Quantum computation as geometry

Quantum computers hold great promise for solving interesting computational problems, but it remains a challenge to find efficient quantum circuits that can perform these complicated tasks. Here we show that finding optimal quantum circuits is essentially equivalent to finding the shortest path between two points in a certain curved geometry. By recasting the problem of finding quantum circuits as a geometric problem, we open up the possibility of using the mathematical techniques of Riemannian geometry to suggest new quantum algorithms or to prove limitations on the power of quantum computers.

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.

The octahedral manipulator: Geometry and mobility

In most Of the practical six-actuator in-parallel manipulators, the octahedral form is either taken as it stands or is approximated. Yet considerable theoretical attention is paid in the literature to more general forms. Here we touch on the general form, and describe some aspects of its behavior that vitiate strongly against its adoption as a pattern for a realistic manipulate,: We reach the conclusion that the structure for in-parallel manipulators must be triangulated as fully as possible, so leading to the octahedral form. In describing some of the geometrical properties of the general octahedron, we show how they apply to manipulators. We examine in detail the special configurations at which the 6 x 6 matrix of leg lines is singular presenting results from the point of view of geometry in preference to analysis. In extending and enlarging on some known properties, a few behavioral surprises materialize. In studying special configurations, we start with the most general situation, and every other case derives from this. Our coverage is more comprehensive than any that we have found. We bring to light material that is, we think, of significant use to a designer.

Heat flow for Yang-Mills-Higgs fields, part I

The Yang-Mills-Higgs field generalizes the Yang-Mills field. The authors establish the local existence and uniqueness of the weak solution to the heat flow for the Yang-Mills-Higgs field in a vector bundle over a compact Riemannian 4-manifold, and show that the weak solution is gauge-equivalent to a smooth solution and there are at most finite singularities at the maximum existing time.

Gender differences in bone geometry during the adolescent growth spurt

To investigate whether there are gender differences in the bone geometry of the proximal femur during the adolescent years we used an interactive computer program ?Hip Strength Analysis? developed by Beck and associates (Beck et al., Invest Radiol. 1990,25:6-18.) to derive femoral neck geometry parameters from DXA bone scans (Hologic 2000, array mode). We analyzed a longitudinal data-set collected on 70 boys and 68 girls over a seven year period. Distance and velocity curves for height were fitted for each child utilizing a cubic spline procedure and the age of peak height velocity (PHV) was determined. To control for maturational differences between children of the same chronological age and between boys and girls, section modulus (Z) an index of bending strength, cross sectional area of bone (CSA), sub-periosteal width (SPW), and BMD values at the neck and shaft of the proximal femur were determined for points on each individual?s curve at the age of PHV and one and two years on either side of peak. To control for size differences, height and weight were introduced as co-variates in the two-way analyses of variance looking at gender over time measured at the maturational age points (-2, -1, age of PHV, +1, +2). The following figure presents the results of the analyses on two variables, BMD and Z at neck and shaft regions:After the age of peak linear growth (PHV), independent of body size, there was a gender difference in BMD at the shaft but not at the neck. Section modulus at both sites indicated that male bones became significantly stronger after PHV. Underlying these maturational changes, male bones became wider (SPW) after PHV in both the neck and shaft and enclosed more material (CSA) at all maturational age points at both regions. These results call into question the emphasis on using BMD as a measure of skeletal integrity in growing children

Solution structure of the sodium channel antagonist conotoxin GS: A new molecular caliper for probing sodium channel geometry

Background: The venoms of Conus snails contain small, disulfide-rich inhibitors of voltage-dependent sodium channels. Conotoxin GS is a 34-residue polypeptide isolated from Conus geographus that interacts with the extracellular entrance of skeletal muscle sodium channels to prevent sodium ion conduction. Although conotoxin GS binds competitively with mu conotoxin GIIIA to the sodium channel surface, the two toxin types have little sequence identity with one another, and conotoxin GS has a four-loop structural framework rather than the characteristic three-loop mu-conotoxin framework. The structural study of conotoxin GS will form the basis for establishing a structure-activity relationship and understanding its interaction with the pore region of sodium channels. Results: The three-dimensional structure of conotoxin GS was determined using two-dimensional NMR spectroscopy. The protein exhibits a compact fold incorporating a beta hairpin and several turns. An unusual feature of conotoxin GS is the exceptionally high proportion (100%) of cis-imide bond geometry for the three proline or hydroxyproline residues. The structure of conotoxin GS bears little resemblance to the three-loop mu conotoxins, consistent with the low sequence identity between the two toxin types and their different structural framework. However, the tertiary structure and cystine-knot motif formed by the three disulfide bonds is similar to that present in several other polypeptide ion channel inhibitors. Conclusions: This is the first three-dimensional structure of a 'four-loop' sodium channel inhibitor, and it represents a valuable new structural probe for the pore region of voltage-dependent sodium channels. The distribution of amino acid sidechains in the structure creates several polar and charged patches, and comparison with the mu conotoxins provides a basis for determining the binding surface of the conotoxin GS polypeptide.

Theory of a mode-locked atom laser with toroidal geometry

We consider a possible technique for mode locking an atom laser, based on the generation of a dark soliton in a ring-shaped Bose-Einstein condensate, with repulsive atomic interactions. The soliton is a kink, with angular momentum per particle equal to (h) over bar /2. It emerges naturally when the condensate is stirred at the soliton velocity and cleansed with a periodic out coupler. The result is a replicating coherent field inside the atom laser, stabilized by topology. We give a numerical demonstration of the generation and stabilization of the soliton.

The effect of channel geometry and wall boundary conditions on the formation of extrusion surface instabilities for LLDPE

It is believed that surface instabilities can occur during the extrusion of linear low density polyethylene due to high extensional stresses at the exit of the die. Local crack development can occur at a critical stress level when melt rupture is reached. This high extensional stress results from the rearrangement of the flow at the boundary transition between the wall exit and the free surface. The stress is highest at the extrudate surface and decreases into the bulk of the material. The location of the region where the critical level is reached can determine the amplitude of the extrudate surface distortion, This paper studies the effect of wall slip on the numerically simulated extensional stress level at the die exit and correlates this to the experimentally determined amplitude of the surface instability. The effect of die exit radius and die wall roughness on extrusion surface instabilities is also correlated to the exit stress level in the same way. Whereas full slip may completely suppress the surface instability, a reduction in the exit stress level and instability amplitude is also shown for a rounded die exit and a slight increase in instability is shown to result from a rough die wall. A surface instability map demonstrates how the shear rate for onset of extrusion surface instabilities can be predicted on the basis of melt strength measurements and simulated stress peaks at the exit of the die. (C) 2001 Elsevier Science B.V. All rights reserved.

Influence of wall stress and left ventricular geometry on the accuracy of Dobutamine stress Echocardiography

OBJECTIVES The goal of this study was to determine whether wall stress at rest and during stress could explain the influence of left ventricular (LV) morphology on the accuracy of dobutamine stress echocardiography (DSE). BACKGROUND The sensitivity of DSE appears to be reduced in patients with concentric remodeling, but the cause of this finding is unclear. METHODS We studied 161 patients without resting wall motion abnormalities who underwent DSE and coronary angiography. Patients were classified into four groups according to relative wan thickness (normal

Neutrophil dysplasia characterised by a pseudo-Pelger-Huet anomaly occurring with the use of mycophenolate mofetil and ganciclovir following renal transplantation: a report of five cases

Aim: The pseudo-Pelger-Huet (PH) anomaly has been associated with a variety of primary haematological disorders, infections and drugs. Recently, the development of dysgranulopoiesis characterised by a pseudo-PH anomaly has been reported in two patients with the use of mycophenolate mofetil (MMF) in the setting of heart and/or lung transplantation. We present a further five cases of MMF-related dysgranulopoiesis characterised by a pseudo-PH anomaly occurring after renal transplantation. Methods: All patients were receiving standard immunosuppression protocols for renal transplantation, including a combination of MMF, steroids and either cyclosporin or tacrolimus. Oral ganciclovir was also used for cytomegalovirus prophylaxis in each case. Results: Development of dysplastic granulopoiesis occurred a median of 96 days (range 66-196 days) after transplantation. Moderate or severe neutropaenia (

Sex- and race-related differences in cross-sectional geometry and bone density of the femoral mid-shaft in older adults

Background : Femoral shaft fracture incidence increases in older adults and is associated with low-energy trauma. Apart from bone density, the distribution and size of bone contributes to its strength. Aim : To examine if bone geometry and density of the femoral mid-shaft in older adults differs by sex and race, we studied 197 White women, 225 Black women, 242 White men, and 148 Black men aged 70-79 years participating in the Health, Aging, and Body Composition study; a prospective cohort study in the USA. A secondary purpose of the study was to examine the association of site-specific muscle and fat to bone geometry and density. Subjects and methods : Subjects were community-dwelling and reported no difficulty walking one-quarter of a mile or climbing stairs. Mid-femoral volumetric bone mineral density (vBMD, mg cm -3 ), total area (TA), cortical area (CA), medullary area (MA), cross-sectional moments of inertia (CSMI: I x , I y , J ), and muscle and fat areas (cm 2 ) were determined by computed tomography (CT; GE CT-9800, 10 mm slice thickness). Results : vBMD was greater in men than women with no difference by race ( p < 0.001). Bone areas and area moments of inertia were also greater in men than women ( p < 0.001), with Black women having higher values than White women for TA and CA. Standardizing geometric parameters for body size differences by dividing by powers of femur length did not negate the sex difference for TA and MA. Significant differences ( p < 0.05) among the four groups also remained for I x and J . Mid-thigh muscle area was an independent contributor to TA in all groups (Std beta = 0.181-0.351, p < 0.05) as well as CA in women (Std beta = 0.246-0.254, p < 0.01) and CSMI in White women (Std beta = 0.175-0.185, p < 0.05). Further, muscle area was a significant contributor to vBMD in Black women. Conclusion : These results indicate that bone geometry and density of the femoral diaphysis differs primarily by sex, rather than race, in older well-functioning adults. In addition, site-specific muscle area appears to have a potential contributory role to bone geometry parameters, especially in women.

Pseudo-randomness of round-off errors in discretized linear maps on the plane

We analyze the sequences of round-off errors of the orbits of a discretized planar rotation, from a probabilistic angle. It was shown [Bosio & Vivaldi, 2000] that for a dense set of parameters, the discretized map can be embedded into an expanding p-adic dynamical system, which serves as a source of deterministic randomness. For each parameter value, these systems can generate infinitely many distinct pseudo-random sequences over a finite alphabet, whose average period is conjectured to grow exponentially with the bit-length of the initial condition (the seed). We study some properties of these symbolic sequences, deriving a central limit theorem for the deviations between round-off and exact orbits, and obtain bounds concerning repetitions of words. We also explore some asymptotic problems computationally, verifying, among other things, that the occurrence of words of a given length is consistent with that of an abstract Bernoulli sequence.

Pseudo memory effects, majorization and entropy in quantum random walks

A quantum random walk on the integers exhibits pseudo memory effects, in that its probability distribution after N steps is determined by reshuffling the first N distributions that arise in a classical random walk with the same initial distribution. In a classical walk, entropy increase can be regarded as a consequence of the majorization ordering of successive distributions. The Lorenz curves of successive distributions for a symmetric quantum walk reveal no majorization ordering in general. Nevertheless, entropy can increase, and computer experiments show that it does so on average. Varying the stages at which the quantum coin system is traced out leads to new quantum walks, including a symmetric walk for which majorization ordering is valid but the spreading rate exceeds that of the usual symmetric quantum walk.