Tree tensor networks for high-dimensional quantum systems and beyond

This thesis presents the development of a numerical simulation technique, the Tree Tensor Network, aiming to overcome current limitations in the simulation of two- and higher-dimensional quantum many-body systems. The development and application of methods based on Tensor Networks (TNs) for such systems are one of the most relevant challenges of the current decade with the potential to promote research and technologies in a broad range of fields ranging from condensed matter physics, high-energy physics, and quantum chemistry to quantum computation and quantum simulation. The particular challenge for TNs is the combination of accuracy and scalability which to date are only met for one-dimensional systems by other established TN techniques. This thesis first describes the interdisciplinary field of TN by combining mathematical modelling, computational science, and quantum information before it illustrates the limitations of standard TN techniques in higher-dimensional cases. Following a description of the newly developed Tree Tensor Network (TTN), the thesis then presents its application to study a lattice gauge theory approximating the low-energy behaviour of quantum electrodynamics, demonstrating the successful applicability of TTNs for high-dimensional gauge theories. Subsequently, a novel TN is introduced augmenting the TTN for efficient simulations of high-dimensional systems. Along the way, the TTN is applied to problems from various fields ranging from low-energy to high-energy up to medical physics.In dieser Arbeit wird die Entwicklung einer numerischen Simulationstechnik, dem Tree Tensor Network (TTN), vorgestellt, die darauf abzielt, die derzeitigen Limitationen bei der Simulation von zwei- und höherdimensionalen Quanten-Vielteilchensystemen zu überwinden. Die Weiterentwicklung von auf Tensor-Netzwerken (TN) basierenden Methoden für solche Systeme ist eine der aktuellsten und relevantesten Herausforderungen. Sie birgt das Potential, Forschung und Technologien in einem breiten Spektrum zu fördern, welches sich von der Physik der kondensierten Materie, der Hochenergiephysik und der Quantenchemie bis hin zur Quantenberechnung und Quantensimulation erstreckt. Die besondere Herausforderung für TN ist die Kombination von Genauigkeit und Skalierbarkeit, die bisher nur für eindimensionale Systeme erfüllt wird. Diese Arbeit beschreibt zunächst das interdisziplinäre Gebiet der TN als eine Kombination von mathematischer Modellierung, Computational Science und Quanteninformation, um dann die Grenzen der Standard-TN-Techniken in höherdimensionalen Fällen aufzuzeigen. Nach einer Beschreibung des neu entwickelten TTN stellt die Arbeit dessen Anwendung zur Untersuchung einer Gittereichtheorie vor, die das Niederenergieverhalten der Quantenelektrodynamik approximiert und somit die erfolgreiche Anwendbarkeit von TTNs für hochdimensionale Eichtheorien demonstriert. Anschließend wird ein neuartiges TN eingeführt, welches das TTN für effiziente Simulationen hochdimensionaler Systeme erweitert. Zusätzlich wird das TTN auf diverse Probleme angewandt, die von Niederenergie- über Hochenergie- bis hin zur medizinischen Physik reichen

Quantum-inspired Machine Learning on high-energy physics data

Tensor Networks, a numerical tool originally designed for simulating quantum many-body systems, have recently been applied to solve Machine Learning problems. Exploiting a tree tensor network, we apply a quantum-inspired machine learning technique to a very important and challenging big data problem in high energy physics: the analysis and classification of data produced by the Large Hadron Collider at CERN. In particular, we present how to effectively classify so-called b-jets, jets originating from b-quarks from proton-proton collisions in the LHCb experiment, and how to interpret the classification results. We exploit the Tensor Network approach to select important features and adapt the network geometry based on information acquired in the learning process. Finally, we show how to adapt the tree tensor network to achieve optimal precision or fast response in time without the need of repeating the learning process. These results pave the way to the implementation of high-frequency real-time applications, a key ingredient needed among others for current and future LHCb event classification able to trigger events at the tens of MHz scale.Comment: 13 pages, 4 figure

Two-Dimensional Quantum-Link Lattice Quantum Electrodynamics at Finite Density

We present an unconstrained tree-tensor-network approach to the study of lattice gauge theories in two spatial dimensions, showing how to perform numerical simulations of theories in the presence of fermionic matter and four-body magnetic terms, at zero and finite density, with periodic and open boundary conditions. We exploit the quantum-link representation of the gauge fields and demonstrate that a fermionic rishon representation of the quantum links allows us to efficiently handle the fermionic matter while finite densities are naturally enclosed in the tensor network description. We explicitly perform calculations for quantum electrodynamics in the spin-one quantum-link representation on lattice sizes of up to 16 716 sites, detecting and characterizing different quantum regimes. In particular, at finite density, we detect signatures of a phase separation as a function of the bare mass values at different filling densities. The presented approach can be extended straightforwardly to three spatial dimensions

On the descriptive power of Neural Networks as constrained Tensor Networks with exponentially large bond dimension

none4noneMario Collura, Luca Dell'Anna, Timo Felser, Simone MontangeroCollura, Mario; Dell'Anna, Luca; Felser, Timo; Montangero, Simon