The term ‘matrix-product state’ (MPS) is introduced in quantum physics (see, e.g., Verstraete-Cirac , [105, Eq. (2)]). The related tensor representation can be found already in Vidal  without a special naming of the representation. The method has been reinvented by Oseledets and Tyrtyshnikov (, , ) and called ‘TT decomposition’.
We start in
with the finite dimensional case. In
we show that the TT representation is a special form of the hierarchical format. Finally three conversions are considered: conversion from
-term format to TT format (cf. §12.3.1), from TT format into hierarchical format with a general tree
(cf. §12.3.2), and vice versa, from general hierarchical format into TT format (cf. §12.3.3). A closely related variant of the TT format is the cyclic matrix product format. As we shall see in
, the change from the tree structure to a proper graph structure may have negative consequences.
The algorithms for obtaining HOSVD bases and for truncations are mentioned only briefly. The reason is the equivalence to the hierarchical format, so that the algorithms defined there can be easily transferred. The interested reader finds such algorithms in .