935 research outputs found

    Classical and quantum algorithms for scaling problems

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    This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Beam scanning by liquid-crystal biasing in a modified SIW structure

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    A fixed-frequency beam-scanning 1D antenna based on Liquid Crystals (LCs) is designed for application in 2D scanning with lateral alignment. The 2D array environment imposes full decoupling of adjacent 1D antennas, which often conflicts with the LC requirement of DC biasing: the proposed design accommodates both. The LC medium is placed inside a Substrate Integrated Waveguide (SIW) modified to work as a Groove Gap Waveguide, with radiating slots etched on the upper broad wall, that radiates as a Leaky-Wave Antenna (LWA). This allows effective application of the DC bias voltage needed for tuning the LCs. At the same time, the RF field remains laterally confined, enabling the possibility to lay several antennas in parallel and achieve 2D beam scanning. The design is validated by simulation employing the actual properties of a commercial LC medium

    Decision-making with gaussian processes: sampling strategies and monte carlo methods

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    We study Gaussian processes and their application to decision-making in the real world. We begin by reviewing the foundations of Bayesian decision theory and show how these ideas give rise to methods such as Bayesian optimization. We investigate practical techniques for carrying out these strategies, with an emphasis on estimating and maximizing acquisition functions. Finally, we introduce pathwise approaches to conditioning Gaussian processes and demonstrate key benefits for representing random variables in this manner.Open Acces

    Quadratic quantum speedup in evaluating bilinear risk functions

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    Computing nonlinear functions over multilinear forms is a general problem with applications in risk analysis. For instance in the domain of energy economics, accurate and timely risk management demands for efficient simulation of millions of scenarios, largely benefiting from computational speedups. We develop a novel hybrid quantum-classical algorithm based on polynomial approximation of nonlinear functions and compare different implementation variants. We prove a quadratic quantum speedup, up to polylogarithmic factors, when forms are bilinear and approximating polynomials have second degree, if efficient loading unitaries are available for the input data sets. We also enhance the bidirectional encoding, that allows tuning the balance between circuit depth and width, proposing an improved version that can be exploited for the calculation of inner products. Lastly, we exploit the dynamic circuit capabilities, recently introduced on IBM Quantum devices, to reduce the average depth of the Quantum Hadamard Product circuit. A proof of principle is implemented and validated on IBM Quantum systems

    Electron Thermal Runaway in Atmospheric Electrified Gases: a microscopic approach

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    Thesis elaborated from 2018 to 2023 at the Instituto de AstrofĂ­sica de AndalucĂ­a under the supervision of Alejandro Luque (Granada, Spain) and Nikolai Lehtinen (Bergen, Norway). This thesis presents a new database of atmospheric electron-molecule collision cross sections which was published separately under the DOI : With this new database and a new super-electron management algorithm which significantly enhances high-energy electron statistics at previously unresolved ratios, the thesis explores general facets of the electron thermal runaway process relevant to atmospheric discharges under various conditions of the temperature and gas composition as can be encountered in the wake and formation of discharge channels

    Field theory of interacting polaritons under drive and dissipation

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    This thesis explores systems that exhibit strong coupling between an optical cavity field and a many-particle system. To treat the drive and dissipative nature of the cavity on the same footing as the dynamics of the many-particle system, we use a non-equilibrium field theoretic approach. The first system considered is an ultracold bosonic gas trapped inside a cavity. The dispersive coupling between the cavity field and the atoms' motion leads to the formation of a polariton. We show how a modulation of the pump laser on the energy scale of the transverse cavity mode splitting can be used to create effective interactions between different cavity modes. This effective interaction results in the polariton acquiring a multimode nature, exemplified by avoided crossings in the cavity spectrum. As the laser power is increased, the polariton softens and at a critical power becomes unstable. This instability signals the transition into a superradiant state. If the multimode polariton contains a cavity mode with an effective negative detuning, then the transition does not happen through a mode softening but at a finite frequency. To investigate this, classical non-linear equations are constructed from the action and from these we derive the critical couplings and frequencies. It is shown how the superradiant transition happening at a finite frequency is a consequence of a competition between the negatively and the positively detuned cavity modes making up the polariton. The finite-frequency transition is found to be equivalent to a Hopf bifurcation and leads to the emergence of limit cycles. Our analysis shows that the system can exhibit both bistabilities and evolution constricted to a two-torus. We end the investigation by showing how interactions among the atoms combined with the emerging limit cycle open new phonon scattering channels. The second system considered in the thesis is inspired by the recent experiments on gated Transition-metal dichalcogenides (TMD) monolayers inside cavities. An exciton within the TMD can couple strongly to the cavity and, due to the electronic gating, also interact strongly with the conduction electrons. To treat the strong interactions of the excitons with both cavity and electrons, we solve the coupled equations for the correlation functions non-perturbatively within a ladder approximation. The strong interactions give rise to new quasiparticles known as polaron-polaritons. By driving the system through the cavity, we show how the competition between electron-induced momentum relaxation and cavity loss leads to the accumulation of polaritons at a small but finite momentum, which is accompanied by significant decrease of the polariton linewidth Due to the hybrid nature of the polaron-polariton, we show that this behavior can by qualitatively modified by changing the cavity detuning

    Geometric optimization problems in quantum computation and discrete mathematics: Stabilizer states and lattices

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    This thesis consists of two parts: Part I deals with properties of stabilizer states and their convex hull, the stabilizer polytope. Stabilizer states, Pauli measurements and Clifford unitaries are the three building blocks of the stabilizer formalism whose computational power is limited by the Gottesman- Knill theorem. This model is usually enriched by a magic state to get a universal model for quantum computation, referred to as quantum computation with magic states (QCM). The first part of this thesis will investigate the role of stabilizer states within QCM from three different angles. The first considered quantity is the stabilizer extent, which provides a tool to measure the non-stabilizerness or magic of a quantum state. It assigns a quantity to each state roughly measuring how many stabilizer states are required to approximate the state. It has been shown that the extent is multiplicative under taking tensor products when the considered state is a product state whose components are composed of maximally three qubits. In Chapter 2, we will prove that this property does not hold in general, more precisely, that the stabilizer extent is strictly submultiplicative. We obtain this result as a consequence of rather general properties of stabilizer states. Informally our result implies that one should not expect a dictionary to be multiplicative under taking tensor products whenever the dictionary size grows subexponentially in the dimension. In Chapter 3, we consider QCM from a resource theoretic perspective. The resource theory of magic is based on two types of quantum channels, completely stabilizer preserving maps and stabilizer operations. Both classes have the property that they cannot generate additional magic resources. We will show that these two classes of quantum channels do not coincide, specifically, that stabilizer operations are a strict subset of the set of completely stabilizer preserving channels. This might have the consequence that certain tasks which are usually realized by stabilizer operations could in principle be performed better by completely stabilizer preserving maps. In Chapter 4, the last one of Part I, we consider QCM via the polar dual stabilizer polytope (also called the Lambda-polytope). This polytope is a superset of the quantum state space and every quantum state can be written as a convex combination of its vertices. A way to classically simulate quantum computing with magic states is based on simulating Pauli measurements and Clifford unitaries on the vertices of the  Lambda-polytope. The complexity of classical simulation with respect to the polytope   is determined by classically simulating the updates of vertices under Clifford unitaries and Pauli measurements. However, a complete description of this polytope as a convex hull of its vertices is only known in low dimensions (for up to two qubits or one qudit when odd dimensional systems are considered). We make progress on this question by characterizing a certain class of operators that live on the boundary of the  Lambda-polytope when the underlying dimension is an odd prime. This class encompasses for instance Wigner operators, which have been shown to be vertices of  Lambda. We conjecture that this class contains even more vertices of  Lambda. Eventually, we will shortly sketch why applying Clifford unitaries and Pauli measurements to this class of operators can be efficiently classically simulated. Part II of this thesis deals with lattices. Lattices are discrete subgroups of the Euclidean space. They occur in various different areas of mathematics, physics and computer science. We will investigate two types of optimization problems related to lattices. In Chapter 6 we are concerned with optimization within the space of lattices. That is, we want to compare the Gaussian potential energy of different lattices. To make the energy of lattices comparable we focus on lattices with point density one. In particular, we focus on even unimodular lattices and show that, up to dimension 24, they are all critical for the Gaussian potential energy. Furthermore, we find that all n-dimensional even unimodular lattices with n   24 are local minima or saddle points. In contrast in dimension 32, there are even unimodular lattices which are local maxima and others which are not even critical. In Chapter 7 we consider flat tori R^n/L, where L is an n-dimensional lattice. A flat torus comes with a metric and our goal is to approximate this metric with a Hilbert space metric. To achieve this, we derive an infinite-dimensional semidefinite optimization program that computes the least distortion embedding of the metric space R^n/L into a Hilbert space. This program allows us to make several interesting statements about the nature of least distortion embeddings of flat tori. In particular, we give a simple proof for a lower bound which gives a constant factor improvement over the previously best lower bound on the minimal distortion of an embedding of an n-dimensional flat torus. Furthermore, we show that there is always an optimal embedding into a finite-dimensional Hilbert space. Finally, we construct optimal least distortion embeddings for the standard torus R^n/Z^n and all 2-dimensional flat tori

    Discontinuous Galerkin Methods for the Linear Boltzmann Transport Equation

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    Radiation transport is an area of applied physics that is concerned with the propagation and distribution of radiative particle species such as photons and electrons within a material medium. Deterministic models of radiation transport are used in a wide range of problems including radiotherapy treatment planning, nuclear reactor design and astrophysics. The central object in many such models is the (linear) Boltzmann transport equation, a high-dimensional partial integro-differential equation describing the absorption, scattering and emission of radiation. In this thesis, we present high-order discontinuous Galerkin finite element discretisations of the time-independent linear Boltzmann transport equation in the spatial, angular and energetic domains. Efficient implementations of the angular and energetic components of the scheme are derived, and the resulting method is shown to converge with optimal convergence rates through a number of numerical examples. The assembly of the spatial scheme on general polytopic meshes is discussed in more detail, and an assembly algorithm based on employing quadrature-free integration is introduced. The quadrature-free assembly algorithm is benchmarked against a standard quadrature-based approach, and an analysis of the algorithm applied to a more general class of discontinuous Galerkin discretisations is performed. In view of developing efficient linear solvers for the system of equations resulting from our discontinuous Galerkin discretisation, we exploit the variational structure of the scheme to prove convergence results and derive a posteriori solver error estimates for a family of iterative solvers. These a posteriori solver error estimators can be used alongside standard implementations of the generalised minimal residual method to guarantee that the linear solver error between the exact and approximate finite element solutions (measured in a problem-specific norm) is below a user-specified tolerance. We discuss a family of transport-based preconditioners, and our linear solver convergence results are benchmarked through a family of numerical examples

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum
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