Ranks of Tensor Networks for Eigenspace Projections and the Curse of Dimensionality
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The hierarchical (multi-linear) rank of an order-d tensor is key in determining the cost of representing a tensor as a (tree) Tensor Network (TN). In general, it is known that, for a fixed accuracy, a tensor with random entries cannot be expected to be efficiently approximable without the curse of dimensionality, i.e., a complexity growing exponentially with d. In this work, we show that the ground state projection (GSP) of a class of unbounded Hamiltonians can be approximately represented as an operator of low effective dimensionality that is independent of the (high) dimension d of the GSP. This allows to approximate the GSP without the curse of dimensionality.
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