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Topological order in a 2d quantum matter can be determined by the topological contribution to the entanglement R\'enyi entropies. However, when close to a quantum phase transition, its calculation becomes cumbersome. Here we show how…
Tensor networks are a powerful tool for many-body ground states with limited entanglement. These methods can nonetheless fail for certain time-dependent processes - such as quantum transport or quenches - where entanglement growth is linear…
The exact treatment of Markovian models of complex systems requires knowledge of probability distributions exponentially large in the number of components $n$. Mean-field approximations provide an effective reduction in complexity of the…
Tensorial neural networks (TNNs) combine the successes of multilinear algebra with those of deep learning to enable extremely efficient reduced-order models of high-dimensional problems. Here, I describe a deep neural network architecture…
Tensor network methods, most prominently matrix product states (MPS), have become fundamental tools in modern quantum many-body physics. While MPS and extensions like the multiscale entanglement renormalization ansatz (MERA) and tree tensor…
Tensor network states are expected to be good representations of a large class of interesting quantum many-body wave functions. In higher dimensions, their utility is however severely limited by the difficulty of contracting the tensor…
We present a tensor network model (TNM) for forecasting nonlinear and chaotic dynamics, bridging quantum many-body methods with classical complex systems. The TNM leverages hierarchical tensor contractions to encode non-Markovian temporal…
We prove a conjecture of W. Hackbusch about tensor network states related to a perfect binary tree and train track tree. Tensor network states are used to present seemingly complicated tensors in a relatively simple and efficient manner.…
This paper is concerned with the approximation of high-dimensional functions in a statistical learning setting, by empirical risk minimization over model classes of functions in tree-based tensor format. These are particular classes of…
The tensor network states (TNS) methods combined with Monte Carlo (MC) techniques have been proved a powerful algorithm for simulating quantum many-body systems. However, because the ground state energy is a highly non-linear function of…
Tensor network states provide successful descriptions of strongly correlated quantum systems with applications ranging from condensed matter physics to cosmology. Any family of tensor network states possesses an underlying entanglement…
Tensor networks, which have been traditionally used to simulate many-body physics, have recently gained significant attention in the field of machine learning due to their powerful representation capabilities. In this work, we propose a…
Tensor networks represent the state-of-the-art in computational methods across many disciplines, including the classical simulation of quantum many-body systems and quantum circuits. Several applications of current interest give rise to…
Tensor network methods are powerful tools for studying quantum many-body systems. In this paper, we investigate the emergent statistical properties of random high-dimensional tensor-network states and the trainability of variational tensor…
Neural networks have achieved state of the art results in many areas, supposedly due to parameter sharing, locality, and depth. Tensor networks (TNs) are linear algebraic representations of quantum many-body states based on their…
Tree tensor networks (TTNs) offer powerful models for image classification. While these TTN image classifiers already show excellent performance on classical hardware, embedding them into quantum neural networks (QNNs) may further improve…
The tensor network representation of a state in higher dimensions, say a projected entangled-pair state (PEPS), is typically obtained indirectly through variational optimization or imaginary-time Hamiltonian evolution. Here, we propose a…
Tensor network states form a variational ansatz class widely used, both analytically and numerically, in the study of quantum many-body systems. It is known that if the underlying graph contains a cycle, e.g. as in projected entangled pair…
We analyze and discuss convergence properties of a numerically exact algorithm tailored to study the dynamics of interacting two-dimensional lattice systems. The method is based on the application of the time-dependent variational principle…
The hierarchical equations of motion (HEOM) method is a numerically exact open quantum system dynamics approach. The method is rooted in an exponential expansion of the bath correlation function, which in essence strategically reshapes a…