The Power of Strong Fourier Sampling: Quantum Algorithms for Affine Groups and Hidden Shifts

Many quantum algorithms, including Shor's celebrated factoring and discrete log algorithms, proceed by reduction to a hidden subgroup problem, in which an unknown subgroup $H$ of a group $G$ must be determined from a quantum state $\psi$ over $G$ that is uniformly supported on a left coset of $H$. These hidden subgroup problems are typically solved by Fourier sampling: the quantum Fourier transform of $\psi$ is computed and measured. When the underlying group is nonabelian, two important variants of the Fourier sampling paradigm have been identified: the weak standard method, where only representation names are measured, and the strong standard method, where full measurement (i.e., the row and column of the representation, in a suitably chosen basis, as well as its name) occurs. It has remained open whether the strong standard method is indeed stronger, that is, whether there are hidden subgroups that can be reconstructed via the strong method but not by the weak, or any other known, method. In this article, we settle this question in the affirmative. We show that hidden subgroups $H$ of the $q$-hedral groups, i.e., semidirect products ${\mathbb Z}_q \ltimes {\mathbb Z}_p$, where $q \mid (p-1)$, and in particular the affine groups $A_p$, can be information-theoretically reconstructed using the strong standard method. Moreover, if $|H| = p/ {\rm polylog}(p)$, these subgroups can be fully reconstructed with a polynomial amount of quantum and classical computation. We compare our algorithms to two weaker methods that have been discussed in the literature—the “forgetful” abelian method, and measurement in a random basis—and show that both of these are weaker than the strong standard method. Thus, at least for some families of groups, it is crucial to use the full power of representation theory and nonabelian Fourier analysis, namely, to measure the high-dimensional representations in an adapted basis that respects the group's subgroup structure. We apply our algorithm for the hidden subgroup problem to new families of cryptographically motivated hidden shift problems, generalizing the work of van Dam, Hallgren, and Ip on shifts of multiplicative characters. Finally, we close by proving a simple closure property for the class of groups over which the hidden subgroup problem can be solved efficiently

Finite-Size-Scaling at the Jamming Transition: Corrections to Scaling and the Correlation Length Critical Exponent

We carry out a finite size scaling analysis of the jamming transition in frictionless bi-disperse soft core disks in two dimensions. We consider two different jamming protocols: (i) quench from random initial positions, and (ii) quasistatic shearing. By considering the fraction of jammed states as a function of packing fraction for systems with different numbers of particles, we determine the spatial correlation length critical exponent $\nu\approx 1$, and show that corrections to scaling are crucial for analyzing the data. We show that earlier numerical results yielding $\nu<1$ are due to the improper neglect of these corrections.Comment: 5 pages, 4 figures -- slightly revised version as accepted for Phys. Rev. E Rapid Communication

Restoration of axon conduction and motor deficits by therapeutic treatment with glatiramer acetate.

Glatiramer acetate (GA; Copaxone) is an approved drug for the treatment of multiple sclerosis (MS). The underlying multifactorial anti-inflammatory, neuroprotective effect of GA is in the induction of reactive T cells that release immunomodulatory cytokines and neurotrophic factors at the injury site. These GA-induced cytokines and growth factors may have a direct effect on axon function. Building on previous findings that suggest a neuroprotective effect of GA, we assessed the therapeutic effects of GA on brain and spinal cord pathology and functional correlates using the chronic experimental autoimmune encephalomyelitis (EAE) mouse model of MS. Therapeutic regimens were utilized based on promising prophylactic efficacy. More specifically, C57BL/6 mice were treated with 2 mg/mouse/day GA for 8 days beginning at various time points after EAE post-induction day 15, yielding a thorough, clinically relevant assessment of GA efficacy within the context of severe progressive disease. Therapeutic treatment with GA significantly decreased clinical scores and improved rotorod motor performance in EAE mice. These functional improvements were supported by an increase in myelinated axons and fewer amyloid precursor protein-positive axons in the spinal cords of GA-treated EAE mice. Furthermore, therapeutic GA decreased microglia/macrophage and T cell infiltrates and increased oligodendrocyte numbers in both the spinal cord and corpus callosum of EAE mice. Finally, GA improved callosal axon conduction and nodal protein organization in EAE. Our results demonstrate that therapeutic GA treatment has significant beneficial effects in a chronic mouse model of MS, in which its positive effects on both myelinated and non-myelinated axons results in improved axon function

Pulmonary ventilation–perfusion mismatch : a novel hypothesis for how diving vertebrates may avoid the bends

© The Author(s), 2018. This article is distributed under the terms of the Creative Commons Attribution License. The definitive version was published in Proceedings of the Royal Society B: Biological Sciences 285 (2018): 20180482, doi:10.1098/rspb.2018.0482.Hydrostatic lung compression in diving marine mammals, with collapsing alveoli blocking gas exchange at depth, has been the main theoretical basis for limiting N2 uptake and avoiding gas emboli (GE) as they ascend. However, studies of beached and bycaught cetaceans and sea turtles imply that air-breathing marine vertebrates may, under unusual circumstances, develop GE that result in decompression sickness (DCS) symptoms. Theoretical modelling of tissue and blood gas dynamics of breath-hold divers suggests that changes in perfusion and blood flow distribution may also play a significant role. The results from the modelling work suggest that our current understanding of diving physiology in many species is poor, as the models predict blood and tissue N2 levels that would result in severe DCS symptoms (chokes, paralysis and death) in a large fraction of natural dive profiles. In this review, we combine published results from marine mammals and turtles to propose alternative mechanisms for how marine vertebrates control gas exchange in the lung, through management of the pulmonary distribution of alveolar ventilation (Embedded Image) and cardiac output/lung perfusion (Embedded Image), varying the level of Embedded Image in different regions of the lung. Man-made disturbances, causing stress, could alter the Embedded Image mismatch level in the lung, resulting in an abnormally elevated uptake of N2, increasing the risk for GE. Our hypothesis provides avenues for new areas of research, offers an explanation for how sonar exposure may alter physiology causing GE and provides a new mechanism for how air-breathing marine vertebrates usually avoid the diving-related problems observed in human divers.Funding to support a portion of this work was obtained by the Fundación Oceanogràfic and by the Office of Naval Research (ONR YIP Award no. N000141410563 and Award no. N000140811220)

An Objective Assessment of the Variability in Number of Drops per Bottle of Glaucoma Medication

Background: The purpose of this study is to evaluate the number of eyedrops available per bottle of a variety of commonly prescribed glaucoma medications. Methods: Six bottles of each glaucoma medication were tested: three each in the vertical and horizontal orientations. Bottles were housed in a customized force gauge apparatus designed to mimic ballpoint fingertip contact with a bottle. At a standard rate, all drops were expressed from each bottle and counted with an automated drop counter. Simultaneously, bottle volume was measured and drop size and number were also estimated. The main outcome measures were: total number of drops, volume per bottle and drops per milliliter (mL) of glaucoma medication. Results: A total of 192 bottles from 32 bottle designs and manufacturers were tested. Twenty-two of the 32 bottle designs had a significantly different mean number of drops in the vertical and horizontal positions, with 10 designs have more drops dispensed in the horizontal orientation and 12 in the vertical orientation. Six of the 32 bottle designs had a significantly different mean total bottle volume in the vertical and horizontal positions, with all designs having greater volume in the vertical position. An adjusted ratio of mean number of drops/mean bottle volume demonstrated a range from 20.9 drops/mL to 40.8 drops/mL. Conclusions: There is significant variability in drops and volume available per bottle of glaucoma medication depending on both the bottle position and manufacturer. These data point to the need for circumspection in prescribing glaucoma medications and caution in evaluating therapeutic outcomes

The power of basis selection in fourier sampling: hidden subgroup problems in affine groups

Many quantum algorithms, including Shor's celebrated factoring and discrete log algorithms, proceed by reduction to a hidden subgroup problem, in which a unknown subgroup H of a group G must be determined from a quantum state ψ over G that is uniformly supported on a left coset of H. These hidden subgroup problems are typically solved by Fourier sampling: the quantum Fourier transform of ψ is computed and measured. When the underlying group is nonabelian, two important variants of the Fourier sampling paradigm have been identified: the weak standard method, where only representation names are measured, and the strong standard method, where full measurement (i.e., the row and column of the representation as well as its name) occurs. It has remained open whether the strong method is indeed stronger, that is, whether there are hidden subgroups that can be reconstructed via the strong method but not by the weak, or any other known, method.In this article, we settle this question in the affirmative. We show that hidden subgroups of semidirect products of the form ℤq × ℤp, where q | (p - 1) and q = p/polylog(p), can be efficiently determined by the strong standard method. Furthermore, the weak standard method and the "forgetful" abelian method are insufficient for these groups so that, in fact, it appears that use of the corresponding nonabelian representation theory is crucial. We extend this to an informationtheoretic solution for the hidden subgroup problem over the groups ℤq × ℤp where q | (p - 1) and, in particular, the affine groups Ap. Finally, we prove a simple closure property for the class of groups over which the hidden subgroup problem can be solved efficiently

Chaos in computer performance

Modern computer microprocessors are composed of hundreds of millions of transistors that interact through intricate protocols. Their performance during program execution may be highly variable and present aperiodic oscillations. In this paper, we apply current nonlinear time series analysis techniques to the performances of modern microprocessors during the execution of prototypical programs. Our results present pieces of evidence strongly supporting that the high variability of the performance dynamics during the execution of several programs display low-dimensional deterministic chaos, with sensitivity to initial conditions comparable to textbook models. Taken together, these results show that the instantaneous performances of modern microprocessors constitute a complex (or at least complicated) system and would benefit from analysis with modern tools of nonlinear and complexity science