A tensor network is a representation of a high-dimensional array or multilinear map as a collection of smaller tensors whose indices are contracted according to a graph: vertices represent tensors, an edge joining two vertices represents an index that is summed over, and an open edge represents a free index of the resulting object; the dimension of an internal edge is called its bond dimension. Tensor networks can represent many-body wave functions and density operators, partition functions, Hamiltonians and other linear maps, probability distributions, and quantum circuits.
The usefulness of a tensor network representation rests on two pillars: compression and contraction. When the required bond dimensions remain moderate and the graph admits a favourable contraction order, the storage and computational costs can be exponentially smaller than for the explicit high-dimensional object. Neither condition holds in general — the required bond dimensions can grow rapidly, and the exact contraction of a general network is computationally intractable — so the power of the formalism lies in identifying the structured problems for which compression and contraction are both efficient.
Tensor network states are the special case in which a tensor network represents a variational wave function of a quantum many-body system — one of the principal settings in which the formalism was developed. The best-known examples are matrix product states (MPS) in one dimension, projected entangled pair states (PEPS) in two and higher dimensions, tree tensor networks (TTN), and the multiscale entanglement renormalization ansatz (MERA). Tensor networks provide the theoretical and computational framework underlying the density matrix renormalization group (DMRG), which effectively turned the simulation of quantum spin chains from a problem of exponential complexity into one of polynomial complexity, and remains widely used for simulating strongly correlated quantum systems.
A central idea behind tensor-network methods is that many physically relevant low-energy states of local quantum Hamiltonians occupy a comparatively small, weakly entangled part of the exponentially large many-body Hilbert space; in important settings their entanglement obeys an area law. Tensor network states are designed to efficiently parameterize such states: the entanglement between different regions is routed through auxiliary "virtual" indices whose dimension (the bond dimension) controls the amount of entanglement the ansatz can accommodate, and which serves as a systematically improvable refinement parameter. A second defining feature of the formalism is its diagrammatic calculus, descending from the graphical notation introduced by Roger Penrose, in which multilinear-algebraic manipulations are performed by deforming pictures of the network.
Beyond condensed matter physics, tensor networks are used in numerical multilinear algebra and scientific computing, fluid dynamics, quantum chemistry, quantum field theory and lattice gauge theory, classical statistical mechanics, probabilistic inference, quantum error correction, machine learning, and the classical simulation of quantum computers. In computational fluid dynamics, tensor-train and matrix-product-state representations have been used to compress and simulate turbulent flows and to represent high-dimensional turbulence probability distributions. In numerical analysis, tensor cross interpolation (TCI) constructs low-rank tensor-train or matrix-product-state representations by adaptively sampling selected entries of a high-dimensional tensor, with applications including multidimensional integration, partial differential equations, partition functions and matrix product operators. Tensor network contraction techniques have also been used to reproduce — and in several cases surpass — the results of leading quantum supremacy experiments on classical hardware.
General ideas The Hilbert space of a quantum many-body system grows exponentially with the number of its constituents: a lattice of spin-1/2 particles requires complex amplitudes for a complete description. A generic state in this space exhibits near-maximal entanglement between any subsystem and its complement. Ground states of local Hamiltonians are, however, far from generic. For gapped one-dimensional systems it has been proven that the entanglement entropy of a contiguous block saturates — an area law — rather than growing with the block size, and any state whose Rényi entanglement entropies scale at most logarithmically can be represented faithfully and efficiently as a matrix product state, with an error that can be controlled polynomially in the bond dimension. The precise conditions on entropy scaling under which such efficient approximability holds or fails have been delineated in detail. The idea that physical states occupy only a tiny corner of Hilbert space was made precise using tools of quantum information theory: the overwhelming majority of Hilbert space is physically inaccessible, and the full Hilbert space is in this sense a "convenient illusion". The program of constructing variational classes of states that parameterize precisely the physical, low-entanglement corner of Hilbert space — with the entanglement structure built in from the outset — was formulated in the original projected entangled pair states paper. By construction, the entanglement entropy of a region in an MPS or PEPS is bounded by the virtual bonds crossing its boundary, so these ansätze satisfy corresponding area-law bounds automatically.
The same considerations apply to the classical simulation of quantum circuits. Quantum circuits that generate limited amounts of entanglement — or whose geometry permits efficient contraction — can be simulated classically using tensor network methods, and those have become the standard tool for probing the claims of quantum supremacy experiments. Tensor network contraction algorithms have indeed been used to solve the sampling problem of Google's Sycamore random-circuit experiment on classical hardware, to perform a real-time simulation of the same circuits on the Sunway supercomputer (awarded the 2021 Gordon Bell Prize), and to reproduce and surpass, on modest classical resources, the results of IBM's 127-qubit "quantum utility" experiment.
Graphical calculus Tensor networks are almost universally manipulated in a diagrammatic notation that goes back to Penrose: a tensor is drawn as a shape with one leg per index, and the contraction of two indices is drawn by joining the corresponding legs. A matrix product state is then a chain of three-legged tensors, a PEPS a grid of five-legged tensors, an expectation value a closed ("fully contracted") diagram, and so on. Much of the power of the formalism resides in this graphical calculus: identities of multilinear algebra that would be opaque as nested index sums become topological deformations of a picture, and many key theorems — gauge freedom, canonical forms, transfer matrix manipulations, the behaviour of symmetries — are most naturally stated and proved diagrammatically. The same diagrams describe quantum circuits, quantum channels, and partition functions of classical statistical models, which is what allows concepts and algorithms to be transferred so easily among these settings.
A closely related but historically later development in quantum information theory is the ZX-calculus, a complete graphical rewrite system for qubit tensor networks generated by so-called Z- and X-spiders. The ZX-calculus can be regarded as a highly simplified species of tensor network, specialized to qubits and generated by a small set of spider tensors with phase parameters; within this restricted setting the diagrammatic rewrite rules can be made complete — complete axiomatisations are known for universal pure-qubit quantum mechanics as well as for important restricted fragments — and it is widely used for quantum circuit verification, optimization and fault-tolerance constructions. Common network geometries The same graphical rules apply to many network topologies; the names in the following table distinguish the most common geometries and choices of open indices. The geometries differ not only in expressive power but also in contraction cost: chains and trees admit efficient exact contraction at fixed bond dimension, whereas networks containing large loops generally require approximate contraction.
Form Graph and interpretation Typical uses Matrix product state (MPS) / tensor train (TT) A chain of tensors, of rank three in the interior, with the physical or data indices left open One-dimensional quantum states; compression of high-order arrays Matrix product operator (MPO) / tensor-train operator A chain with one input and one output index at each site Hamiltonians, transfer matrices, density operators, channels Tree tensor network (TTN) A loop-free hierarchy of tensors combining the branches of a tree; closely related to the hierarchical Tucker format Hierarchical correlations, multiscale approximation, high-dimensional numerical problems Projected entangled pair state (PEPS) Tensors on a two- or higher-dimensional lattice or general graph, with one physical index per site Higher-dimensional quantum states; classical partition functions Multiscale entanglement renormalization ansatz (MERA) A layered network of isometries and disentanglers with a causal-cone structure Critical systems; real-space renormalization Continuous matrix product state (cMPS) A continuum limit of the MPS construction One-dimensional quantum fields
History
Statistical mechanics origins The prehistory of tensor networks lies in classical statistical mechanics. In 1941 Kramers and Wannier introduced the transfer matrix treatment of the two-dimensional Ising model together with a variational method for its leading eigenvector, based on a class of trial states that are recognizable in hindsight as precursors of matrix product states. In the language of the modern field, their transfer matrices are matrix product operators and their variational ansatz an MPS with a bond dimension equal to the physical dimension. Baxter developed these ideas much further, introducing the corner transfer matrix formalism and using matrix-product-like variational calculations to compute thermodynamic quantities of classical lattice models to high precision.
This variational transfer-matrix tradition was carried on through the 1970s: Kelland applied it to the Potts model, Tsang studied square-lattice variational approximations to the Ising model, and Baxter and Enting used corner transfer matrices to generate efficient series expansions; the corner transfer matrix formalism is treated in Baxter's classic monograph. Nightingale and Blöte generalized Baxter's variational method to quantum Hamiltonians, using a matrix-product-type trial state as the guiding function of a Monte Carlo computation of the Haldane gap of the spin-1 Heisenberg chain.
A separate thread originated with Wilson's numerical renormalization group (NRG), devised in the 1970s to solve the Kondo problem. It was understood only three decades later that NRG is itself a variational method within the class of matrix product states, which allowed its systematic improvement and unification with DMRG.
Matrix product states in mathematical physics