Quantum computing algorithms for inverse problems on graphs and an NP-complete inverse problem

We consider an inverse problem for a finite graph $(X,E)$ where we are given a subset of vertices $B\subset X$ and the distances $d_{(X,E)}(b_1,b_2)$ of all vertices $b_1,b_2\in B$. The distance of points $x_1,x_2\in X$ is defined as the minimal number of edges needed to connect two vertices, so all edges have length 1. The inverse problem is a discrete version of the boundary rigidity problem in Riemannian geometry or the inverse travel time problem in geophysics. We will show that this problem has unique solution under certain conditions and develop quantum computing methods to solve it. We prove the following uniqueness result: when $(X,E)$ is a tree and $B$ is the set of leaves of the tree, the graph $(X,E)$ can be uniquely determined in the class of all graphs having a fixed number of vertices. We present a quantum computing algorithm which produces a graph $(X,E)$, or one of those, which has a given number of vertices and the required distances between vertices in $B$. To this end we develop an algorithm that takes in a qubit representation of a graph and combine it with Grover's search algorithm. The algorithm can be implemented using only $O(|X|^2)$ qubits, the same order as the number of elements in the adjacency matrix of $(X,E)$. It also has a quadratic improvement in computational cost compared to standard classical algorithms. Finally, we consider applications in theory of computation, and show that a slight modification of the above inverse problem is NP-complete: all NP-problems can be reduced to a discrete inverse problem we consider

Reconstruction of generic anisotropic stiffness tensors from partial data around one polarization

We study inverse problems in anisotropic elasticity using tools from algebraic geometry. The singularities of solutions to the elastic wave equation in dimension $n$ with an anisotropic stiffness tensor have propagation kinematics captured by so-called slowness surfaces, which are hypersurfaces in the cotangent bundle of $\mathbb{R}^n$ that turn out to be algebraic varieties. Leveraging the algebraic geometry of families of slowness surfaces we show that, for tensors in a dense open subset in the space of all stiffness tensors, a small amount of data around one polarization in an individual slowness surface uniquely determines the entire slowness surface and its stiffness tensor. Such partial data arises naturally from geophysical measurements or geometrized versions of seismic inverse problems. Additionally, we explain how the reconstruction of the stiffness tensor can be carried out effectively, using Gr\"obner bases. Our uniqueness results fail for very symmetric (e.g., fully isotropic) materials, evidencing the counterintuitive claim that inverse problems in elasticity can become more tractable with increasing asymmetry.Comment: 39 pages, 4 figures. Computer Code included in the ancillary files folde

Polygenic risk provides biological validity for the ICHD-3 criteria among Finnish migraine families

Background Migraine is diagnosed using the extensively field-tested International Classification of Headache Disorders (ICHD-3) consensus criteria derived by the International Headache Society. To evaluate the criteria in respect to a measurable biomarker, we studied the relationship between the main ICHD-3 criteria and the polygenic risk score, a measure of common variant burden in migraine. Methods We used linear mixed models to study the correlation of ICHD-3 diagnostic criteria, underlying symptoms, and main diagnoses with the polygenic risk score of migraine in a cohort of 8602 individuals from the Finnish Migraine Genome Project. Results Main diagnostic categories and all underlying diagnostic criteria formed a consistent continuum along the increasing polygenic burden. Polygenic risk was associated with the heterogeneous clinical picture starting from the non-migraine headache (mean 0.07; 95% CI 0.02-0.12; p = 0.008 compared to the non-headache group), to probable migraine (mean 0.13; 95% CI 0.08-0.18; p < 0.001), migraine headache (mean 0.17; 95% CI 0.14-0.21; p < 0.001) and migraine with typical visual aura (mean 0.29; 95% CI 0.26-0.33; p < 0.001), all the way to the hemiplegic aura (mean 0.37; 95% CI 0.31-0.43; p < 0.001). All individual ICHD-3 symptoms and the total number of reported symptoms, a surrogate of migraine complexity, demonstrated a clear inclination with an increasing polygenic risk. Conclusions The complex migraine phenotype progressively follows the polygenic burden from individuals with no headache to non-migrainous headache and up to patients with attacks manifesting all the features of the ICHD-3 headache and aura. Results provide further biological support for the ICHD-3 diagnostic criteria.Peer reviewe

Food Web Responses to Artificial Mixing in a Small Boreal Lake

In order to simulate food web responses of small boreal lakes to changes in thermal stratification due to global warming, a 4 year whole-lake manipulation experiment was performed. Within that time, period lake mixing was intensified artificially during two successive summers. Complementary data from a nearby lake of similar size and basic water chemistry were used as a reference. Phytoplankton biomass and chlorophyll a did not respond to the greater mixing depth but an increase was observed in the proportional abundance of diatoms, and the proportional abundance of cryptophytes also increased immediately after the onset of mixing. Obligate anoxic green sulphur bacteria vanished at the onset of mixing but gradually recovered after re-establishment of hypolimnetic anoxic conditions. No major effect on crustacean zooplankton was found, but their diversity increased in the metalimnion. During the mixing, the density of rotifers declined but protozoan density increased in the hypolimnion. Littoral benthic invertebrate density increased during the mixing due to Ephemeroptera, Asellus aquaticus and Chironomidae, whereas the density of Chaoborus larvae declined during mixing and lower densities were still recorded one year after the treatment. No structural changes in fish community were found although gillnet catches increased after the onset of the study. The early growth of perch (Perca fluviatilis) increased compared to the years before the mixing and in comparison to the reference lake, suggesting improved food availability in the experimental lake. Although several food web responses to the greater mixing depth were found, their persistence and ecological significance were strongly dependent on the extent of the disturbance. To better understand the impacts of wind stress on small lakes, long term whole-lake experiments are needed.Peer reviewe

Common Variant Burden Contributes to the Familial Aggregation of Migraine in 1,589 Families

Complex traits, including migraine, often aggregate in families, but the underlying genetic architecture behind this is not well understood. The aggregation could be explained by rare, penetrant variants that segregate according to Mendelian inheritance or by the sufficient polygenic accumulation of common variants, each with an individually small effect, or a combination of the two hypotheses. In 8,319 individuals across 1,589 migraine families, we calculated migraine polygenic risk scores (PRS) and found a significantly higher common variant burden in familial cases (n = 5,317, OR = 1.76, 95% CI = 1.71-1.81, p = 1.7 × 10-109) compared to population cases from the FINRISK cohort (n = 1,101, OR = 1.32, 95% CI = 1.25-1.38, p = 7.2 × 10-17). The PRS explained 1.6% of the phenotypic variance in the population cases and 3.5% in the familial cases (including 2.9% for migraine without aura, 5.5% for migraine with typical aura, and 8.2% for hemiplegic migraine). The results demonstrate a significant contribution of common polygenic variation to the familial aggregation of migraine

The contribution of CACNA1A, ATP1A2 and SCN1A mutations in hemiplegic migraine : A clinical and genetic study in Finnish migraine families

Objective To study the position of hemiplegic migraine in the clinical spectrum of migraine with aura and to reveal the importance of CACNA1A, ATP1A2 and SCN1A in the development of hemiplegic migraine in Finnish migraine families. Methods The International Classification of Headache Disorders 3rd edition criteria were used to determine clinical characteristics and occurrence of hemiplegic migraine, based on detailed questionnaires, in a Finnish migraine family collection consisting of 9087 subjects. Involvement of CACNA1A, ATP1A2 and SCN1A was studied using whole exome sequencing data from 293 patients with hemiplegic migraine. Results Overall, hemiplegic migraine patients reported clinically more severe headache and aura episodes than non-hemiplegic migraine with aura patients. We identified two mutations, c.1816G>A (p.Ala606Thr) and c.1148G>A (p.Arg383His), in ATP1A2 and one mutation, c.1994C>T (p.Thr665Met) in CACNA1A. Conclusions The results highlight hemiplegic migraine as a clinically and genetically heterogeneous disease. Hemiplegic migraine patients do not form a clearly separate group with distinct symptoms, but rather have an extreme phenotype in the migraine with aura continuum. We have shown that mutations in CACNA1A, ATP1A2 and SCN1A are not the major cause of the disease in Finnish hemiplegic migraine patients, suggesting that there are additional genetic factors contributing to the phenotype.Peer reviewe