Tensor network

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.…

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.

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