Related papers: Low-rank Tree Tensor Network Operators for Long-Ra…
We show how to construct relevant families of matrix product operators in one and higher dimensions. Those form the building blocks for the numerical simulation methods based on matrix product states and projected entangled pair states. In…
Tree tensor networks (TTNs) are widely used in low-rank approximation and quantum many-body simulation. In this work, we present a formal analysis of the differential geometry underlying TTNs. Building on this foundation, we develop…
A deep neural network is a parametrization of a multilayer mapping of signals in terms of many alternatively arranged linear and nonlinear transformations. The linear transformations, which are generally used in the fully connected as well…
Tensor networks are a class of algorithms aimed at reducing the computational complexity of high-dimensional problems. They are used in an increasing number of applications, from quantum simulations to machine learning. Exploiting data…
In this paper we review basic and emerging models and associated algorithms for large-scale tensor networks, especially Tensor Train (TT) decompositions using novel mathematical and graphical representations. We discus the concept of…
Tensor networks have in recent years emerged as the powerful tools for solving the large-scale optimization problems. One of the most popular tensor network is tensor train (TT) decomposition that acts as the building blocks for the…
Tensor network (TN), a young mathematical tool of high vitality and great potential, has been undergoing extremely rapid developments in the last two decades, gaining tremendous success in condensed matter physics, atomic physics, quantum…
Dynamical low-rank approximation by tree tensor networks is studied for the data-sparse approximation to large time-dependent data tensors and unknown solutions of tensor differential equations. A time integration method for tree tensor…
It has been recently shown how the tensorial nature of real-time path integrals involving the Feynman-Vernon influence functional can be utilized using matrix product states, taking advantage of the finite length of the non-Markovian…
Tensor Networks (TNs) are a computational paradigm used for representing quantum many-body systems. Recent works have shown how TNs can also be applied to perform Machine Learning (ML) tasks, yielding comparable results to standard…
By a tensor we mean an element of a tensor product of vector spaces over a field. Up to a choice of bases in factors of tensor products, every tensor may be coordinatized, that is, represented as an array consisting of numbers. This note is…
Hybrid Tensor Networks (hTN) offer a promising solution for encoding variational quantum states beyond the capabilities of efficient classical methods or noisy quantum computers alone. However, their practical usefulness and many…
We present a compendium of numerical simulation techniques, based on tensor network methods, aiming to address problems of many-body quantum mechanics on a classical computer. The core setting of this anthology are lattice problems in low…
We investigate the disordered spin-$\frac12$Heisenberg model in two dimensions and employ tree tensor networks (TTNs) with a physics-informed structural optimization of the tree layout, to simulate dynamics in the many-body localization…
Tensor networks provide compact and scalable representations of high-dimensional data, enabling efficient computation in fields such as quantum physics, numerical partial differential equations (PDEs), and machine learning. This paper…
The groundbreaking performance of deep neural networks (NNs) promoted a surge of interest in providing a mathematical basis to deep learning theory. Low-rank tensor decompositions are specially befitting for this task due to their close…
There is a significant expansion in both volume and range of applications along with the concomitant increase in the variety of data sources. These ever-expanding trends have highlighted the necessity for more versatile analysis tools that…
Low-rank tensor approximation techniques attempt to mitigate the overwhelming complexity of linear algebra tasks arising from high-dimensional applications. In this work, we study the low-rank approximability of solutions to linear systems…
Tensor networks are a compressed format for multi-dimensional data. One-dimensional tensor networks -- often referred to as tensor trains (TT) or matrix product states (MPS) -- are increasingly being used as a numerical ansatz for continuum…
Variational Quantum Algorithms (VQAs) are being highlighted as key quantum algorithms for demonstrating quantum advantage on Noisy Intermediate-Scale Quantum (NISQ) devices, which are limited to executing shallow quantum circuits because of…