Many complex chemical problems encoded in terms of physics-based models become computationally intractable for traditional numerical approaches due to their unfavorable scaling with increasing molecular size. Tensor decomposition techniques can overcome such challenges by decomposing unattainably large numerical representations of chemical problems into smaller, tractable ones. In the first two decades of this century, algorithms based on such tensor factorizations have become state-of-the-art methods in various branches of computational chemistry, ranging from molecular quantum dynamics to electronic structure theory and machine learning. Here, we consider the role that tensor decomposition schemes have played in expanding the scope of computational chemistry. We relate some of the most prominent methods to their common underlying tensor network formalisms, providing a unified perspective on leading tensor-based approaches in chemistry and materials science.
Many complex chemical problems encoded in terms of physics-based models become computationally intractable for traditional numerical approaches due to their unfavorable scaling with increasing molecular size. Tensor decomposition techniques can overcome such challenges by decomposing unattainably large numerical representations of chemical problems into smaller, tractable ones. In the first two decades of this century, algorithms based on such tensor factorizations have become state-of-the-art methods in various branches of computational chemistry, ranging from molecular quantum dynamics to electronic structure theory and machine learning. Here, we consider the role that tensor decomposition schemes have played in expanding the scope of computational chemistry. We relate some of the most prominent methods to their common underlying tensor network formalisms, providing a unified perspective on leading tensor-based approaches in chemistry and materials science.
Many complex chemical problems encoded in terms of physics-based models become computationally intractable for traditional numerical approaches due to their unfavorable scaling with increasing molecular size. Tensor decomposition techniques can overcome such challenges by decomposing unattainably large numerical representations of chemical problems into smaller, tractable ones. In the first two decades of this century, algorithms based on such tensor factorizations have become state-of-the-art methods in various branches of computational chemistry, ranging from molecular quantum dynamics to electronic structure theory and machine learning. Here, we consider the role that tensor decomposition schemes have played in expanding the scope of computational chemistry. We relate some of the most prominent methods to their common underlying tensor network formalisms, providing a unified perspective on leading tensor-based approaches in chemistry and materials science.
Tensor networks were developed in the context of many-body physics as compressed representations of multiparticle quantum states. These representations mitigate the exponential complexity of many-body systems by capturing only the most relevant dependencies. Due to the formal similarity between quantum entanglement and statistical correlations, tensor networks have recently been integrated in machine learning, operating both as alternative learning architectures and as decompositions of components of neural networks. The expectation is that the theoretical understanding of tensor networks developed within quantum many-body physics leads to novel methods that offer advantages in terms of computational efficiency, explainability, or privacy. Here we review the use of tensor networks in the context of machine learning, providing a critical assessment of the state of the art, the potential advantages, and the challenges that must be overcome.
Predicting the properties of correlated materials and molecules often requires a many-body description that goes beyond single-particle and mean-field approximations. Tensor networks address this challenge by exploiting the entanglement structure of quantum states to construct compact, systematically improvable representations. Although developed largely for idealized model Hamiltonians, these methods are increasingly being integrated into material-specific calculations. This Perspective introduces their underlying principles and examines how network geometry, bond-dimension truncation, and approximate contraction determine the accuracy and computational reach of these approaches. We survey applications across regimes of increasing complexity, ranging from ground-state properties of quantum magnets and itinerant-fermion systems to excitation spectra, nonequilibrium dynamics, finite-temperature and open systems, and molecular electronic structure. We then examine four routes connecting these methods to real materials, including experimentally constrained models, downfolded Hamiltonians, dynamical mean-field theory with tensor network impurity solvers, and quantum embedding. Throughout, we evaluate the maturity of existing capabilities and their principal limitations, synthesizing this assessment into a practical method-selection guide. We argue that reliable materials predictions require quantifying uncertainties introduced at each stage, from constructing material-specific Hamiltonians to extracting experimentally accessible observables. Finally, we identify the algorithmic advances and reproducible computational practices needed to establish tensor networks as a routine tool for predictive materials science and quantum chemistry.
The conventional numerical methods when applied to multidimensional problems suffer from the so-called "curse of dimensionality", that cannot be eliminated by using parallel architectures and high performance computing. The novel tensor numerical methods are based on a "smart" rank-structured tensor representation of the multivariate functions and operators discretized on Cartesian grids thus reducing solution of the multidimensional integral-differential equations to 1D calculations. We explain basic tensor formats and algorithms and show how the orthogonal Tucker tensor decomposition originating from chemometrics made a revolution in numerical analysis, relying on rigorous results from approximation theory. Benefits of tensor approach are demonstrated in ab-initio electronic structure calculations. Computation of the 3D convolution integrals for functions with multiple singularities is replaced by a sequence of 1D operations, thus enabling accurate MATLAB calculations on a laptop using 3D uniform tensor grids of the size up to 10 15 . Fast tensor-based Hartree-Fock solver, incorporating the grid-based low-rank factorization of the two-electron integrals, serves as a prerequisite for economical calculation of the excitation energies of molecules. Tensor approach suggests efficient grid-based numerical treatment of the long-range electrostatic potentials on large 3D finite lattices with defects.The novel range-separated tensor format applies to interaction potentials of multi-particle systems of general type opening the new prospects for tensor methods in scientific computing. This research monograph presenting the modern tensor techniques applied to problems in quantum chemistry may be interesting for a wide audience of students and scientists working in computational chemistry, material science and scientific computing.
A tensor network is a type of decomposition used to express and approximate large arrays of data.A given dataset, quantum state, or higher-dimensional multilinear map is factored and approximated by a composition of smaller multilinear maps.This is reminiscent to how a Boolean function might be decomposed into a gate array: this represents a special case of tensor decomposition, in which the tensor entries are replaced by 0, 1 and the factorisation becomes exact.The associated techniques are called tensor network methods: the subject developed independently in several distinct fields of study, which have more recently become interrelated through the language of tensor networks.The tantamount questions in the field relate to expressability of tensor networks and the reduction of computational overheads.A merger of tensor networks with machine learning is natural.On the one hand, machine learning can aid in determining a factorisation of a tensor network approximating a data set.On the other hand, a given tensor network structure can be viewed as a machine learning model.Herein the tensor network parameters are adjusted to learn or classify a data-set.In this survey we review the basics of tensor networks and explain the ongoing effort to develop the theory of tensor networks in machine learning.1
Abstract Linear combination of unitaries (LCU) decompositions have become a key tool for encoding operators on quantum computers, enabling efficient implementations of arbitrary operators. In particular, LCU methods provide a way to encode the electronic structure Hamiltonian into quantum circuits. Over the years, various decomposition techniques have been developed for this purpose. Here, we introduce the Majorana tensor decomposition, a framework that unifies existing LCU approaches and introduces novel decompositions using low-rank tensor factorizations. We benchmark a range of decomposition techniques on small molecular systems and hydrogen chains of increasing sizes, evaluating their performance across different LCU methods.