The complexity of approximately counting in 2-spin systems on kk-uniform bounded-degree hypergraphs

One of the most important recent developments in the complexity of approximate counting is the classification of the complexity of approximating the partition functions of antiferromagnetic 2-spin systems on bounded-degree graphs. This classification is based on a beautiful connection to the so-called uniqueness phase transition from statistical physics on the infinite $\Delta$-regular tree. Our objective is to study the impact of this classification on unweighted 2-spin models on $k$-uniform hypergraphs. As has already been indicated by Yin and Zhao, the connection between the uniqueness phase transition and the complexity of approximate counting breaks down in the hypergraph setting. Nevertheless, we show that for every non-trivial symmetric $k$-ary Boolean function $f$ there exists a degree bound $\Delta_0$ so that for all $\Delta \geq \Delta_0$ the following problem is NP-hard: given a $k$-uniform hypergraph with maximum degree at most $\Delta$, approximate the partition function of the hypergraph 2-spin model associated with $f$. It is NP-hard to approximate this partition function even within an exponential factor. By contrast, if $f$ is a trivial symmetric Boolean function (e.g., any function $f$ that is excluded from our result), then the partition function of the corresponding hypergraph 2-spin model can be computed exactly in polynomial time

Quantum many-body systems in thermal equilibrium

The thermal or equilibrium ensemble is one of the most ubiquitous states of matter. For models comprised of many locally interacting quantum particles, it describes a wide range of physical situations, relevant to condensed matter physics, high energy physics, quantum chemistry and quantum computing, among others. We give a pedagogical overview of some of the most important universal features about the physics and complexity of these states, which have the locality of the Hamiltonian at its core. We focus on mathematically rigorous statements, many of them inspired by ideas and tools from quantum information theory. These include bounds on their correlations, the form of the subsystems, various statistical properties, and the performance of classical and quantum algorithms. We also include a summary of a few of the most important technical tools, as well as some self-contained proofs.Comment: 42 Pages + References, 7 Figures. Parts of these notes were the basis for a lecture series within the "Quantum Thermodynamics Summer School 2021" during August 2021 in Les Diablerets, Switzerlan

Approximate counting via complex zero-free regions and spectral independence

This thesis investigates fundamental problems in approximate counting that arise in the field of statistical mechanics. Building upon recent advancements in the area, our research aims to enhance our understanding of the computational complexity of sampling from the Ising and Potts models, as well as the random $k$-SAT model. The $q$-state Potts model is a spin model in which each particle is randomly assigned a spin (out of $q$ possible spins), where the probability of a certain assignment depends on how many adjacent particles present the same spin. The edge interaction of the model is a parameter that quantifies the strength of interaction between two adjacent particles. The Ising model corresponds to the Potts model with $q = 2$. Sampling from these models is inherently connected to approximating the partition function of the model, a graph polynomial that encodes several aggregate thermodynamic properties of the system. In addition to classical connections with quantum computing and phase transitions in statistical physics, recent work in approximate counting has shown that the behaviour in the complex plane of these partition functions, and more precisely the location of zeros, is strongly connected with the complexity of the approximation problem, even for positive real-valued parameters. Thus, following this trend in both statistical physics and algorithmic research, we allow the edge interaction to be any complex number. First, we study the complexity of approximating the partition function of the $q$-state Potts model and the closely related Tutte polynomial for complex values of the underlying parameters. Previous work in the complex plane by Goldberg and Guo focused on $q=2$; for $q>2$, the behaviour in the complex plane is not as well understood and most work applies only to the real-valued Tutte plane. Our main result is a complete classification of the complexity of the approximation problems for all non-real values of the parameters, by establishing \#P-hardness results that apply even when restricted to planar graphs. Our techniques apply to all $q\geq 2$ and further complement/refine previous results both for the Ising model and the Tutte plane, answering in particular a question raised by Bordewich, Freedman, Lov\'{a}sz and Welsh in the context of quantum computations. Secondly, we investigate the complexity of approximating the partition function \ising(G; \beta) of the Ising model in terms of the relation between the edge interaction $\beta$ and a parameter $\Delta$ which is an upper bound on the maximum degree of the input graph $G$. In this thesis we establish both new tractability and inapproximability results. Our tractability results show that \ising(-; \beta) has an FPTAS when $\beta \in \mathbb{C}$ and $\lvert \beta - 1 \rvert / \lvert \beta + 1 \rvert 1 / \sqrt{\Delta - 1}$. These are the first results to show intractability of approximating \ising(-, \beta) on bounded degree graphs with complex $\beta$. Moreover, we demonstrate situations in which zeros of the partition function imply hardness of approximation in the Ising model. Finally, we exploit the recently successful framework of spectral independence to analyse the mixing time of a Markov chain, and we apply it in order to sample satisfying assignments of $k$-CNF formulas. Our analysis leads to a nearly linear-time algorithm to approximately sample satisfying assignments in the random $k$-SAT model when the density of the random formula $\alpha=m/n$ scales exponentially with $k$, where $n$ is the number of variables and $m$ is the number of clauses. The best previously known sampling algorithm for the random $k$-SAT model applies when the density $\alpha=m/n$ of the formula is less than $2^{k/300}$ and runs in time $n^{\exp(\Theta(k))}$. Our algorithm achieves a significantly faster running time of $n^{1 + o_k(1)}$ and samples satisfying assignments up to density $\alpha\leq 2^{0.039 k}$. The main challenge in our setting is the presence of many variables with unbounded degree, which causes significant correlations within the formula and impedes the application of relevant Markov chain methods from the bounded-degree setting

Nonlinear rheology of colloidal dispersions

Colloidal dispersions are commonly encountered in everyday life and represent an important class of complex fluid. Of particular significance for many commercial products and industrial processes is the ability to control and manipulate the macroscopic flow response of a dispersion by tuning the microscopic interactions between the constituents. An important step towards attaining this goal is the development of robust theoretical methods for predicting from first-principles the rheology and nonequilibrium microstructure of well defined model systems subject to external flow. In this review we give an overview of some promising theoretical approaches and the phenomena they seek to describe, focusing, for simplicity, on systems for which the colloidal particles interact via strongly repulsive, spherically symmetric interactions. In presenting the various theories, we will consider first low volume fraction systems, for which a number of exact results may be derived, before moving on to consider the intermediate and high volume fraction states which present both the most interesting physics and the most demanding technical challenges. In the high volume fraction regime particular emphasis will be given to the rheology of dynamically arrested states.Comment: Review articl

Machine Learning for Fluid Mechanics

The field of fluid mechanics is rapidly advancing, driven by unprecedented volumes of data from field measurements, experiments and large-scale simulations at multiple spatiotemporal scales. Machine learning offers a wealth of techniques to extract information from data that could be translated into knowledge about the underlying fluid mechanics. Moreover, machine learning algorithms can augment domain knowledge and automate tasks related to flow control and optimization. This article presents an overview of past history, current developments, and emerging opportunities of machine learning for fluid mechanics. It outlines fundamental machine learning methodologies and discusses their uses for understanding, modeling, optimizing, and controlling fluid flows. The strengths and limitations of these methods are addressed from the perspective of scientific inquiry that considers data as an inherent part of modeling, experimentation, and simulation. Machine learning provides a powerful information processing framework that can enrich, and possibly even transform, current lines of fluid mechanics research and industrial applications.Comment: To appear in the Annual Reviews of Fluid Mechanics, 202

Theoretical approaches to realistic strongly correlated nanosystems

Recent developments in methods and computational power render it possible to realistically simulate nanoscopic systems such as surfaces, two-dimensional materials, and nanodots including strong electronic correlations. Nanoscopic structuring enables the tailoring of the electronic structure which can be the basis of future electronic devices. This thesis addresses method developments and applications at the interface of ab-initio methods and model based many-body methods for the case of nanoscopically structured systems with strong correlations. In contrast to bulk materials, low-dimensional materials exhibit long-range interactions due to reduced screening. In this work, the general question how these long-range interactions affect electronic properties is investigated. To this end, a variational approach which approximates models with long-range interactions by models with only local interactions is introduced. For the case of an ab-initio derived model of graphene it is found that nonlocal interactions stabilize the semimetallic phase. The quality of this approach is discussed using a simple test case. Realistic models of interacting impurities embedded in an extended solid involve a large amount of bath sites and low symmetries regarding the impurity, which renders an exact treatment impossible. A variational algorithm is presented which optimizes corresponding exactly solvable effective models. Thus, the method is a proposition to an unambiguous solution of the so-called bath-discretization problem in exact diagonalization approaches to the Anderson impurity model. The method is benchmarked for a simple test case and applied to realistic models of Co atoms in Cu hosts and Fe atoms on alkali surfaces. Finally, the (001) surface of Cr is investigated by incorporating local correlation effects into a material realistic description derived from density functional theory. To this end, the LDA DMFT method is used to calculate spectral functions, which are compared to spectroscopic experimental data. So far open experimental features are thereby clarified. Cr(001) exemplifies a situation where correlation effects are determined by the geometric structure of a material: While correlations effects are weak in bulk Cr, they are key for the electronic structure of the Cr(001) surface

Miscibility properties of bosonic binary mixtures in ring lattices

L'abstract è presente nell'allegato / the abstract is in the attachmen