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The numerical solution of algebraic tensor equations is a largely open and challenging task. Assuming that the operator is symmetric and positive definite, we propose two new gradient-descent type methods for tensor equations that…
This paper proposes a novel formulation of the tensor completion problem to impute missing entries of data represented by tensors. The formulation is introduced in terms of tensor train (TT) rank which can effectively capture global…
We have discussed the tensor-network representation of classical statistical or interacting quantum lattice models, and given a comprehensive introduction to the numerical methods we recently proposed for studying the tensor-network…
The Tensor-Train (TT) format is a highly compact low-rank representation for high-dimensional tensors. TT is particularly useful when representing approximations to the solutions of certain types of parametrized partial differential…
The study of tensor network theory is an important field and promises a wide range of experimental and quantum information theoretical applications. Matrix product state is the most well-known example of tensor network states, which…
Numerically exact simulations of quantum reaction dynamics, including non-adiabatic effects in excited electronic states, are essential to gain fundamental insights into ultrafast chemical reactivity and rigorous interpretations of…
The term Tensor Network States (TNS) refers to a number of families of states that represent different ans\"atze for the efficient description of the state of a quantum many-body system. Matrix Product States (MPS) are one particular case…
Simulation of quantum systems is challenging due to the exponential size of the state space. Tensor networks provide a systematically improvable approximation for quantum states. 2D tensor networks such as Projected Entangled Pair States…
Recent years have seen rapid advances in the data-driven analysis of dynamical systems based on Koopman operator theory and related approaches. On the other hand, low-rank tensor product approximations -- in particular the tensor train (TT)…
This work presents a comparative study of new and existing optimization and diagonalization methods for solving time-independent partial differential equations (PDEs) using matrix product states (MPS) in the quantized tensor-train formalism…
Quantum computer simulators are an indispensable tool for prototyping quantum algorithms and verifying the functioning of existing quantum computer hardware. The current largest quantum computers feature more than one thousand qubits,…
We propose a multilevel tensor-train (TT) framework for solving nonlinear partial differential equations (PDEs) in a global space-time formulation. While space-time TT solvers have demonstrated significant potential for compressed…
Tensor network methods as presented in our open source Matrix Product States software have opened up the possibility to study many-body quantum physics in one and quasi-one-dimensional systems in an easily accessible package similar to…
Over the last decade tensor network states (TNS) have emerged as a powerful tool for the study of quantum many body systems. The matrix product states (MPS) are one particular case of TNS and are used for the simulation of 1+1 dimensional…
Transport through correlated nanoscale systems underpins the operation of quantum-dot and molecular-scale devices, yet accurate simulations of large open quantum systems remain computationally challenging as system size increases.…
We present a tree-tensor-network-based method to study strongly correlated systems with nonlocal interactions in higher dimensions. Although the momentum-space and quantum-chemistry versions of the density matrix renormalization group…
We introduce a fully discrete scheme to solve a class of high-dimensional Mean Field Games systems. Our approach couples semi-Lagrangian (SL) time discretizations with Tensor-Train (TT) decompositions to tame the curse of dimensionality. By…
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…
The simulation of lattice gauge theories with tensor network (TN) methods is becoming increasingly fruitful. The vision is that such methods will, eventually, be used to simulate theories in $(3+1)$ dimensions in regimes difficult for other…
We present a brief survey on the modern tensor numerical methods for multidimensional stationary and time-dependent partial differential equations (PDEs). The guiding principle of the tensor approach is the rank-structured separable…