by specific quantitative real time reverse transcriptase PCR (RT-qPCR) assays of similar relative efficiencies developed for direct comparative analysis. However, small interfering RNAs (siRNAs) specific for the PVY sequence of the transgene and none specific for the LNYV spacer sequence were detected. Following infection with Cucumber mosaic virus (CMV), which suppresses dsRNA-induced RNA silencing, transcript levels of PVY-NIa as well as spacer sequence increased manifold with the same time course. The cellular abundance of the single-stranded (ss) spacer sequence was consistently higher than that of PVY dsRNA in all cases. The results show that during RNA silencing and its suppression of a hairpin transcript in transgenic tobacco, the ssRNA spacer sequence is affected differently than the dsRNA. In PVY-silenced plants. the spacer is efficiently degraded by a mechanism not involving the accumulation of siRNAs, while following suppression of RNA silencing by CMV, the spacer appears protected from degradation. Crown Copyright (c) 2006 Published by Elsevier B.V. All rights reserved.

Optimal control, geometry, and quantum computing

We prove upper and lower bounds relating the quantum gate complexity of a unitary operation, U, to the optimal control cost associated to the synthesis of U. These bounds apply for any optimal control problem, and can be used to show that the quantum gate complexity is essentially equivalent to the optimal control cost for a wide range of problems, including time-optimal control and finding minimal distances on certain Riemannian, sub-Riemannian, and Finslerian manifolds. These results generalize the results of [Nielsen, Dowling, Gu, and Doherty, Science 311, 1133 (2006)], which showed that the gate complexity can be related to distances on a Riemannian manifold.

Positive solutions of a Neumann problem with competing critical nonlinearities

We present existence results for a Neumann problem involving critical Sobolev nonlinearities both on the right hand side of the equation and at the boundary condition.. Positive solutions are obtained through constrained minimization on the Nehari manifold. Our approach is based on the concentration 'compactness principle of P. L. Lions and M. Struwe.

A Bethe ansatz solvable model for superpositions of Cooper pairs and condensed molecular bosons

We introduce a general Hamiltonian describing coherent superpositions of Cooper pairs and condensed molecular bosons. For particular choices of the coupling parameters, the model is integrable. One integrable manifold, as well as the Bethe ansatz solution, was found by Dukelsky et al. [J. Dukelsky, G.G. Dussel, C. Esebbag, S. Pittel, Phys. Rev. Lett. 93 (2004) 050403]. Here we show that there is a second integrable manifold, established using the boundary quantum inverse scattering method. In this manner we obtain the exact solution by means of the algebraic Bethe ansatz. In the case where the Cooper pair energies are degenerate we examine the relationship between the spectrum of these integrable Hamiltonians and the quasi-exactly solvable spectrum of particular Schrodinger operators. For the solution we derive here the potential of the Schrodinger operator is given in terms of hyperbolic functions. For the solution derived by Dukelsky et al., loc. cit. the potential is sextic and the wavefunctions obey PT-symmetric boundary conditions. This latter case provides a novel example of an integrable Hermitian Hamiltonian acting on a Fock space whose states map into a Hilbert space of PE-symmetric wavefunctions defined on a contour in the complex plane. (c) 2006 Elsevier B.V. All rights reserved.

Towards a framework for combining stochastic and deterministic descriptions of nonstationary financial time series

We present in this paper ideas to tackle the problem of analysing and forecasting nonstationary time series within the financial domain. Accepting the stochastic nature of the underlying data generator we assume that the evolution of the generator's parameters is restricted on a deterministic manifold. Therefore we propose methods for determining the characteristics of the time-localised distribution. Starting with the assumption of a static normal distribution we refine this hypothesis according to the empirical results obtained with the methods anc conclude with the indication of a dynamic non-Gaussian behaviour with varying dependency for the time series under consideration.

Hierarchical GTM: Constructing localized non-linear projection manifolds in a principled way

It has been argued that a single two-dimensional visualization plot may not be sufficient to capture all of the interesting aspects of complex data sets, and therefore a hierarchical visualization system is desirable. In this paper we extend an existing locally linear hierarchical visualization system PhiVis ¸iteBishop98a in several directions: bf(1) We allow for em non-linear projection manifolds. The basic building block is the Generative Topographic Mapping. bf(2) We introduce a general formulation of hierarchical probabilistic models consisting of local probabilistic models organized in a hierarchical tree. General training equations are derived, regardless of the position of the model in the tree. bf(3) Using tools from differential geometry we derive expressions for local directional curvatures of the projection manifold. Like PhiVis, our system is statistically principled and is built interactively in a top-down fashion using the EM algorithm. It enables the user to interactively highlight those data in the parent visualization plot which are captured by a child model. We also incorporate into our system a hierarchical, locally selective representation of magnification factors and directional curvatures of the projection manifolds. Such information is important for further refinement of the hierarchical visualization plot, as well as for controlling the amount of regularization imposed on the local models. We demonstrate the principle of the approach on a toy data set and apply our system to two more complex 12- and 19-dimensional data sets.

Using directional curvatures to visualize folding patterns of the GTM projection manifolds

In data visualization, characterizing local geometric properties of non-linear projection manifolds provides the user with valuable additional information that can influence further steps in the data analysis. We take advantage of the smooth character of GTM projection manifold and analytically calculate its local directional curvatures. Curvature plots are useful for detecting regions where geometry is distorted, for changing the amount of regularization in non-linear projection manifolds, and for choosing regions of interest when constructing detailed lower-level visualization plots.

Hierarchical GTM: constructing localized non-linear projection manifolds in a principled way

It has been argued that a single two-dimensional visualization plot may not be sufficient to capture all of the interesting aspects of complex data sets, and therefore a hierarchical visualization system is desirable. In this paper we extend an existing locally linear hierarchical visualization system PhiVis ¸iteBishop98a in several directions: bf(1) We allow for em non-linear projection manifolds. The basic building block is the Generative Topographic Mapping (GTM). bf(2) We introduce a general formulation of hierarchical probabilistic models consisting of local probabilistic models organized in a hierarchical tree. General training equations are derived, regardless of the position of the model in the tree. bf(3) Using tools from differential geometry we derive expressions for local directional curvatures of the projection manifold. Like PhiVis, our system is statistically principled and is built interactively in a top-down fashion using the EM algorithm. It enables the user to interactively highlight those data in the ancestor visualization plots which are captured by a child model. We also incorporate into our system a hierarchical, locally selective representation of magnification factors and directional curvatures of the projection manifolds. Such information is important for further refinement of the hierarchical visualization plot, as well as for controlling the amount of regularization imposed on the local models. We demonstrate the principle of the approach on a toy data set and apply our system to two more complex 12- and 18-dimensional data sets.

A principled approach to interactive hierarchical non-linear visualization of high-dimensional data

Hierarchical visualization systems are desirable because a single two-dimensional visualization plot may not be sufficient to capture all of the interesting aspects of complex high-dimensional data sets. We extend an existing locally linear hierarchical visualization system PhiVis [1] in several directions: bf(1) we allow for em non-linear projection manifolds (the basic building block is the Generative Topographic Mapping -- GTM), bf(2) we introduce a general formulation of hierarchical probabilistic models consisting of local probabilistic models organized in a hierarchical tree, bf(3) we describe folding patterns of low-dimensional projection manifold in high-dimensional data space by computing and visualizing the manifold's local directional curvatures. Quantities such as magnification factors [3] and directional curvatures are helpful for understanding the layout of the nonlinear projection manifold in the data space and for further refinement of the hierarchical visualization plot. Like PhiVis, our system is statistically principled and is built interactively in a top-down fashion using the EM algorithm. We demonstrate the visualization system principle of the approach on a complex 12-dimensional data set and mention possible applications in the pharmaceutical industry.

Constructing localized non-linear projection manifolds in a principled way:hierarchical generative topographic mapping

It has been argued that a single two-dimensional visualization plot may not be sufficient to capture all of the interesting aspects of complex data sets, and therefore a hierarchical visualization system is desirable. In this paper we extend an existing locally linear hierarchical visualization system PhiVis (Bishop98a) in several directions: 1. We allow for em non-linear projection manifolds. The basic building block is the Generative Topographic Mapping. 2. We introduce a general formulation of hierarchical probabilistic models consisting of local probabilistic models organized in a hierarchical tree. General training equations are derived, regardless of the position of the model in the tree. 3. Using tools from differential geometry we derive expressions for local directionalcurvatures of the projection manifold. Like PhiVis, our system is statistically principled and is built interactively in a top-down fashion using the EM algorithm. It enables the user to interactively highlight those data in the parent visualization plot which are captured by a child model.We also incorporate into our system a hierarchical, locally selective representation of magnification factors and directional curvatures of the projection manifolds. Such information is important for further refinement of the hierarchical visualization plot, as well as for controlling the amount of regularization imposed on the local models. We demonstrate the principle of the approach on a toy data set andapply our system to two more complex 12- and 19-dimensional data sets.

Visual data mining: integrating machine learning with information visualization

Today, the data available to tackle many scientific challenges is vast in quantity and diverse in nature. The exploration of heterogeneous information spaces requires suitable mining algorithms as well as effective visual interfaces. Most existing systems concentrate either on mining algorithms or on visualization techniques. Though visual methods developed in information visualization have been helpful, for improved understanding of a complex large high-dimensional dataset, there is a need for an effective projection of such a dataset onto a lower-dimension (2D or 3D) manifold. This paper introduces a flexible visual data mining framework which combines advanced projection algorithms developed in the machine learning domain and visual techniques developed in the information visualization domain. The framework follows Shneiderman’s mantra to provide an effective user interface. The advantage of such an interface is that the user is directly involved in the data mining process. We integrate principled projection methods, such as Generative Topographic Mapping (GTM) and Hierarchical GTM (HGTM), with powerful visual techniques, such as magnification factors, directional curvatures, parallel coordinates, billboarding, and user interaction facilities, to provide an integrated visual data mining framework. Results on a real life high-dimensional dataset from the chemoinformatics domain are also reported and discussed. Projection results of GTM are analytically compared with the projection results from other traditional projection methods, and it is also shown that the HGTM algorithm provides additional value for large datasets. The computational complexity of these algorithms is discussed to demonstrate their suitability for the visual data mining framework.

The acceptability of alcohol fuels for automobile engines

The aim of this project was to carry out an investigastion into suitable alternatives to gasoline for use in modern automobiles. The fuel would provide the western world with a means of extending the natural gasoline resources and the third world a way of cutting down their dependence on the oil producing countries for their energy supply. Alcohols, namely methanol and ethanol, provide this solution. They can be used as gasoline extenders or as fuels on their own.In order to fulfil the aims of the project a literature study was carried out to investigate methods and costs of producing these fuels. An experimental programme was then set up in which the performance of the alcohols was studied on a conventional engine. The engine used for this purpose was the Fiat 127 930cc four cylinder engine. This engine was used because of its popularity in the European countries. The Weber fixed jet carburettor, since it was designed to be used with gasoline, was adapted so that the alcohol fuels and the blends could be used in the most efficient way. This was mainly to take account of the lower heat content of the alcohols. The adaptation of the carburettor was in the form of enlarging the main metering jet. Allowances for the alcohol's lower specfic gravity were made during fuel metering.Owing to the low front end volatility of methanol and ethanol, it was expected that `start up' problems would occur. An experimental programme was set up to determine the temperature range for a minimum required percentage `take off' that would ease start-up since it was determined that a `take off' of about 5% v/v liquid in the vapour phase would be sufficient for starting. Additions such as iso-pentane and n-pentane were used to improve the front end volatility. This proved to be successful.The lower heat content of the alcohol fuels also meant that a greater charge of fuel would be required. This was seen to pose further problems with fuel distribution from the carburettor to the individual cylinders on a multicylinder engine. Since it was not possible to modify the existing manifold on the Fiat 127 engine, experimental tests on manifold geometry were carried out using the Ricardo E6 single cylinder variable compression engine. Results from these tests showed that the length, shape and cross-sectional area of the manifold play an important part in the distribution of the fuel entering the cylinder, ie. vapour phase, vapour/small liquid droplet/liquid film phase, vapour/large liquid droplet/liquid film phase etc.The solvent properties of the alcohols and their greater electrical conductivity suggested that the materials used on the engine would be prone to chemical attack. In order to determine the type and rate of chemical attack, an experimental programme was set up whereby carburettor and other components were immersed in the alcohols and in blends of alcohol with gasoline. The test fuels were aerated and in some instances kept at temperatures ranging from 50oC to 90oC. Results from these tests suggest that not all materials used in the conventional engine are equally suitable for use with alcohols and alcohol/gasoline blends. Aluminium for instance was severely attacked by methanol causing pitting and pin-holing in the surface.In general this whole experimental programme gave valuable information on the acceptability of substitute fuels. While the long term effects of alcohol use merit further study, it is clear that methanol and ethanol will be increasingly used in place of gasoline.

Outlier detection in scatterometer data:Neural network approaches

Satellite-borne scatterometers are used to measure backscattered micro-wave radiation from the ocean surface. This data may be used to infer surface wind vectors where no direct measurements exist. Inherent in this data are outliers owing to aberrations on the water surface and measurement errors within the equipment. We present two techniques for identifying outliers using neural networks; the outliers may then be removed to improve models derived from the data. Firstly the generative topographic mapping (GTM) is used to create a probability density model; data with low probability under the model may be classed as outliers. In the second part of the paper, a sensor model with input-dependent noise is used and outliers are identified based on their probability under this model. GTM was successfully modified to incorporate prior knowledge of the shape of the observation manifold; however, GTM could not learn the double skinned nature of the observation manifold. To learn this double skinned manifold necessitated the use of a sensor model which imposes strong constraints on the mapping. The results using GTM with a fixed noise level suggested the noise level may vary as a function of wind speed. This was confirmed by experiments using a sensor model with input-dependent noise, where the variation in noise is most sensitive to the wind speed input. Both models successfully identified gross outliers with the largest differences between models occurring at low wind speeds. © 2003 Elsevier Science Ltd. All rights reserved.

Characterization of the hairpin vortex solution in plane Couette flow

Quantitative evidence that establishes the existence of the hairpin vortex state (HVS) in plane Couette flow (PCF) is provided in this work. The evidence presented in this paper shows that the HVS can be obtained via homotopy from a flow with a simple geometrical configuration, namely, the laterally heated flow (LHF). Although the early stages of bifurcations of LHF have been previously investigated, our linear stability analysis reveals that the root in the LHF yields multiple branches via symmetry breaking. These branches connect to the PCF manifold as steady nonlinear amplitude solutions. Moreover, we show that the HVS has a direct bifurcation route to the Rayleigh-Bénard convection. © 2010 The American Physical Society.