Lesson plan
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.…
Article summary
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.…
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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Key claims and bibliography are generated from the same tracked claims and sources as the article page.