Tensor network

Common card · No. 483374

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...

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