The Symmetric Group Defies Strong Fourier Sampling

The dramatic exponential speedups of quantum algorithms over their best existing classical counterparts were ushered in by the technique of Fourier sampling, introduced by Bernstein and Vazirani and developed by Simon and Shor into an approach to the hidden subgroup problem. This approach has proved successful for abelian groups, leading to efficient algorithms for factoring, extracting discrete logarithms, and other number-theoretic problems. We show, however, that this method cannot resolve the hidden subgroup problem in the symmetric groups, even in the weakest, information-theoretic sense. In particular, we show that the Graph Isomorphism problem cannot be solved by this approach. Our work implies that any quantum approach based upon the measurement of coset states must depart from the original framework by using entangled measurements on multiple coset states

Solid-phase microextraction for bioconcentration studies according to OECD TG 305

An important aim of the European Community Regulation on chemicals and their safe use is the identification of (very) persistent, (very) bioaccumulative, and toxic substances. In other regulatory chemical safety assessments (pharmaceuticals, biocides, pesticides), the identification of such (very) persistent, (very) bioaccumulative, and toxic substances is of increasing importance. Solid-phase microextraction is especially capable of extracting total water concentrations as well as the freely dissolved fraction of analytes in the water phase, which is available for bioconcentration in fish. However, although already well established in environmental analyses to determine and quantify analytes mainly in aqueous matrices, solid-phase microextraction is still a rather unusual method in regulatory ecotoxicological research. Here, the potential benefits and drawbacks of solid-phase microextraction are discussed as an analytical routine approach for aquatic bioconcentration studies according to OECD TG 305, with a special focus on the testing of hydrophobic organic compounds characterized by log KOW > 5

Modelling the evolution of biological complexity with a two-dimensional lattice self-assembly process

Self-assembling systems are prevalent across numerous scales of nature, lying at the heart of diverse physical and biological phenomena. Individual protein subunits self-assembling into complexes is often a vital first step of biological processes. Errors during protein assembly, due to mutations or misfolds, can have devastating effects and are responsible for an assortment of protein diseases, known as proteopathies. With proteins exhibiting endless layers of complexity, building any all-encompassing model is unrealistic. Coarse-grained models, despite not faithfully capturing every detail of the original system, have massive potential to assist understanding complex phenomenon. A principal actor in self-assembly is the binding interactions between subunits, and so geometric constraints, polarity, kinetic forces, etc. can often be marginalised. This work explores how self-assembly and its outcomes are inextricably tied to the involved interactions through the use of a two-dimensional lattice polyomino model. %Armed with this tractable model, we can probe how dynamics acting on evolution are reflected in interaction properties. First, this thesis addresses how the interaction characteristics of self-assembly building blocks determine what structures they form. Specifically, if the same structures are consistently produced and remain finite in size. Assembly graphs store subunit interaction information and are used in classifying these two properties, the determinism and boundedness respectively. Arbitrary sets of building blocks are classified without the costly overhead of repeated stochastic assembling, improving both the analysis speed and accuracy. Furthermore, assembly graphs naturally integrate combinatorial and graph techniques, enabling a wider range of future polyomino studies. The second part narrows in on implications of nondeterministic assembly on interaction strength evolution. Generalising subunit binding sites with mutable binary strings introduces such interaction strengths into the polyomino model. Deterministic assemblies obey analytic expectations. Conversely, interactions in nondeterministic assemblies rapidly diverge from equilibrium to minimise assembly inconsistency. Optimal interaction strengths during assembly are also reflected in evolution. Transitions between certain polyominoes are strongly forbidden when interaction strengths are misaligned. The third aspect focuses on genetic duplication, an evolutionary event observed in organisms across all taxa. Through polyomino evolutions, a duplication-heteromerisation pathway emerges as an efficient process. This pathway exploits the advantages of both self-interactions and pairwise-interactions, and accelerates evolution by avoiding complexity bottlenecks. Several simulation predictions are successfully validated against a large data set of protein complexes. These results focus on coarse-grained models rather than quantified biological insight. Despite this, they reinforce existing observations of protein complexes, as well as posing several new mechanisms for the evolution of biological complexity

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

Competition and dispersal in the regulation of plant species richness on Carex stricta tussocks

Many wetland plant species can be found growing on Carex stricta Lam. (tussock sedge) tussocks in freshwater marshes. Based on Grime\u27s model of plant species richness, the objectives of this research were to: (1) examine if dispersal characteristics vary among C. stricta marshes in a manner that could potentially influence species richness on individual tussocks, and (2) examine how variation in propagule availability may interact with standing crop and leaf litter to regulate species richness on individual tussocks. All of the research was conducted in southeastern New Hampshire. Dispersal characteristics were quantified in five wetlands representing a broad range of average species richness per tussock. In each wetland, I observed patterns of plant colonization on 50 artificial tussocks (10 per site) over a one year period. In wetlands with high numbers of species per C. stricta tussock, species arrived at artificial tussocks at higher rates than at sites with few species per C. stricta tussock. Therefore, it was possible that variation in dispersal characteristics among wetlands could contribute to the observed differences in average species richness per C. stricta tussock. In addition, I found that the variation among wetlands in the rates at which species anived at artificial tussocks was due primarily to variation in numbers of dispersing species (species pool) rather than to variation in the densities of dispersing propagules per species. In order to examine how variation in propagule availabilities may interact with standing crop and leaf litter to regulate species richness on C. stricta tussocks, I experimentally manipulated these factors using a factorial design involving 168 tussocks in three wetlands. Clipping of live C. stricta, removal of leaf litter, and addition of seeds from tussock inhabiting species all increased species richness on tussocks. Moreover, the magnitude of the limitation imposed by each was strongly dependent on the levels of each of the other factors. All of these relationships were consistent with Grime\u27s model