Tensor networks provide succinct representations of quantum many-body states and are an important computational tool for strongly correlated quantum systems. Their expressive and computational power is characterized by an underlying entanglement structure, on a lattice or more generally a (hyper)graph, with virtual entangled pairs or multipartite entangled states associated to (hyper)edges. Changing this underlying entanglement structure into another can lead to both theoretical and computational benefits. We study a natural resource theory which generalizes the notion of bond dimension to entanglement structures using multipartite entanglement. It is a direct extension of resource theories of tensors studied in the context of multipartite entanglement and algebraic complexity theory, allowing for the application of the sophisticated methods developed in these fields to tensor networks. The resource theory of tensor networks concerns both the local entanglement structure of a quantum many-body state and the (algebraic) complexity of tensor network contractions using this entanglement structure. We show that there are transformations between entanglement structures which go beyond edge-by-edge conversions, highlighting efficiency gains of our resource theory that mirror those obtained in the search for better matrix multiplication algorithms. We also provide obstructions to the existence of such transformations by extending a variety of methods originally developed in algebraic complexity theory for obtaining complexity lower bounds. The resource theory of tensor networks allows to compare different entanglement structures and should lead to more efficient tensor network representations and contraction algorithms.
Tensor networks provide succinct representations of quantum many-body states and are an important computational tool for strongly correlated quantum systems. Their expressive and computational power is characterized by an underlying entanglement structure, on a lattice or more generally a (hyper)graph, with virtual entangled pairs or multipartite entangled states associated to (hyper)edges. Changing this underlying entanglement structure into another can lead to both theoretical and computational benefits. We study a natural resource theory which generalizes the notion of bond dimension to entanglement structures using multipartite entanglement. It is a direct extension of resource theories of tensors studied in the context of multipartite entanglement and algebraic complexity theory, allowing for the application of the sophisticated methods developed in these fields to tensor networks. The resource theory of tensor networks concerns both the local entanglement structure of a quantum many-body state and the (algebraic) complexity of tensor network contractions using this entanglement structure. We show that there are transformations between entanglement structures which go beyond edge-by-edge conversions, highlighting efficiency gains of our resource theory that mirror those obtained in the search for better matrix multiplication algorithms. We also provide obstructions to the existence of such transformations by extending a variety of methods originally developed in algebraic complexity theory for obtaining complexity lower bounds. The resource theory of tensor networks allows to compare different entanglement structures and should lead to more efficient tensor network representations and contraction algorithms.
Tensor networks provide descriptions of strongly correlated quantum systems based on an underlying entanglement structure given by a graph of entangled states along the edges that identify the indices of the local tensors to be contracted. Considering a more general setting, where entangled states on edges are replaced by multipartite entangled states on faces, allows us to employ the geometric properties of multipartite entanglement in order to obtain representations in terms of superpositions of tensor network states with smaller effective dimension, leading to computational savings.
We establish a direct connection between general tensor networks and deep feed-forward artificial neural networks. The core of our results is the construction of neural-network layers that efficiently perform tensor contractions and that use commonly adopted nonlinear activation functions. The resulting deep networks feature a number of edges that closely match the contraction complexity of the tensor networks to be approximated. In the context of many-body quantum states, this result establishes that neural-network states have strictly the same or higher expressive power than practically usable variational tensor networks. As an example, we show that all matrix product states can be efficiently written as neural-network states with a number of edges polynomial in the bond dimension and depth that is logarithmic in the system size. The opposite instead does not hold true, and our results imply that there exist quantum states that are not efficiently expressible in terms of matrix product states or projected entangled pair states but that are instead efficiently expressible with neural network states.
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.
Tensor networks have emerged as a powerful theoretical and computational framework for understanding and simulating quantum many-body systems. Their efficiency in representing quantum states, particularly those satisfying area laws of entanglement, is intimately linked to the structure of their constituent tensors. A fundamental aspect of these networks is their spectral properties, which manifest in various forms, such as the singular values of bond tensors or the eigenvalues of associated transfer matrices. Concurrently, entanglement measures provide quantitative insights into the quantum correlations present within these systems. This paper explores the intricate relationship between the spectra arising from tensor network representations and the different entanglement measures used to characterize quantum states. We delve into how the decay rates of singular values directly impact entanglement entropy, how entanglement spectra derived from Schmidt decompositions can reveal topological order, and how the spectral gaps of transfer matrices are connected to correlation lengths. By analyzing various tensor network architectures, including Matrix Product States (MPS) and Projected Entangled Pair States (PEPS), we elucidate the mechanisms through which their spectral features encode profound information about quantum entanglement. The study highlights how a deeper understanding of these spectral properties can not only guide the construction of more efficient tensor network algorithms but also provide fundamental insights into the nature of quantum phases and phase transitions.
Exact many-body quantum problems are known to be computationally hard due to the exponential scaling of the numerical resources required. Since the advent of the Density Matrix Renormalization Group, it became clear that a successful strategy to work around this obstacle was to develop numerical methods based on the well-known theoretical renormalization group. In recent years, it was realized that quantum states engineered via numerical renormalization allow a variational representation in terms of a tensor network picture. The discovery provided a further boost to the effectiveness of these techniques, not only due to the increased flexibility and manipulability, but also because tensor network states embed a direct interface to the entanglement they carry, so that one can directly address many-body quantum correlations within these variational ansatz states. This lead to the application of several numerical tools, originally developed in the field of quantum-information, to approach condensed matter problems.
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 information science, statistical physics, and so on. In this lecture notes, we focus on the contraction algorithms of TN as well as some of the applications to the simulations of quantum many-body systems. Starting from basic concepts and definitions, we first explain the relations between TN and physical problems, including the TN representations of classical partition functions, quantum many-body states (by matrix product state, tree TN, and projected entangled pair state), time evolution simulations, etc. These problems, which are challenging to solve, can be transformed to TN contraction problems. We present then several paradigm algorithms based on the ideas of the numerical renormalization group and/or boundary states, including density matrix renormalization group, time-evolving block decimation, coarse-graining/corner tensor renormalization group, and several distinguished variational algorithms. Finally, we revisit the TN approaches from the perspective of multi-linear algebra (also known as tensor algebra or tensor decompositions) and quantum simulation. Despite the apparent differences in the ideas and strategies of different TN algorithms, we aim at revealing the underlying relations and resemblances in order to present a systematic picture to understand the TN contraction approaches.